EQUIVARIANT ALGEBRAIC COBORDISM AND DOUBLE POINT
RELATIONS
By
Chun Lung Liu

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILSOPHY
Mathematics
2012

ABSTRACT
EQUIVARIANT ALGEBRAIC COBORDISM AND DOUBLE POINT RELATIONS
By
Chun Lung Liu
For a reductive connected group or a finite group over a field of characteristic zero, we
define an equivariant algebraic cobordism theory by a generalized version of the double point
relation of Levine-Pandharipande. We prove basic properties and the well-definedness of a
canonical fixed point map. We also find explicit generators of the algebraic cobordism ring
of the point when the group is finite abelian.

ACKNOWLEDGMENTS

I would like to thank my advisor G. Pappas for useful conversations and helpful comments,
P. Brosnan for a useful discussion and M. Levine for his interest. This research was partially
supported by NSF grants DMS-1102208 and DMS-0802686.

iii

TABLE OF CONTENTS
§1. Introduction

1

§2. Notations and assumptions

11

§3. Geometric equivariant algebraic cobordism UG

14

§4. The Chern class operator c(L)

49

§5. More properties for UG

74

§6. Generators for the equivariant algebraic cobordism ring

79

§7. Fixed point map

110

REFERENCES

128

iv

1. Introduction
Cobordism is a deep and well-developed theory in topology. According to Thom’s definition, two dimension d smooth oriented manifolds M, N are said to be cobordant if there exists
a dimension d + 1 smooth oriented manifold with boundary M

(−N ) (Negative sign means

opposite orientation). By definition, the set of all cobordism classes, with addition given by
disjoint union and multiplication given by product, is called the oriented bordism ring U∗
(grading given by dimension). This ring was well-studied. For instance, Thom showed that
the torsion free part can be described by U∗ ⊗Z Q ∼ Q[x4k | k ≥ 1]. In addition, Milnor and
=
Wall showed that all torsion has order 2. The main technique involved was the use of the
Thom spectrum which we will briefly explain below.
Consider a SO(n)-bundle E over a manifold X. Let D be the set of all vectors (fiberwise)
with length ≤ 1 and S be the set of all vectors with length 1. Then, the Thom space is
defined as the quotient space D/S and denoted by T (E). Now consider the classifying space
BSO(n) with universal SO(n)-bundle En . Denote the Thom space T (En ) by M SO(n).
Notice that En × R1 becomes a SO(n + 1)-bundle over BSO(n) and hence induces the
classifying map BSO(n) → BSO(n + 1) and En × R1 → En+1 . Apply the Thom space
construction on both sides of the second map, we get
M SO(n) ∧ S 1 ∼ T (En × R1 ) → T (En+1 ) = M SO(n + 1).
=
That defines the Thom spectrum M SO. We can then consider the homotopy groups of M SO,
def

namely πk (M SO) = lim πn+k (M SO(n)). The importance of the Thom spectrum comes
from the isomorphism

−→
n
Uk →
˜

πk (M O) which is given by the Pontrjagin-Thom construction

(see [St]).
More generally, for a smooth oriented manifold X, we say two maps f1 : Y1 → X and
f2 : Y2 → X, where Y1 , Y2 are both dimension d smooth oriented manifolds, are cobordant if
there exists a map F : Z → X such that Z is a dimension d+1 smooth oriented manifold with
boundary and F |∂Z = f1

(−f2 ) (Negative sign means opposite orientation on domain).

The set of all cobordism classes with addition given by disjoint union is denoted by U∗ (X)
(grading given by dimension of the domain of the map). That is the oriented bordism group.

1

Other than oriented bordism theory, there are other bordism (or cobordism) theories. For
example, for a stably complex manifold X, we define
def

M Uk (X) = lim [S 2n+k , M U (n) ∧ X]
−→
n

and
def

M U k (X) = lim [S 2n−k ∧ X, M U (n)]
−→
n

where M U (n) is the Thom space of the universal U (n)-bundle over the classifying space
BU (n). This way, one defines the complex bordism theory (given by M U∗ (X)) and the
complex cobordism theory (given by M U ∗ (X)). Milnor showed [Mil] that the complex
bordism ring M U∗ is just a polynomial ring Z[x2k | k ≥ 1] and M U ∗ ∼ M U−∗ .
=
Moreover, this complex cobordism theory is equipped with Chern classes and it leads to
what is called the formal group law. More precisely, for each complex vector bundle E over
X of rank r, there are Chern classes ci (E) ∈ M U 2i (X) for 1 ≤ i ≤ r associated to it (see
[CoF]). It turns out the complex cobordism group M U ∗ (CP∞ ) is given by the power series
ring M U ∗ [[s]] and the tensor product map CP∞ × CP∞ → CP∞ will define a Hopf-algebra
structure on M U ∗ (CP∞ ). Thus, we obtain a map
ˆ
M U ∗ [[s]] ∼ M U ∗ (CP∞ ) → M U ∗ (CP∞ ×CP∞ ) ∼ M U ∗ (CP∞ )⊗M U ∗ (CP∞ ) ∼ M U ∗ [[u, v]].
=
=
=
Denote the image of s by F ∈ M U ∗ [[u, v]]. Since CP∞ is the classifying space for U (1) and
the elements s, u and v correspond to c1 (O(1)), c1 (O(1, 0)) and c1 (O(0, 1)) respectively, we
obtain the following relation for any pairs of complex line bundles L1 , L2 over X :
c1 (L1 ⊗ L2 ) = F (c1 (L1 ), c1 (L2 ))
as elements inside M U ∗ (X). This power series F is called a formal group law over M U ∗
(see [Q]).
Unfortunately, because of the lack of the notion of boundary in the category of algebraic
varieties, an algebraic version of cobordism theory can not be defined in a similar manner.
There is a naive approach which turns out to be unsuccessful. We may define two dimension
d smooth projective varieties X, X to be cobordant if there exists a morphism Y → P1
2

where Y is a dimension d + 1 smooth projective variety such that X, X are the fibers over
0, 1 respectively. This approach was also addressed by M. Levine and F. Morel (see remark
1.2.9 in [LeMo] for more detail). Consider the case when d = 1. Since the genus and the
number of connected components are invariant under this concept of cobordism, we can not
decompose a smooth genus g curve. Hence, the cobordism group of curves will be much
bigger than Z, which is what we expect from the theory of complex cobordism.
Nevertheless, in [LeMo], Levine and Morel managed to define an algebraic cobordism
theory Ω, which is an analog of the complex cobordism theory, in spite of the absence of
notion of boundary. However, the definition is relatively complicated. Roughly speaking, if
X is a separated scheme of finite type over the ground field k, then we consider elements of
the form (f : Y → X, L1 , . . . , Lr ) where f is projective, Y is an irreducible smooth variety
over k and the sheaves Li are line bundles over Y (the order of Li does not matter and
the number r of line bundles can be zero). The dimension of (f : Y → X, L1 , . . . , Lr ) is
defined to be dim Y − r. There is a natural notion of isomorphism on elements of this form.
Denote the free abelian group generated by isomorphism classes of elements of this form by
Z(X). Let Ω(X) be the quotient of Z(X) by the subgroup corresponding to imposing the
axioms (Dim) and (Sect) (following the notations in [LeMo]). The algebraic cobordism
group Ω(X) is defined to be the quotient of Ω(X) ⊗Z L, where L is the Lazard ring, by the
L-submodule corresponding to imposing the formal group law (FGL).
This cobordism theory satisfies a number of basic properties, (D1)-(D4), (A1)-(A8),
(Dim), (Sect) and (FGL) (following the notation in [LeMo]). It also satisfies some more
advanced properties, for example, the localization property and the homotopy invariance
property. Moreover, the cobordism ring Ω(Spec k) will be isomorphic to the Lazard ring
L when the characteristic of k is 0, which is what we expect from the complex cobordism
theory (see Corollary 1.2.11 and Theorem 4.3.7 in [LeMo]).
One may wonder if it is possible to construct an algebraic cobordism theory via a more
geometric approach. Suppose X is a smooth variety over k. We may consider the abelian
group M (X)+ generated by isomorphism classes over X of projective morphisms f : Y → X
where Y is a smooth variety over k. The hope is that by imposing some reasonable relations,
we will obtain an algebraic cobordism theory that also satisfies some previously mentioned
3

properties. Such a construction was introduced by M. Levine and R. Pandharipande in
[LeP]. A relation called “double point relation” was introduced and it was shown that the
theory ω obtained by imposing this relation is canonically isomorphic to the theory Ω under
the assumption that the characteristic of k is 0 (see Theorem 1 of [LeP]).
More precisely, let φ : Y → X×P1 be a projective morphism where Y is an equidimensional
smooth variety over k. Consider the fibers for the composition Y → X × P1 → P1 . Suppose
the fiber Yξ is a smooth divisor on Y and the fiber Y0 can be expressed as the union of two
smooth divisors A, B such that A intersects B transversely. Then, the double point relation
is
[Yξ → Y → X] = [A → Y → X] + [B → Y → X]
− [P(O ⊕ NA∩B →A ) → A ∩ B → Y → X].
The objective of the current paper is to develop an algebraic cobordism theory of varieties
with group action that assembles the theories of Levine-Morel and Levine-Pandharipande.
For this, we go back to topology for inspiration. In topology, for a compact Lie group G,
the concept of G-equivariant bordism was first studied by Conner and Floyd (see [Co] or
[H]). In their approach, for a G-space X, we consider the set of maps Y → X where Y is
a stable almost complex G-manifold. Define the notion of G-bordism similarly to form the
G
geometric unitary bordism group of X, denoted by U∗ (X). Another approach was pursued
G
by Tom Dieck [T]. Let ξn → BU (n, G) be the universal unitary n-dimensional G-bundle

and M U (n, G) be its Thom space. Then, the homotopy theoretic unitary G-bordism group
of X is defined by
def

G
M U2k (X) = lim [ S V , M U (dimC V − k, G) ∧ X ]G
−→
V

and
def

G
M U2k+1 (X) = lim [ S V ∧ S 1 , M U (dimC V − k, G) ∧ X ]G
−→
V

where V runs through all unitary G-representations (see [T]). Inspired by the isomorphism between the principal G-bordism group over a point and the oriented bordism group
M SO(BG) (where EG → BG is the universal G-bundle) when G is finite (see [Co]), there
4

is also a third G-equivariant bordism theory defined by the following equation :
G,h

M U∗

def

(X) = M U∗ ((X × EG)/G).

In the case when X is a point, there are some maps relating the three theories.
a

b

G,h

G
G
U∗ → M U ∗ → M U ∗

The map a is given by the same Pontrjagin-Thom construction, but an inverse can not be
constructed in the same manner due to the lack of transversality when there is group action.
Indeed, the map a is never surjective (unless the group G is trivial) because there are nonG
trivial elements in the negative degrees of M U∗ . However, the injectivity of the map a

was shown by Loffler and Comezana when G is abelian (see [Lo] and [Ma]). On the other
G
hand, when G is abelian, the map b identifies the I-adic completion of M U∗ , where I is the
G,h

augmentation ideal, to M U∗

(see [GrMa]).

There are some computational results on different versions of equivariant bordism ring.
In [Ko], Kosniowski gave a list of G-spaces which multiplicatively generate the geometric
G
unitary bordism ring U∗ over M U∗ when G is a cyclic group of prime order. When G is

an abelian compact Lie group, Sinha gave a list of elements and relations that describe the
G
structure of the homotopy theoretic unitary bordism ring M U∗ as a M U∗ -algebra (see [Si]).
G,h

Since M U∗

G
can be identified with the I-adic completion of M U∗ when G is abelian, we
G,h

also obtain the structure of M U∗

.

Following this pattern, we can expect to also have several different approaches to equivariant algebraic cobordism theory. In order to define an analog of the homotopy theoretic
bordism theory M U G in the algebraic geometry setup, one possible way is through Voevodsky’s machinery of A1 -homotopy theory (see [MoV]). A (non-equivariant) algebraic
cobordism theory defined this way is discussed in [V], but, to our knowledge, an equivariant
version of this has not yet been considered.
To define an analog of the theory M U G,h , one can employ Totaro’s approximation of EG.
In [EG], Edidin and Graham successfully defined an equivariant Chow ring following this line
of thought. For a given dimension n algebraic space X with G-action and for a fixed integer

5

i, pick a G-representation V and an invariant open set U inside such that G acts freely on
U and the codimension of V − U is larger than n − i. Then, X × U → (X × U )/G will
be a principle G-bundle. Moreover, the Chow group Ai+dim V −dim G ((X × U )/G) is indeed
independent of the choice of the pair (V, U ). Hence, the equivariant Chow group AG (X) is
i
defined to be Ai+dim V −dim G ((X × U )/G).
Unfortunately, since the independence of choice relies on the fact that the negative (cohomological) degrees of Chow groups are always zero, i.e. Ai = 0 whenever i < 0, equivariant
algebraic cobordism theory can not be defined in the exact same manner. One approach is
by considering a whole system of good pairs {(V, U )} and define the equivariant algebraic
cobordism group Ωi (X) to be the inverse limit of Ωi ((X × U )/G) (see [HeLop] for more
G
details). Another, possibly equivalent, approach was pursued by Krishna [Kr].
Aside from these two homotopical approaches, one can also define an equivariant algebraic
cobordism theory analogously to the geometric bordism theory U G , namely by considering
varieties with G-action and imposing the G-action also on the double point relation. Suppose
G is an algebraic group over k and X is a smooth G-variety over k. This is what we do in
this paper. We can consider the abelian group MG (X)+ generated by isomorphism classes
of G-equivariant projective morphism f : Y → X where Y is also a smooth G-variety. For
a morphism φ : Y → X × P1 where Y is a smooth G-variety, P1 is equipped with the trivial
action and φ is projective and G-equivariant satisfying the same conditions on the fibers
Yξ and Y0 as before, we impose the exact same equation with all objects involved equipped
with their naturally inherited G-actions. Then, all morphisms involved will also be naturally
equivariant.
For technical reasons, we focus on the case when the characteristic of k is zero and G is
either a finite group or a connected reductive group. Observe that if there is a projective,
G-equivariant morphism Y → X and smooth G-invariant divisors Yξ , A, B on Y satisfying
the conditions in the double point relation, then Yξ is equivariantly linearly equivalent to
A + B and Yξ + A + B is a reduced strict normal crossing divisor. Suppose we are given
a smooth, G-invariant, very ample divisor C on Y . Due to the lack of transversality in the
equivariant setting, the choice of the pairs of smooth G-invariant divisors A, B such that
C ∼ A + B and A + B + C is a reduced strict normal crossing divisor may become seriously
6

limited, if not impossible. To remedy this, it is preferable to impose a more general relation
which we call generalized double point relation.
More precisely, suppose X, Y are both smooth varieties with G-action and φ : Y → X is
an equivariant projective morphism. Assume there are smooth invariant divisors A1 , . . . , An ,
B1 , . . . , Bm on Y such that A1 +· · ·+An is equivariantly linearly equivalent to B1 +· · ·+Bm
and A1 + · · · + An + B1 + · · · + Bm is a reduced strict normal crossing divisor. Then, the
generalized double point relation GDP R(n, m) we will impose is of the form
[A1 → Y → X] + [A2 → Y → X] + · · · + [An → Y → X] + extra terms
= [B1 → Y → X] + [B2 → Y → X] + · · · + [Bm → Y → X] + extra terms
where the extra terms are of the form [P → C → Y → X] such that C is the intersection of some of the divisors A1 , . . . , An , B1 , . . . , Bm and P → C is an admissible tower
(see subsection 6.3 for the definition). Denote the left hand side of the above equation
by L(φ, A1 , . . . , An , B1 , . . . , Bm ) and the right hand side by R(φ, A1 , . . . , An , B1 , . . . , Bm ).
Hence, we define the (geometric) equivariant algebraic cobordism group, denoted by UG (X),
to be the quotient of MG (X)+ by the abelian subgroup generated by
L(φ, A1 , . . . , An , B1 , . . . , Bm ) − R(φ, A1 , . . . , An , B1 , . . . , Bm )
for all equivariant projective morphisms φ : Y → X and all possible set of invariant divisors
A1 , . . . , An , B1 , . . . , Bm satisfying the conditions described above. We conjecture that the
generalized double point relation is indeed stronger than the double point relation (See
Remark 6.23 in the text).
An important observation is that the generalized double point relation actually holds in
the non-equivariant theory ω. In other words, our equivariant algebraic cobordism theory in
the case when G is trivial coincides with the non-equivariant algebraic cobordism theory, i.e
. U{1} (X) ∼ ω(X) for all smooth varieties X. That means this theory UG can be thought as
=
a generalization of ω. In addition, although the generalized double point relation may look
tedious, it is actually easier to use because of the freedom of the number of divisors involved.

7

Using this theory, we are able to define a “fixed-point map” which is similar to a wellknown construction in topology. Recall the definition of the fixed point map in topology (see
[T]). For simplicity, suppose G is a finite group of prime order p. Then, there are exactly
p non-isomorphic irreducible complex G-representations. Denote them by V1 , . . . , Vp . For
a unitary G-manifold M , let F be a component of the fixed point set M G . The normal
p

bundle of F inside M can be written as ⊕i=1 Vi ⊗ Ni for some complex vector bundles
Ni over F with no G-action. Compose the classifying map of Ni with the natural map
BU (rank of Ni ) → BU . We get a map F → BU which we will denote by fi . Thus, the
fixed point map

p

φ:

G
U∗

→

M U∗ (BU )
i=1

is given by sending [M ] to the sum of
([f1 : F → BU ], . . . , [fp : F → BU ])
over all components F . If we add up the elements [fi ] and push them down to the bordism
ring, we obtain a map
p
G
U∗

→

M U∗ (BU ) → M U∗ (BU ) → M U∗
i=1

given by
[M ] →

([f1 ], . . . , [fp ]) →
F

[F ] = p [M G ].

[fi ] →
F,i

F,i

Assume the ground field k has characteristic 0 as before. If the group G is finite, then
the fixed point locus of any smooth variety over k is again smooth (Proposition 3.4 in [Ed]).
The same statement also holds when G is reductive (by Proposition 7.1). So, for a smooth
variety X, we have an abelian group homomorphism from MG (X)+ to M (X G )+ defined by
sending [Y → X] to [Y G → X G ], which we will also call fixed point map. One of our main
results is the following Theorem (Corollary 7.3 in the text) which can be considered as an
analog of the topological fixed point map.

8

Theorem 1. For any smooth G-variety X, sending [Y → X] to [Y G → X G ] defines an
abelian group homomorphism
F : UG (X) → ω(X G ).
We also managed to find a set of generators for the equivariant algebraic cobordism ring of
the point Spec k when G is a finite abelian group with exponent e and k contains a primitive
e-th root of unity. We can naturally embed the non-equivariant algebraic cobordism ring
L ∼ ω(Spec k) ∼ U{1} (Spec k)
=
=
inside the equivariant algebraic cobordism ring UG (Spec k) (by assigning trivial G-action)
(see Corollary 7.4). This construction provides UG (Spec k) with a L-algebra structure. Then,
the following Theorem describes a set of generators of UG (Spec k) (see Theorem 6.22 for more
detail).
Theorem 2.

Suppose G is a finite abelian group with exponent e and k contains a

primitive e-th root of unity. Then, the equivariant algebraic cobordims ring UG (Spec k) is
generated by the set of exceptional objects
{En,H,H | n ≥ 0 and G ⊇ H ⊇ H }
and the set of admissible towers over Spec k as a L-algebra.
Here is the definition of the exceptional objects. For an integer n ≥ 0 and a pair of
subgroups G ⊇ H ⊇ H , since G is abelian, we can write
H/H ∼ H1 × · · · × Ha
=
m
where Hi is a cyclic group of order pi i for a prime pi . Define an (H/H )-action on

Proj k[x0 , . . . , xn , v1 , . . . , va ]
by assigning Hi to act faithfully on k−span{vi } and trivially on other generators, for all
1 ≤ i ≤ a. Then, the exceptional object is defined as, with the natural G-action,
def

En,H,H = G/H × Proj

m
p1 1
k[x0 , . . . , xn , v1 , . . . , va ] / (v1

9

pma

− g1 , . . . , v a a

− ga )

m
where gi ∈ k[x0 , . . . , xn ] is homogeneous with degree pi i such that En,H,H is smooth with

dimension n ([En,H,H ] ∈ UG (Spec k) is independent of the choice of {gi }).
Let us now give a brief outline of this paper. In section 2, we state some basic notions
and assumptions we will be using throughout the paper. In section 3, we give the precise
definition of generalized double point relation and also the definition of our equivariant
algebraic cobordism theory UG . We also show that the generalized double point relation
holds in the non-equivariant theory ω. Then, a number of basic properties, namely (D1)(D4) and (A1)-(A8) that does not involve the first Chern class operator, will be stated
and proved. The last subsection will be devoted to the investigation of the case when the
action is free. In this case, we show an isomorphism ω(X/G) ∼ UG (X).
=
In section 4, we handle the (first) Chern class operator. We first define the notion of
“nice” G-linearized invertible sheaves. Then, we define the Chern class operator c(L) for
all such sheaves and prove the most important property of this operator : formal group
law (FGL). Next, we extend the definition to arbitrary G-linearized invertible sheaves with
stronger assumptions on G and k (in particular, G is a finite abelian group).
In section 5, we will first prove the rest of the list of basic properties, i.e. (D1)-(D4) and
(A1)-(A8) that involve the Chern class operator. Then, we will show the properties (Dim)
and (FGL).
The whole section 6 will be devoted to proving the Theorem about the set of generators
of the equivariant cobordism ring UG (Spec k) as a L-algebra. The first subsection in section
6 will be dedicated to an interesting general technique which we will call splitting principle
by blowing up along invariant smooth centers. Finally, in the last section, we will prove the
well-definedness of the fixed point map, i.e. Theorem 7.2.

10

2. Notations and assumptions
Throughout this paper, we work over a field k with characteristic 0. We will denote by
Sm the category of smooth quasi-projective schemes over k. We will often refer to this as
varieties even though they do not have to be irreducible. The identity morphism will be
denoted by IX : X → X. The groups which act on varieties are either reductive connected
groups or finite groups over k. So, they are always affine over Spec k. We will often use the
symbol πk to denote the structure morphism X → Spec k and the symbol πi to denote the
projection of X1 × · · · × Xn onto its i-th component Xi .
As in [MuFoKi], an action of a group scheme G on a variety X is by definition a morphism
σ : G × X → X such that
1.

The two morphisms σ ◦ (IG × σ) and σ ◦ (µ × IX ) from G × G × X to X
agree, where µ : G × G → G is the group law of G.

2.

The composition
e×I

σ

X
X → Spec k × X −→ G × X −→ X
˜

is equal to IX , where e is the identity morphism.
For any α ∈ G and x ∈ X, we will denote σ(α, x) by α · x, or simply αx if there is no
confusion. We will say that the action is proper if the morphism G × X → X × X given by
(α, x) → (α · x, x) is proper. Similarly, we will say the action is free if the above map is a
closed immersion. This notion is stronger than the notion “set-theoretically free”. According
to Lemma 8 of [EG], set-theoretically free and proper implies free. In the case when G is a
finite group scheme, the two morphisms σ, π2 : G×X → X are both proper. That means the
morphism G × X → X × X above is proper. Hence, in this case, “set-theoretically free” is
equivalent to free. Morphisms between G-varieties are always assumed to be G-equivariant
unless specified otherwise. We will denote by G-Sm the category with objects in Sm with G
action and
MorG-Sm (X, Y ) = {f : X → Y | f is G-equivariant}.
If X is in G-Sm and E is a locally free coherent sheaf on X with rank r, then a Glinearization of E is a collection of isomorphisms {φα : α∗ E → E | α ∈ G} that satisfies the
˜

11

cocycle condition :
φαβ = φβ ◦ (β ∗ φα ),
as isomorphisms from (αβ)∗ E to E, for all α, β ∈ G. There is a natural definition of isomorphism associated to it. The set of isomorphism classes of invertible sheaves on X with a
G-linearization will be denoted by PicG (X).
If X, Y are two objects in G-Sm, then X × Y is considered to be in G-Sm with G
acting diagonally. An object Y ∈ G-Sm is called G-irreducible if there exists an irreducible
component Y of Y such that G·Y = Y . The set of isomorphism classes of invertible sheaves
on X with a G-linearization will be denoted by PicG (X). For a locally free sheaf E of rank
r over a k-scheme X, the corresponding vector bundle E over X will be given by
def
E = Spec Sym E ∨ .

The same applies to the case that X is a G-scheme over k and E is G-linearized.
Recall the definition of transversality from [LeP]. For objects A, B, C ∈ Sm and morphisms
f : A → C and g : B → C, we say f, g are transverse if A ×C B is smooth and for all
irreducible components A ⊆ A and B ⊆ B such that f (A ), g(B ) are both contained in
the same irreducible component C ⊆ C, we have either
dim A ×C B = dim A + dim B − dim C
or A ×C B = ∅. If A, B are both subschemes of C, we say that A, B are transverse if the
inclusion morphisms are transverse. If f : A → C and x is point in C, we say that x is a
regular value of f if the inclusion morphism x → C and f are transverse.
Also recall the definition of principal G-bundle from [EG]. A morphism f : X → Y is
called a principal G-bundle if G acts on X, the morphism f is flat, surjective, G-equivariant
for the trivial G-action on Y and the morphism
G × X → X ×Y X,
defined by (α, x) → (α · x, x), is an isomorphism.

12

For a morphism f : X → Y and a point y ∈ Y , we denote the fiber product Spec k(y) ×Y
X by f −1 (y) where k(y) is the residue field of y and Spec k(y) → Y is the morphism
corresponding to y. Similarly, if Z is a subscheme of Y , then we denote Z ×Y X by f −1 (Z).
If A, B are both subschemes of X, then we denote A ×X B by A ∩ B.
In this paper, for a G-irreducible object X ∈ G-Sm, a G-invariant divisor D on X is a
Weil divisor of the form

i mi Di

where Di are distinct, G-invariant, G-irreducible, reduced,

codimension 1, closed subscheme of X. We call such a divisor smooth if all the multiplicities
mi are 1 and Di are smooth and disjoint. We call a G-invariant divisor A1 + · · · + An
reduced strict normal crossing divisor if each Ai is a smooth G-invariant divisor and, for
each I ⊆ {1, . . . , n}, the closed subscheme ∩i∈I Ai is smooth with codimension |I| in X.

13

3. Geometric equivariant algebraic cobordism UG
3.1. Preliminaries. Before digging into the equivariant algebraic cobordism theory, we need
to understand more about G-invariant divisors and G-linearized invertible sheaves.
Weil Divisors :
Let X be a G-irreducible object in G-Sm. A G-invariant, G-irreducible reduced closed
subscheme D ⊆ X with codimension 1 will be called a prime G-invariant Weil divisor. A
G-invariant Weil divisor is a finite Z-linear combination of prime divisors, i.e. D =

ni Di .

A G-invariant Weil divisor D is called effective if ni are all non-negative. Let K be the sheaf
of total quotient rings of OX , which has its natural G-action. We say that two G-invariant
Weil divisors D, D are G-equivariantly linearly equivalent, denoted by D ∼ D , if there is
an element f ∈ H0 (X, K∗ )G such that D − D = div f where div f is defined in the usual
way.
Cartier Divisors :
Similar to the definition of Cartier divisors in Ch II, section 6 in [Ha], a G-invariant Cartier
divisor is an element in H0 (X, K∗ /O∗ )G . We say two G-invariant Cartier divisors D, D are
G-equivariantly linearly equivalent if D − D is in the image of
H0 (X, K∗ )G → H0 (X, K∗ /O∗ )G .
As usual, we will represent a G-invariant Cartier divisor by {(Ui , fi )} where {Ui } is an open
cover of X and fi ∈ H0 (Ui , K∗ ). The (left) G-action on the sheaf K (or the sheaf O) is given
explicitly by
(α · f )(x) = f (α−1 · x)
for any f ∈ K (or in O) and α ∈ G. Then, the Cartier divisor D being G-invariant implies
{(Ui , fi )} = {(α · Ui , α · fi )}
as elements in H0 (X, K∗ /O∗ ) for all α ∈ G. In other words, (α·fi )/fj is a unit in O(α·U )∩U
i
j
for all i, j. Since X is smooth, we have a one-to-one correspondence between the set of Ginvariant Weil divisors and the set of G-invariant Cartier divisors by the same construction
as in [Ha]. Moreover, the notion of G-equivariantly linearly equivalent is also preserved.
14

Hence, from now on, we will use the two notions interchangeably. Furthermore, divisors are
always assumed to be G-invariant unless specified otherwise and linear equivalence means
G-equivariant linear equivalence.
G-linearized invertible sheaves :
For a given G-invariant divisor D on a smooth G-variety X, we can construct a G-linearized
invertible sheaf naturally. We will denote it by OX (D). Here is the construction.
The underlying invertible sheaf structure is given by the natural definition as in Ch II,
section 6 in [Ha] : if D is represented by {(Ui , fi )}, then we define OX (D) by the following
equation :
def

OX (D)|Ui = OUi fi−1
for all i.
The G-linearization of OX (D) can be defined as the following. Consider the case when D is
a prime G-invariant divisor. Then, it defines an ideal sheaf I which is naturally G-linearized.
Then, the natural isomorphism OX (−D) ∼ I induces a G-linearization on OX (−D). Hence,
=
def

we can define the G-linearization by taking the dual, namely, OX (D) = OX (−D)∨ . In
general, if D =

def

ni Di for some prime G-invariant divisors Di , then we define OX (D) =

⊗ OX (Di )⊗ni .
The G-linearization structure on OX (D) can be explicitly given as the following. For a
given α ∈ G, we will define an isomorphism φα : α∗ O(D) → O(D). Let us consider the
restriction on Ui , the domain becomes
−1
(α∗ O(D))|Ui = α∗ (O(D)|αUi ) = α∗ (OUj ∩αUi fj )

(further restricted on Uj ∩ αUi )
−1
= OU ∩α−1 U α−1 · fj .
i
j

On the other hand, the codomain becomes OU ∩α−1 U fi−1 when restricted on Ui ∩ α−1 Uj .
i
j
Then, we define
−1
φα |U ∩α−1 U : O α−1 · fj → Ofi−1
i
j

15

−1
by sending α−1 · fj to (fi / (α−1 · fj )) fi−1 . Since φα |U ∩α−1 U is an identity map, φα is
i
j
well-defined and is an isomorphism.

We need to check the cocycle condition
φαβ = φβ ◦ (β ∗ φα ) : (αβ)∗ O(D) → O(D).
For simplicity, we will denote OX (or other base) by simply O. Notice that, by the above
−1
definition, φβ : β ∗ O(D) → O(D) corresponds to O β −1 · fj → Ofi−1 and φα : α∗ O(D) →
−1
−1
O(D) corresponds to O α−1 · fk → Ofj . So, the morphism β ∗ φα : β ∗ α∗ O(D) → β ∗ O(D)
−1
−1
corresponds to O β −1 α−1 ·fk → O β −1 ·fj . On the other hand, φαβ : (αβ)∗ O(D) → O(D)
−1
corresponds to O β −1 α−1 · fk → Ofi−1 . Thus, the domains and codomains of φαβ and

φβ ◦ (β ∗ φα ) are represented by the same generators and all the morphisms are identities.
Hence, they commute.
It remains to check its independence of the choice of representations {(Ui , fi )} of the
Cartier divisor. In other words, if D is represented by {(Ui , fi )} where fi ∈ H0 (Ui , O∗ ), then
the G-linearized invertible sheaf it defined will be G-equivariantly isomorphic to the structure
sheaf. To see this, we define a morphism from O(D) to O by patching the morphisms
Ofi−1 → O in which we send fi−1 to fi−1 . Then, it is a well-defined isomorphism. The
commutativity of the following diagram implies that this morphism is G-equivariant.
−1 − →
O α−1 · fj − − O



Ofi−1

−− O
−→

This natural construction also takes G-equivariantly linearly equivalent divisors to isomorphic G-linearized invertible sheaves, i.e. if f is in H0 (X, K∗ )G , then Of −1 → O by sending
˜
f −1 to 1.
Unfortunately, we do not have the one-to-one correspondence between the set of Ginvariant divisor classes and the set of isomorphism classes of G-linearized invertible sheaves.
Here is a simple reason. If the G-action on X is trivial, then the G-action on any G-invariant
divisor will be trivial too. Hence, the G-action on the line bundle corresponding to O(D)
must be trivial. But, there are certainly G-equivariant line bundles over X with non-trivial
fiberwise G-actions.
16

The following are some basic properties of G-invariant divisors.

Proposition 3.1. Suppose X, Y are objects in G-Sm.
(1) If f : X → Y is a morphism in G-Sm and D is a G-invariant divisor on
Y such that f ∗ D is a G-invariant divisor on X, then f ∗ O(D) ∼ O(f ∗ D).
=
(2) If D is a G-invariant divisor on X and Z is a closed subscheme of X such
that Z ∩ SuppD is empty, then OX (D)|Z ∼ OZ .
=
(3) If D is a G-invariant divisor on X, then OX (D) ∼ OX if and only if
=
D ∼ 0.

Proof. (1) Suppose D is represented by {(Ui , gi )}. Then, the G-invariant divisor f ∗ D can
be represented by {(f −1 (Ui ), f ∗ gi )}. Thus,
−1
−1
(f ∗ O(D))|f −1 (U ) = f ∗ (OUi gi ) = Of −1 (U ) f ∗ gi .
i
i
−1
On the other hand, O(f ∗ D)|f −1 (U ) = Of −1 (U ) f ∗ gi . So they are isomorphic. The
i
i
compatibility of the G-action is easy to check.

(2) Suppose D is represented by {(Ui , gi )} and i : Z → X is the closed immersion. Since
Z ∩ SuppD = ∅, by refinement, we can assume Ui either has empty intersection with Z or,
otherwise, gi is a unit in OUi . Thus, i∗ D is a G-invariant divisor on Z and can be represented
by {(Ui ∩ Z, gi |Z )}, or simply {(Z, 1)} by the independence of representation. That means
OX (D)|Z ∼ OZ (i∗ D) ∼ OZ .
=
=
(3) As mentioned before, if D and D are G-equivariantly linearly equivalent, then they
define the same G-linearized invertible sheaf, i.e. OX (D) ∼ OX (D ). So, it is enough to
=
show if OX (D) ∼ OX , then D ∼ 0. Suppose D is represented by {(Ui , gi )}. Then, the
=
−1
∗
isomorphism OX → OX (D) over Ui is given by sending 1 to ai gi for some ai ∈ OU . The
i

fact that the isomorphism is globally defined implies that ai (gj /gi ) = aj . Thus,
def

h =

aj
ai
=
∈ H0 (X, K∗ ).
gi
gj
17

−1
Since ai gi corresponds to 1, the G-action on h is trivial. Hence, the two G-invariant

divisors {(Ui , gi )} and {(Ui , ai )} are G-equivariantly linearly equivalent. The result then
∗
follows from the fact that ai ∈ OU .
i

Remark 3.2. By property (3), we can consider the set of G-invariant divisor classes on X
as a natural subgroup of PicG (X).
We will also use the following fact about projective bundles from time to time.
Proposition 3.3. For an object X ∈ G-Sm, suppose L is in PicG (X) and E is a G-linearized
locally free sheaf of rank r over X. Then P(E) and P(E ⊗ L) are naturally isomorphic as
G-equivariant projective bundles over X.
Proof. First of all, we define a morphism from P(E ⊗ L) to P(E) without considering the
group action. Let {Ui } be an open cover of X such that E|Ui is trivial and L|Ui ∼ OUi li .
=
Then, we define a morphism
γ : E|Ui → E ⊗ L|Ui
by e → e ⊗ li . This induces a morphism
f |Ui : P(E ⊗ L|Ui ) = Proj Sym E ⊗ L|Ui → Proj Sym E|Ui = P(E|Ui ).
We claim that {f |Ui } will patch together to form a morphism from P(E ⊗ L) to P(E) and it
will be an isomorphism of projective bundles over X.
Let σE , σL be the transition functions of E, L respectively from Ui to Uj . Then, we have
∗
σL (li ) = a lj for some a ∈ OU and the transition function for E ⊗ L will be σE ⊗ σL . Then,
j

(σE ⊗ σL ) ◦ γ(e) = (σE ⊗ σL )(e ⊗ li )
= σE (e) ⊗ σL (li )
= σE (e) ⊗ a lj
= a (σE (e) ⊗ lj ).
On the other hand,
γ ◦ σE (e) = σE (e) ⊗ lj .
18

If we consider σE (e) ⊗ lj and a (σE (e) ⊗ lj ) as elements in Sym E ⊗ L, then they agree, in
homogeneous coordinates. Hence, {f |Ui } patch together to form a morphism f . Moreover,
it is obviously an isomorphism and a projective bundle morphism.
It remains to check if f is G-equivariant. The G-linearization on L is described by a
set of isomorphisms {φL,α : α∗ L → L}. When restricted on Ui ∩ α−1 Uj , φL,α defines an
˜
isomorphism from Olj to Oli . So, φL,α (lj ) = bα li for some bα ∈ O∗
. Similarly, if
Ui ∩α−1 Uj
{φE,α } and {φE⊗L,α } defines the G-linearizations on E and E ⊗ L respectively, then
γ ◦ φE,α (e) = φE,α (e) ⊗ li .
On the other hand,
φE⊗L,α ◦ γ(e) = φE⊗L,α (e ⊗ lj )
= φE,α (e) ⊗ φL,α (lj )
= φE,α (e) ⊗ bα li
= bα (φE,α (e) ⊗ li ).
For the same reason, they agree in homogeneous coordinates and hence, f is G-equivariant.

3.2. Generalized double point relation. In [LeP] (Definition 0.2), the graded cobordism
group ω∗ (X) is defined as the quotient of the free abelian group generated by symbols
[f : Y → X] where Y is an object in Sm and f is a projective morphism, by an equivalence
relation called double point relation. More precisely, suppose Y ∈ Sm is equidimensional
and there is a projective morphism φ : Y → X × P1 such that a closed point 0 = ξ ∈ P1 is a
def

regular value of π2 ◦ φ (in other words, Yξ = (π2 ◦ φ)−1 (ξ) is a smooth divisor on Y ), while
the fiber Y0 = A ∪ B for some smooth divisors A, B and A + B is a reduced strict normal
crossing divisor. Then, the double point relation is
[Yξ → Y → X] = [A → Y → X] + [B → Y → X]
− [P(O ⊕ O(A)) → A ∩ B → Y → X].
19

We refer the reader to section 0 in [LeP] for more details.
In addition, in section 5.2 in [LeP], a relation called extended double point relation is
also introduced. Suppose Y ∈ Sm is equidimensional and there is a projective morphism
φ : Y → X. In addition, suppose we have divisors A, B, C on Y such that A + B + C is
a reduced strict normal crossing divisor and C ∼ A + B. Then, the extended double point
relation is
[C → Y → X] = [A → Y → X] + [B → Y → X]
− [P(O ⊕ O(A)) → A ∩ B → Y → X]
+ [P(O ⊕ O(1)) → P(O(−B) ⊕ O(−C)) → A ∩ B ∩ C → Y → X]
− [P(O ⊕ O(−B) ⊕ O(−C)) → A ∩ B ∩ C → Y → X].
On one hand, if we assume C does not intersect A ∪ B, then this is the same as the double
point relation. On the other hand, Lemma 5.2 in [LeP] shows that the extended double point
relation holds in the theory ω defined by the double point relation. One may then expect the
existence of similar formulas when Y0 = A1 ∪ A2 ∪ A3 in the double point relation setup, or
when B ∼ A1 + A2 + A3 in the extended double point relation setup. Indeed, it is possible
to write a formula for arbitrary number of divisors. For induction purposes, we will consider
the extended double point relation setup.
More precisely, suppose X is a separated scheme of finite type over k and φ : Y → X is a
projective morphism with Y ∈ Sm such that Y is equidimensional. Moreover, suppose there
are divisors A1 , . . . , An , B1 , . . . , Bm on Y such that
A1 + · · · + An ∼ B1 + · · · + Bm
and A1 + · · · + An + B1 + · · · + Bm is a reduced strict normal crossing divisor. Then, we
expect a formula of the form
[A1 → X] + · · · + [An → X] + extra terms = [B1 → X] + · · · + [Bm → X] + extra terms.
We will give such a formula inductively. For this purpose, we will consider the following.

20

Definition 3.4. Define a polynomial ring over Z with commuting variables :
def

p

q

R = Z[{Xi , Yj , Uk , Vl }]
where i, j, k, l ≥ 1 and 1 ≤ p, q ≤ 3.

Then, we define some elements in R inductively.

def

X
X
Definition 3.5. Let E1 , F1 = 0. For n ≥ 2, define
X
En

def

X
X
1
X
En−1 − (X1 + · · · + Xn−1 + En−1 )Xn Un−1 − Xn Fn−1

X
Fn

def

X
X
2
3
Fn−1 + (X1 + · · · + Xn−1 + En−1 )Xn (Un − Un ).

=

=

X
X
Y
Y
Similarly, define En , Fn by replacing X by Y and U by V in En , Fn respectively, namely,
Y
Y
E1 , F1

def

Y
En

def

Y
Y
1
Y
En−1 − (Y1 + · · · + Yn−1 + En−1 )Yn Vn−1 − Yn Fn−1

Y
Fn

def

Y
Y
2
3
Fn−1 + (Y1 + · · · + Yn−1 + En−1 )Yn (Vn − Vn )

=

=

=

0

for n ≥ 2. Also, for n, m ≥ 1, define the elements GX as the following :
n,m
def

X
X
Y X
GX = X1 + · · · + Xn + En + (Y1 + · · · + Ym )Fn + Em Fn .
n,m

Finally, define GY by interchanging X and Y in GX , namely,
n,m
n,m
def

X Y
Y
Y
GY
n,m = Y1 + · · · + Yn + En + (X1 + · · · + Xm )Fn + Em Fn .

For a projective morphism φ : Y → X, such that Y is equidimensional, and divisors
A1 , . . . , An , B1 , . . . , Bm on Y such that A1 + · · · + An ∼ B1 + · · · + Bm and A1 + · · · +
An + B1 + · · · + Bm is a reduced strict normal crossing divisor, we define an abelian group
homomorphism
G : R → ω(X)
as the following.
21

p

First of all, any term with Xi such that i > n, or Yj such that j > m, or Uk such that
q

k > n, or Vl such that l > m is sent to 0. Then, we send
1 → [Y → X]
Xi → [Ai → Y → X]
Yj → [Bj → Y → X]
1
Uk → [P(O ⊕ O(D)) → Y → X]
def

1
where D = A1 + · · · + Ak . Denote it by [Pk → X] for simplicity.
2
Uk → [P(O ⊕ O(1)) → P(O(−Ak ) ⊕ O(−D)) → Y → X]
2
Denote it by [Pk → X].
3
Uk → [P(O ⊕ O(−Ak ) ⊕ O(−D)) → Y → X]
3
Denote it by [Pk → X].
q

Vl

q

→ [Ql → Y → X]
q

q

where Ql is defined in the same manner as Pl with divisors Bl and
D = B1 + · · · + Bl instead.
p

q

Finally, we send the general term Xi · · · Yj · · · Uk · · · Vl · · · to
p

q

[Ai ×Y · · · ×Y Bj ×Y · · · ×Y Pk ×Y · · · ×Y Ql ×Y · · · → Y → X].
In order for the homomorphism G to be well-defined, we need to check that, in general,
the morphism
p

q

Ai ×Y · · · ×Y Bj ×Y · · · ×Y Pk ×Y · · · ×Y Ql ×Y · · · → Y → X
is projective and its domain is smooth. Notice that
n
G(Xi ) = [Ai ×Y · · · ×Y Ai → X] = [Ai → X],

22

which is projective and Ai is smooth. Since A1 + · · · + An + B1 + · · · + Bm is a reduced strict
normal crossing divisor, the same is true for the value of G at any term with Xi , Yj only. In
p

q

addition, the morphisms Pk → Y and Ql → Y are both projective and smooth. That means
G : R → ω(X) is well-defined. Then, the generalized double point relation GDP R(n, m) is
the equality :
G(GX ) = G(GY ).
n,m
m,n

Remark 3.6. Observe that for any n, m ≥ 1, the terms in GX or GY are always of the
m,n
n,m
form
p

q

Xi · · · Yj · · · Uk · · · Vl · · ·
where the powers for Xi , Yj are either 0 or 1. In other words, self intersection will never
happen in any GDP R(n, m). Moreover, 1 ≤ i, k ≤ n and 1 ≤ j, l ≤ m. In addition,
G(GX ), G(GY ) are both in ωdim Y −1 (X).
m,n
n,m
The generalized double point relation is indeed a generalization of the double point relation
and the extended double point relation. For example, if we apply the definition on the setup
IX : Y = X → X with A1 + A2 ∼ B1 , we get
X
1
E2 = − X1 X2 U1
X
2
3
F2 = X1 X2 (U2 − U2 )
X
X
GX = X1 + X2 + E2 + Y1 F2
2,1
1
2
3
= X1 + X2 − X1 X2 U1 + Y1 X1 X2 (U2 − U2 )

GY = Y1
1,2
Hence, the GDP R(2, 1) is the equality
[A1 → X] + [A2 → X] − [P(O ⊕ O(A1 )) → A1 ∩ A2 → X]
+ [P(O ⊕ O(1)) → P(O(−A2 ) ⊕ O(−A1 − A2 )) → B1 ∩ A1 ∩ A2 → X]
− [P(O ⊕ O(−A2 ) ⊕ O(−A1 − A2 )) → B1 ∩ A1 ∩ A2 → X]
= [B1 → X],
23

which is exactly the extended double point relation as Lemma 5.2 in [LeP]. If we further
assume that B1 is disjoint from A1 , A2 , then we get
[A1 → X] + [A2 → X] − [P(O ⊕ O(A1 )) → A1 ∩ A2 → X] = [B1 → X],
which is the double point relation in [LeP] (because NA1 ∩A2 →A2 ∼ OA1 ∩A2 (A1 )).
=

Our first goal is to prove GDP R(n, m) holds in the theory ω. In other words, we will
show that imposing the generalized double point relation is equivalent to imposing the double
point relation.
To be more precise, suppose X is a separated scheme of finite type over k and φ : Y → X
is a projective morphism such that Y ∈ Sm is equidimensional. Moreover, suppose there
are divisors A1 , . . . , An , B1 , . . . , Bm on Y such that A1 + · · · + An ∼ B1 + · · · + Bm and
A1 + · · · + An + B1 + · · · + Bm is a reduced strict normal crossing divisor. We want to show

G(GX ) = G(GY )
n,m
m,n
where G : R → ω(X) is the corresponding group homomorphism.
First of all, observe that G(GX ) = φ∗ ◦ G (GX ) where G is the map defined by the
n,m
n,m
setup I : Y → Y with the same set of divisors on Y . Similarly, G(GY ) = φ∗ ◦ G (GY ).
m,n
m,n
Hence, we reduce to the case when φ = IX . In particular, we may assume X is in Sm and
is equidimensional.
Suppose X is a separated scheme of finite type over k and L is an invertible sheaf over X.
Recall that in [LeP], there is a corresponding operator c1 (L) ∈ End (ω(X)) which is called
˜
the first Chern class operator. For simplicity, we will denote it by c(L) and call it Chern
class operator for the rest of this paper.
We are going to prove GDP R(n, m) by induction. For this purpose, we need to modify
the definition of G. Suppose X ∈ Sm is equidimensional and there are divisors A1 , . . . , An ,
B1 , . . . , Bm on X such that A1 + · · · + An ∼ B1 + · · · + Bm . Then, we define a ring
homomorphism G : R → End (ω(X)) by
24

Xi → c(O(Ai ))
Yj → c(O(Bj ))
a
U k → pa ∗ p∗
a
a
where pa : Pk → X
∗
Vlb → qb∗ qb

where qb : Qb → X
l
∗
if 1 ≤ i, k ≤ n and 1 ≤ j, l ≤ m (The morphisms pa ∗ , p∗ , qb∗ , qb are all well-defined because
a

pa , qb are both smooth and projective.). Otherwise, send them to zero.
For well-definedness of G, we need to check the commutativity of some endomorphisms.
a
Axiom (A5) in ω implies that c(L)c(L ) = c(L )c(L). In addition, for p : Pk → X, we have

c(L)p∗ p∗ = p∗ c(p∗ L)p∗ = p∗ p∗ c(L)
def

a
and same for q. For the commutativity between p∗ p∗ and (p )∗ (p )∗ where p : P = Pk → X
def

and p : P = P a → X, consider the following commutative diagram :
k

p

−→
P ×X P − − P



p

p

P

p

−− X
−→

By axiom (A2) in ω,
∗

p∗ p∗ (p )∗ (p )∗ = p∗ p ∗ p∗ (p )∗ = (p )∗ p∗ p p∗ = (p )∗ (p )∗ p∗ p∗ .
The commutativity between q and q , p and q follow from similar arguments. Hence, the
ring homomorphism G : R → End (ω(X)) is well-defined.
The statement we are going to prove is G(GX ) = G(GY ) as elements in End (ω(X)).
n,m
m,n
Notice that we do not assume A1 + · · · + An + B1 + · · · + Bm to be a reduced strict normal
crossing divisor in the setup anymore. Moreover, if Ai is a smooth divisor, then
G(Xi )[IX ] = c(O(Ai ))[IX ] = [Ai → X]
25

by the (Sect) axiom in [LeP]. So, the statement corresponding to this modified G is actually
stronger than what we aimed to prove at the beginning (we will make this more precise later).
For simplicity, we will still call this statement GDP R(n, m) within this proof.
Here is the outline of the proof. We will prove that GDP R(n, m) holds by assuming
GDP R(n, 1). Then, for n ≥ 3, we will prove GDP R(n, 1) by assuming GDP R(n − 1, 1)
and GDP R(2, 1). Thus, we reduce the proof of GDP R(n, m) to the proof of GDP R(2, 1),
which is essentially the extended double point relation in [LeP]. But since the definition of
G is modified, GDP R(2, 1) becomes a stronger statement. Hence, there is still some works
needed to be done.
Suppose GDP R(n, 1) holds. Then, for a given equidimensional X ∈ Sm and divisors
def

A1 , . . . , An , B1 , . . . , Bm on X, let C = A1 + · · · + An . Consider the setup corresponding
to A1 + · · · + An ∼ C. From GDP R(n, 1), we get G(GX ) = G(GY ). This means that, as
n,1
1,n
elements in End (ω(X)),
X
X
c(O(C)) = G(X1 + · · · + Xn + En ) + G(Fn )c(O(C)).

Similarly, by considering the setup C ∼ B1 + · · · + Bm , we get
Y
Y
c(O(C)) = G (Y1 + · · · + Ym + Em ) + G (Fm )c(O(C))

with corresponding G .
Now, consider the map G corresponding to the setup A1 + · · · + An ∼ B1 + · · · + Bm .
Then, by observing that G = G on terms without Y or V and G = G on terms without X
or U , we have
c(O(C))
X
X
= G(X1 + · · · + Xn + En ) + G(Fn )c(O(C))
X
X
Y
Y
= G (X1 + · · · + Xn + En ) + G (Fn ) (G (Y1 + · · · + Ym + Em ) + G (Fm )c(O(C)))
X
X
Y
Y
= G (X1 + · · · + Xn + En ) + G (Fn ) (G (Y1 + · · · + Ym + Em ) + G (Fm )c(O(C)))
X Y
= G (GX ) + G (Fn Fm )c(O(C)).
n,m

26

On the other hand,
c(O(C))
Y
Y
= G (Y1 + · · · + Ym + Em ) + G (Fm )c(O(C))
X Y
= G (GY ) + G (Fn Fm )c(O(C)).
m,n
X Y
Then, the result follows from cancelling G (Fn Fm )c(O(C)) on both sides. That means it

is enough to show GDP R(n, 1).
Assume GDP R(n − 1, 1) and GDP R(2, 1) are true. Now, we start with a setup A1 + · · · +
def

def

An ∼ B. Let C = A1 + · · · + An−1 . Consider the setup C + An ∼ B. Define σ = p1 ∗ p∗
1
def

and σ = p2 ∗ p∗ − p3 ∗ p∗ where
2
3
p1 : P(O ⊕ O(C)) → X
p2 : P(O ⊕ O(1)) → P(O(−An ) ⊕ O(−B)) → X
p3 : P(O ⊕ O(−An ) ⊕ O(−B)) → X.
Then, by GDP R(2, 1), we get
(1)

c(O(B)) = c(O(C)) + c(O(An ))
− c(O(C))c(O(An ))σ + c(O(B))c(O(C))c(O(An ))σ .

By GDP R(n − 1, 1) corresponding to the setup A1 + · · · + An−1 ∼ C, we have G (GX
n−1,1 ) =
G (GY
1,n−1 ) where G is the corresponding ring homomorphism. That implies
(2)

X
X
c(O(C)) = G (X1 + · · · + Xn−1 + En−1 ) + c(O(C))G (Fn−1 ).

Now, consider the setup A1 + · · · + An ∼ B and call the corresponding ring homomorphism
p

G. Then, G = G on terms involving only Xi , Uk , if 1 ≤ i, k ≤ n − 1. Also, we have
G(Xn ) = c(O(An )).
For simplicity, we will drop the notation G. Hence, as elements in End (ω(X)),

27

c(O(B))
= c(O(C)) + Xn − c(O(C))Xn σ + c(O(B))c(O(C))Xn σ
(by equation (1))
X
X
= (X1 + · · · + Xn−1 + En−1 + c(O(C))Fn−1 ) + Xn
X
X
− Xn σ (X1 + · · · + Xn−1 + En−1 + c(O(C))Fn−1 )
X
X
+ c(O(B))Xn σ (X1 + · · · + Xn−1 + En−1 + c(O(C))Fn−1 )

(by equation (2))
= X1 + · · · + Xn−1 + Xn
X
X
+ En−1 − (X1 + · · · + Xn−1 + En−1 )Xn σ
X
+ c(O(B))σ Xn (X1 + · · · + Xn−1 + En−1 )
X
X
X
+ c(O(C))Fn−1 − c(O(C))Fn−1 Xn σ + c(O(B))c(O(C))σ Xn Fn−1 ,

which is equal to
X
X
X
X
X1 + · · · + Xn + (En + Xn Fn−1 ) + c(O(B))(Fn − Fn−1 )
X
X
X
+ c(O(C))Fn−1 − c(O(C))Fn−1 Xn σ + c(O(B))c(O(C))σ Xn Fn−1
X
X
1
2
3
by definition of En and Fn and the fact that σ = G(Un−1 ) and σ = G(Un − Un ). Notice

that the last three terms are
X
X
X
c(O(C))Fn−1 − c(O(C))Fn−1 Xn σ + c(O(B))c(O(C))σ Xn Fn−1
X
= (c(O(B)) − Xn + c(O(C))Xn σ − c(O(B))c(O(C))Xn σ ) Fn−1
X
X
− c(O(C))Fn−1 Xn σ + c(O(B))c(O(C))σ Xn Fn−1

(by equation (1))
X
= (c(O(B)) − Xn )Fn−1 .

28

Hence,
X
X
X
X
c(O(B)) = X1 + · · · + Xn + En + Xn Fn−1 + c(O(B))(Fn − Fn−1 )
X
+ (c(O(B)) − Xn )Fn−1
X
X
= X1 + · · · + Xn + En + c(O(B))Fn ,

which is exactly G(GY ) = G(GX ). That means it is enough to show GDP R(2, 1).
n,1
1,n
Suppose X ∈ Sm is equidimensional and L, M are two invertible sheaves over X. Define
an element H(L, M) ∈ End (ω(X)) by :
H(L, M)

def

=

c(L) + c(M) − c(L)c(M)p1 ∗ p1 ∗
+ c(L)c(M)c(L ⊗ M)(p2 ∗ p2 ∗ − p3 ∗ p3 ∗ ) − c(L ⊗ M)
where p1 : P(O ⊕ L) → X
p2 : P(O ⊕ O(1)) → P(M∨ ⊕ (L ⊗ M)∨ ) → X
p3 : P(O ⊕ M∨ ⊕ (L ⊗ M)∨ ) → X.

Observe that if A, B, C are divisors on X such that A + B ∼ C, then
def

H(OX (A), OX (B)) = G(GX ) − G(GY )
2,1
1,2
where G is the ring homomorphism corresponding to the setup A + B ∼ C. In other words,
it is enough to show H(L, M) = 0 for any equidimensional X ∈ Sm and invertible sheaves
L, M over X. For this purpose, we need the following two Lemmas.

Lemma 3.7. Suppose f : X → X is a morphism in Sm such that X, X are both equidimensional and L, M are two invertible sheaves over X.
1. If f is projective, then H(L, M)f∗ = f∗ H(f ∗ L, f ∗ M).
2. If f is smooth, then H(f ∗ L, f ∗ M)f ∗ = f ∗ H(L, M).

Proof. 1. Axiom (A3) in ω implies that c(L)f∗ = f∗ c(f ∗ L). For pi , consider the commutative diagram
29

p

P i ×X X − − X
− i→



f

f

pi

Pi
−− X
−→
∗ f = p f p ∗ = f p p ∗ and the morphisms p are
By axiom (A2), we have pi∗ pi ∗
∗ i∗ i
i∗ ∗ i
i
P 1 ×X X

= P(O ⊕ f ∗ L) → X

P 2 ×X X

= P(O ⊕ O(1)) → P(f ∗ M∨ ⊕ f ∗ (L ⊗ M)∨ ) → X

P 3 ×X X

= P(O ⊕ f ∗ M∨ ⊕ f ∗ (L ⊗ M)∨ ) → X .

2. Similarly, axiom (A4) implies that c(f ∗ L)f ∗ = f ∗ c(L). For pi , we can consider the
same diagram above and we get pi ∗ pi ∗ f ∗ = pi ∗ f ∗ pi ∗ = f ∗ pi∗ pi ∗ .
Lemma 3.8. Suppose X is a smooth k-scheme, L1 , L2 , . . . , Ln are invertible sheaves over
˜ def
X and L1 , L2 , . . . , Ln are the corresponding line bundles over X. Let X = L1 ×X L2 ×X
˜
· · · ×X Ln and π : X → X be the projection. Then, there are canonically defined global
˜
sections si ∈ H0 (X, π ∗ Li ) such that, for each i, the section si will cut out a smooth divisor
˜
Di on X and D1 + · · · + Dn is a reduced strict normal crossing divisor.
˜
˜
Proof. Define si : X → X ×X Li by (x, v1 , . . . , vn ) → (x, v1 , . . . , vn , vi ). This is a canonically
def

defined global section of π ∗ Li . It cuts out the divisor Di = {(x, v1 , . . . , vn ) | vi = 0}.
Moreover, the intersection of Di1 , . . . , Dij is just {(x, v1 , . . . , vn ) | vi1 = vi2 = · · · = vij = 0},
˜
which is smooth and has codimension j in X.
We are now ready to prove H(L, M) = 0. First of all,
H(L, M)[f ] = H(L, M)f∗ [I] = f∗ H(f ∗ L, f ∗ M)[I].
So, it is enough to consider the element [I]. Let L1 , L2 , L3 be the invertible sheaves L, M,
˜
L ⊗ M respectively and π : X → X as in Lemma 3.8. Then, we have
π ∗ H(L, M)[I] = H(π ∗ L, π ∗ M)π ∗ [I] = H(π ∗ L, π ∗ M)[I].
˜
By the extended homotopy property in ω, π ∗ : ω(X) → ω(X) is an isomorphism. That
means it is enough to prove H(L, M)[I] = 0 when there are divisors A, B, C on X such that
30

L ∼ OX (A), M ∼ OX (B), C ∼ A + B and A + B + C is a reduced strict normal crossing
=
=
divisor. In this case,
H(L, M)[I]
= c(O(A))[I] + c(O(B))[I] − c(O(A))c(O(B))p1 ∗ p1 ∗ [I]
+ c(O(A)) c(O(B)) c(O(C)) (p2 ∗ p2 ∗ − p3 ∗ p3 ∗ )[I] − c(O(C))[I]
= [A → X] + [B → X] − p1 ∗ p1 ∗ c(O(A))c(O(B))[I]
+ (p2 ∗ p2 ∗ − p3 ∗ p3 ∗ ) c(O(A)) c(O(B)) c(O(C)) [I] − [C → X]
(by (Sect) axiom in ω)
= [A → X] + [B → X] − p1 ∗ p1 ∗ [A ∩ B → X]
+ (p2 ∗ p2 ∗ − p3 ∗ p3 ∗ )[A ∩ B ∩ C → X] − [C → X]
(by (Sect) axiom)
= [A → X] + [B → X] − [P(O ⊕ O(A)) → A ∩ B → X]
+ [P(O ⊕ O(1)) → P(O(−B) ⊕ O(−C)) → A ∩ B ∩ C → X]
− [P(O ⊕ O(−B) ⊕ O(−C)) → A ∩ B ∩ C → X] − [C → X]
= 0
by the extended double point relation in [LeP] (Lemma 5.2). Hence, we proved the following
Proposition.
Proposition 3.9. Suppose X ∈ Sm is equidimensional and A1 , . . . , An , B1 , . . . , Bm are
divisors on X such that A1 + · · · + An ∼ B1 + · · · + Bm . Let G : R → End (ω(X)) be the
corresponding map constructed before. Then, G(GX ) = G(GY ).
n,m
m,n
We can now apply this statement to prove that the generalized double point relation holds
in ω.
Corollary 3.10. Suppose X is a separated scheme of finite type over k and there is a
projective morphism φ : Y → X such that Y is in Sm and is equidimensional. Moreover,
suppose A1 , . . . , An , B1 , . . . , Bm are divisors on Y such that A1 +· · ·+An ∼ B1 +· · ·+Bm and
31

A1 + · · · + An + B1 + · · · + Bm is a reduced strict normal crossing divisor. Let G : R → ω(X)
be the corresponding map constructed before. Then, G(GX ) = G(GY ).
m,n
n,m

Proof. By definition, G(GX ) = φ∗ ◦ G (GX ) and G(GY ) = φ∗ ◦ G (GY ) where G is
m,n
m,n
n,m
n,m
the map corresponding to the setup IY : Y → Y with the same set of divisors. So, we may
assume φ = IX . Then, it follows from the fact that
G(GX )[IX ]
n,m
(the modified definition G : R → End (ω(X)))
= G(GX )
n,m
(the original definition G : R → ω(X))
and similarly for GY .
m,n

Remark 3.11. Notice that in the generalized double point relation setup φ : Y → X with
A1 + · · · + An ∼ B1 + · · · + Bm on Y , we do not assume Ai or Bj to be nonempty. If G is
the map corresponding to A1 + · · · + An ∼ B1 + · · · + Bm and G is the map corresponding
to

N

A1 + · · · + An +

M

Ci ∼ B1 + · · · + Bm +
i=n+1

Dj
j=m+1

where {Ci , Dj } are zero divisors, then
G(GX ) = G (GX ) = G (GX ) and G(GY ) = G (GY ) = G (GY ).
n,m
n,m
m,n
m,n
N,M
M,N
p

Indeed, notice that if a general term Xi · · · Uk · · · in R contains Xi or Yj with n+1 ≤ i ≤ N
p

or m + 1 ≤ j ≤ M , then G (Xi · · · Uk · · · ) = 0. By definition,
X
X
En+1 = En +

terms with Xn+1 .

Inductively,
X
X
EN = En +

terms with Xi where n + 1 ≤ i ≤ N.
32

X
Y
Y
Similar facts hold for FN , EM and FM . Hence,
X
X
Y X
GX
N,M = X1 + · · · + XN + EN + (Y1 + · · · + YM )FN + EM FN
X
X
Y X
= X1 + · · · + Xn + En + (Y1 + · · · + Ym )Fn + Em Fn

+

terms with Xi where n + 1 ≤ i ≤ N

+

terms with Yj where m + 1 ≤ j ≤ M.

That means G (GX ) = G (GX ) = G(GX ). Similarly, G (GY ) = G (GY ) =
m,n
n,m
n,m
M,N
N,M
G(GY ).
m,n
3.3. Definition and basic properties. Now we will define our equivariant algebraic cobordism theory using the generalized double point relation.
Definition 3.12. For an object X in G-Sm, let MG (X) be the set of isomorphism classes
over X of projective morphisms f : Y → X in G-Sm. Then, MG (X) is a monoid under
disjoint union of domains, i.e.
def

[Y → X] + [Y → X] = [Y

Y → X].

We define the abelian group MG (X)+ as the group completion of MG (X).
The i-th graded piece (cohomological grading) : (MG (X)+ )i , when X is equidimensional,
is given by [Y → X] where Y is equidimensional and i = dim X − dim Y . We also have
homological grading MG (X)+ where i denotes the dimension of Y , if Y is equidimensional.
i
Remark 3.13. The main reason for focusing on quasi-projective X instead of just separated
scheme of finite type over k as in [LeP] is because we will sometimes consider the quotient
X/G and the operation of taking quotient works better in the quasi-projective category.
Next, we will define the notion of equivariant generalized double point relation which is
the equivariant analog of the generalized double point relation we just defined in section 3.2.
To be more precise, we will consider the following setup.
Let φ : Y → X be a projective morphism in G-Sm such that Y is equidimensional. In
addition, A1 , . . . , An , B1 , . . . , Bm are G-invariant divisors on Y such that A1 + · · · + An ∼
33

B1 + · · · + Bm (G-equivariantly linearly equivalent) and A1 + · · · + An + B1 + · · · + Bm
is a reduced strict normal crossing divisor. In this setup, we construct a corresponding
abelian group homomorphism G : R → MG (X)+ by the exact same definition as in section
3.2. Notice that all objects involved are smooth varieties with natural G-action and all
morphisms involved are naturally G-equivariant. We will call the collection of φ : Y → X
together with the divisors as above a generalized double point relation setup over X, or
GDPR setup.
Definition 3.14. The equivariant algebraic cobordism group UG (X) is defined as the quotient of MG (X)+ by the subgroup generated by all expressions G(GX ) − G(GY ) where
n,m
m,n
G corresponds to some GDPR setup over X.
Remark 3.15. As pointed out in remarks 3.6, if φ : Y → X is the morphism defining G,
then G(GX ), G(GY ) both lie in MG (X)+ Y −1 . Hence, if X is equidimensional, we can
n,m
m,n
dim
define a homological (cohomological) grading on UG (X), namely
i
UG (X) =

UG (X) =
i

UiG (X)
i

where UiG (X) is defined as the quotient of MG (X)+ by the subgroup generated by all
i
expressions G(GX ) − G(GY ) such that G corresponds to some GDPR setup over X where
n,m
m,n
i
the dimension of the domain of φ is i + 1. Similarly, the group UG (X) is the quotient of

(MG (X)+ )i with GDPR setups over X when the dimension of the domain of φ is dim X−i+1.
Generalized double point relation is a generalization of the double point relation in the
equivariant configuration.
Proposition 3.16. Suppose φ : Y → X × P1 is a projective morphism in G-Sm (with trivial
G-action on P1 ) such that Y is equidimensional. Let ξ ∈ P1 be a closed point. Assume that
def

the fiber Yξ = (π2 ◦ φ)−1 (ξ) is a smooth G-invariant divisor on Y and there exist smooth
G-invariant divisors A, B on Y such that Y0 = A ∪ B and A, B intersect transversely, then
[Yξ → X] = [A → X] + [B → X] − [P(O ⊕ O(A)) → A ∩ B → X]
as elements in UG (X).
34

Proof. Since Yξ is disjoint from A, B and A, B intersect transversely, Yξ + A + B is a reduced
strict normal crossing divisor on Y . In addition, since P1 has trivial G-action, Yξ ∼ A + B.
That defines a generalized double point relation setup π1 ◦ φ : Y → X with Yξ ∼ A + B.
Thus, we obtain the equality G(GX ) = G(GY ) in UG (X) which is exactly
2,1
1,2
[Yξ → X] = [A → X] + [B → X] − [P(O ⊕ O(A)) → A ∩ B → X].

In [LeP], M. Levine and R. Pandharipande listed several natural axioms and properties
that an algebraic cobordism theory should satisfy. Here, we will show the equivariant version
of some of them.
(D1) If f : X → X in G-Sm is projective, then there is an abelian group homomorphism
G
G
f∗ : U∗ (X) → U∗ (X ).

Moreover, if f, g are both projective, then (g ◦ f )∗ = g∗ ◦ f∗ .

Proof. As in the ω∗ theory of [LeP], the push-forward f∗ is given by sending [h : Y → X] to
[f ◦ h : Y → X ]. We need to check that it preserves the generalized double point relation.
Suppose a generalized double point relation on X is defined by a projective morphism
φ : Y → X in G-Sm with A1 + · · · + An ∼ B1 + · · · + Bm . It defines a homomorphism
G : R → MG (X)+ . We can then consider the generalized double point relation on X given
by f ◦ φ : Y → X with the same set of divisors. This will also define a homomorphism
p
q
G : R → MG (X )+ . Thus, for a general term Xi · · · Yj · · · Uk · · · Vl · · · in R,
p

q

f∗ ◦ G(Xi · · · Yj · · · Uk · · · Vl · · · )
p

q

= f∗ [Ai ×Y · · · ×Y Bj ×Y · · · ×Y Pk ×Y · · · ×Y Ql ×Y · · · → X]
p

q

= [Ai ×Y · · · ×Y Bj ×Y · · · ×Y Pk ×Y · · · ×Y Ql ×Y · · · → X → X ].
On the other hand,
35

p

q

G (Xi · · · Yj · · · Uk · · · Vl · · · )
p

q

= [Ai ×Y · · · ×Y Bj ×Y · · · ×Y Pk ×Y · · · ×Y Ql ×Y · · · → X → X ].
That implies f∗ ◦ G = G . In particular, f∗ ◦ G(GX ) = G (GX ) and f∗ ◦ G(GY ) =
n,m
n,m
m,n
G (GY ), which means f∗ ◦ G(GX ) = f∗ ◦ G(GY ) in UG (X ). So, the group homomorm,n
n,m
m,n
phism f∗ : U G (X) → U G (X ) is well-defined. Clearly, it preserves the homological grading
and (g ◦ f )∗ = g∗ ◦ f∗ .

(D2) If f : X → X in G-Sm is smooth such that X, X are both equidimensional, then
there is an abelian group homomorphism
∗
∗
f ∗ : UG (X) → UG (X ).

Proof. Let [Y → X] be an element UG (X), then we define the pull-back f ∗ [Y → X] as
[Y ×X X → X ]. First of all, Y ×X X is a smooth variety with natural diagonal G-action
and the morphism Y ×X X → X is projective and G-equivariant.
Consider a GDPR setup over X given by φ : Y → X with divisors A1 , . . . , An , B1 , . . . , Bm
on Y and G be the corresponding map. We have the following commutative diagram :

Y

f

def

= Y ×X X − − Y
−→



φ

φ

f

−− X
−→

X

We obtain a generalized double point relation setup over X given by φ : Y → X with
divisors f ∗ A1 , . . . , f ∗ An , f ∗ B1 , . . . , f ∗ Bm on Y . Let G be the corresponding homomorphism. The smoothness of f implies that f ∗ A1 + · · · + f ∗ An + f ∗ B1 + · · · + f ∗ Bm is still a
1
reduced strict normal crossing divisor. Observe that if Pk = P(O ⊕ O(D)) is a G-equivariant
1
projective bundle over Y , then Pk ×Y Y ∼ P(O ⊕ O(f ∗ D)), as G-equivariant projective
=

bundles over Y . So,
1
1
1
1
G (Uk ) = [Pk ×Y Y → Y ] = f ∗ [Pk → Y ] = f ∗ ◦ G(Uk ).

36

p

q

Similar statements with respect to Uk and Vl also hold. For a general term,
p

f ∗ ◦ G(Xi · · · Uk · · · )
p

= f ∗ [Ai ×Y · · · ×Y Pk ×Y · · · → X]
p

= [(Ai ×Y · · · ×Y Pk ×Y · · · ) ×X X → X ].
On the other hand,
p

G (Xi · · · Uk · · · )
p

= [(Ai ×Y Y ) ×Y · · · ×Y (Pk ×Y Y ) ×Y · · · → X ].
p

= [(Ai ×Y · · · ×Y Pk ×Y · · · ) ×Y Y → X ].
p

= [(Ai ×Y · · · ×Y Pk ×Y · · · ) ×X X → X ].
That shows the well-definedness of f ∗ : UG (X) → UG (X ). Since f is smooth, taking fiber
product with f : X → X preserves codimension. Thus, f ∗ preserves the cohomological
grading.
(D3) In [LeP], there is a discussion of the first Chern class operator. This will be addressed
in the next section.
(D4) For each pair (X, X ) of objects in G-Sm, there is a bilinear, graded pairing
G
G
× : UiG (X) × Uj (X ) → Ui+j (X × X )
G
which is commutative, associative and admits a distinguished element 1 ∈ U0 (Spec k) as a

unit.
Proof. The definition is standard. We define
def

[f : Y → X] × [f : Y → X ] = [f × f : Y × Y → X × X ].
Suppose a GDPR setup over X is given by φ : Z → X with divisors A1 , . . . , An ,
B1 , . . . , Bm on Z and G be the corresponding homomorphism. We need to show
G(GX ) × [f : Y → X ] = G(GY ) × [f : Y → X ].
n,m
m,n
37

Without loss of generality, we may assume Y is equidimensional. Consider the GDPR setup
∗
∗
∗
∗
over X ×X given by φ×f : Z×Y → X ×X with divisors π1 A1 , . . . , π1 An , π1 B1 , . . . , π1 Bm

on Z × Y . Let G be the corresponding homomorphism.
∗
1
1
Observe that if Pk = P(OZ ⊕ OZ (D)), then Pk × Y = P(OZ×Y ⊕ OZ×Y (π1 D)). So,
1
1
1
1
G (Uk ) = [Pk × Y → X × X ] = [Pk → X] × [Y → X ] = G(Uk ) × [Y → X ].
p

q

Similar statements with respect to Uk and Vl also hold. For a general term,
p

[f ] × G(Xi · · · Uk · · · )
p

= [f ] × [Ai ×Z · · · ×Z Pk ×Z · · · → X]
p

= [(Ai ×Z · · · ×Z Pk ×Z · · · ) × Y → X × X ].
On the other hand,
p

G (Xi · · · Uk · · · )
p

= [(Ai × Y ) ×Z×Y · · · ×Z×Y (Pk × Y ) ×Z×Y · · · → X × X ].
p

= [(Ai ×Z · · · ×Z Pk ×Z · · · ) × Y → X × X ].
That shows the well-definedness of ×. It is not hard to see that this product is graded,
G
associative and commutative. The unit in U0 (Spec k) is simply [I : Spec k → Spec k].

Remark 3.17. We will refer to
G
G
× : UiG (X) × Uj (X ) → Ui+j (X × X )
G
as the external product. This external product gives U∗ (Spec k) a graded ring structure
G
G
and U∗ (X) a graded U∗ (Spec k)-module structure. In addition, if f : X → X is a proG
G
jective morphism in G-Sm, then the push-forward f∗ : U∗ (X) → U∗ (X ) will be a graded
G
U∗ (Spec k)-module homomorphism. Similarly, if f : X → X in G-Sm is smooth such that
∗
∗
X, X are equidimensional, then the pull-back f ∗ : UG (X ) → UG (X) will be a graded
∗
UG (Spec k)-module homomorphism.

38

The following two properties can be easily derived from the definitions, similarly to [LeP].
(A1) If f : X → X and g : X → X are both smooth and X, X , X are all equidimensional, then
(g ◦ f )∗ = f ∗ ◦ g ∗ .
Moreover, I∗ is the identity homomorphism.

(A2) If f : X → Z is projective and g : Y → Z is smooth such that X, Y , Z are all
equidimensional, then we have g ∗ f∗ = f∗ g ∗ in the pull-back square
g

X ×Z Y − − X
−→



f

f

g

−− Z
−→

Y

(A3), (A4), (A5) in [LeP] are properties involving the Chern class operator. Hence,
they will be addressed in the next section.
(A6) If f, g are projective, then
× ◦ (f∗ × g∗ ) = (f × g)∗ ◦ ×.
Proof. Let f : X → X and g : Z → Z . The statement follows from the commutativity of
the following diagram, which is easy to check.
×

UG (X) × UG (Z) − − UG (X × Z)
−→


f∗ ×g∗

(f ×g)∗

×

UG (X ) × UG (Z ) − − UG (X × Z )
−→

(A7) If f, g are smooth with equidimensional domains and codomains, then
× ◦ (f ∗ × g ∗ ) = (f × g)∗ ◦ ×.
Proof. It follows from the commutativity of the previous diagram with vertical arrows reversed.
39

3.4. Results for free action. Consider the set of objects Y ∈ G-Sm such that the geometric
quotient (definition 0.6 in [MuFoKi] ) Y /G exists as scheme over k, lies in Sm and the map
Y → Y /G is a principal G-bundle. Denote this set of objects by D. We will consider D as
a full subcategory of G-Sm. Suppose X is a variety in D, it turns out that there is a oneto-one correspondence between morphisms Z → X/G in the category Sm and G-equivariant
morphisms Y → X in the category G-Sm. This important observation will lead us to the
proof of the isomorphism
ω(X/G) → UG (X)
˜
for any X ∈ D.
Throughout this paper, we will call going from X to X/G “descent” and going from X/G
to X “ascent”.

Proposition 3.18. If f : Y → X is a morphism in G-Sm and X is in D, then Y is also in
D.

Proof. Recall that the group scheme G we are working with is either a reductive connected
group over k or a finite group.
Consider the case when G is connected and reductive. Since Y is quasi-projective, the
map Y → X is quasi-projective. Then, there exists an invertible sheaf L over Y (may not be
G-linearized) which is very ample relative to X. By Theorem 1.6 in [Su], since Y is normal,
def

there exists a positive integer m such that Lm ( = L⊗m ) admits a G-linearization. Then,
by Proposition 7.1 in [MuFoKi], we have the following commutative diagram in which Y /G
is quasi-projective and Y → Y /G is a principal G-bundle.
Y


−−
−→

X


Y /G − − X/G
−→
Since Y → Y /G is a principal G-bundle, the morphism Y → Y /G is locally trivial in the
´tale topology. That means that Y /G can be covered by ´tale neighborhoods W for which
e
e
we have the following commutative diagram :
40

´tale
e

W ×G −−
−→

W

Y


´tale
e

− − Y /G
−→

Hence, Y is smooth if and only if Y /G is smooth.
For the case when G is finite, just replace Lm by ⊗α∈G α∗ L.
The following is mostly a standard application of descent theory, but we need to make
sure we preserve the smoothness and quasi-projectiveness assumptions.
Proposition 3.19. For any object X ∈ D,
(1) There is a one-to-one correspondence between the set of morphisms f :
Z → X/G in Sm and the set of morphisms g : Y → X in G-Sm, given
by sending Z → X/G to its fiber product with X → X/G. Moreover, its
inverse is given by sending Y → X to Y /G → X/G.
(2) The above map defines a one-to-one correspondence between the set of projective morphisms f : Z → X/G in Sm and the set of projective morphisms
g : Y → X in G-Sm.
(3) The above map defines a one-to-one correspondence between the set of vector bundles E → X/G and the set of G-equivariant vector bundles E → X.
Proof. (1) For ascent, consider the following commutative diagram :
g

Z ×X/G X − −
−→

Z

X


f

− − X/G
−→

There is a natural G-action on Z ×X/G X and g is G-equivariant. Since X, Z are quasiprojective, Z ×X/G X is quasi-projective.
Claim 1 : If X is an object in D, then the morphism X → X/G is smooth.
Since X → X/G is a principal G-bundle, it is flat and locally trivial in the ´tale topology.
e
Thus, we have the following commutative diagram :
´tale
e

W ×G −−
−→

W

´tale
e

X


− − X/G
−→
41

Let x be a point in X/G and K be the algebraic closure of k(x). Then, by taking fiber
product with Spec K → Spec k(x), we have the following commutative diagram :
´tale
e

WK × G − −
−→

WK

XK


´tale
e

− − Spec K
−→

Clearly, dim XK = dim WK × G = dim G and XK is regular. The claim then follows from
Theorem 10.2 in Ch III in [Ha].
Since the morphism X → X/G is smooth and Z is smooth, Z ×X/G X is smooth. That
shows the well-definedness of ascent.
For descent, consider the following commutative diagram :
Y


g

−−
−→

X


def
f = g/G

Y /G − − − − X/G
− − −→
By Proposition 3.18, Y is in D. So, Y /G is in Sm. The fact that these two constructions are
inverse to each other is standard and follows from descent theory.
(2) Ascent clearly preserves projectiveness. For descent, it follows from the descent of
properness (Proposition 2 of [EG]) and the fact that Y /G is quasi-projective.
(3) Ascent clearly takes vector bundles to G-equivariant vector bundles. For descent, it
follows from Lemma 1 of [EG].
We are now ready to prove the following Theorem.
Theorem 3.20. Suppose X is an object in D. Sending [Z → X/G] to [Z ×X/G X → X]
defines an abelian group isomorphism
∗
Ψ : ω ∗ (X/G) → UG (X).

Proof. Define the inverse homomorphism Ψ−1 by sending [Y → X] to [Y /G → X/G]. We
will call Ψ “ascent” and Ψ−1 “descent”.
First of all, we need to prove that Ψ is well-defined. By Proposition 3.19, Ψ is well-defined
at the level of M (X/G)+ . In this proof, we will denote the fiber product with X → X/G by
42

def

a star, i.e. W ∗ = W ×X/G X. We also denote by π : X → X/G the projection. Consider
the following commutative diagram :
φ∗

Y∗ −−
−→


X × P1


φ

Y − − X/G × P1
−→
where φ corresponds to a double point relation setup over X/G (the fiber Yξ is a smooth
divisor, Y0 = A ∪ B for some smooth divisors A, B and A, B intersect transversely).
We want to show that φ∗ gives an equivariant double point relation setup over X. Notice
that Y ∗ is in D because X is in D (Proposition 3.18). So, Y ∗ is smooth and the projection
Y ∗ → Y is smooth (claim 1 in the proof of Proposition 3.19). Then, Y ∗ is equidimensional,
(Yξ )∗ = (Y ∗ )ξ , A∗ and B ∗ are G-invariant divisors on Y ∗ ,
A∗ ∪ B ∗ = (A ∪ B)∗ = (Y0 )∗ = (Y ∗ )0
and A∗ , B ∗ intersect transversely. Clearly, φ∗ is projective. Hence, that gives us an equivariant double point relation setup over X. By Proposition 3.16, we obtain the following
equation in UG (X) :
(3)

[Yξ∗ → X] = [A∗ → X] + [B ∗ → X] − [P(OD∗ ⊕ OD∗ (A∗ )) → X]
def

where D = A ∩ B.
On the other hand, the double point relation on X/G corresponding to φ is
[Yξ → X/G] = [A → X/G] + [B → X/G] − [P(OD ⊕ OD (A)) → X/G].
If we apply Ψ on this equation, we will get
(4)

[Yξ∗ → X] = [A∗ → X] + [B ∗ → X] − [P(OD ⊕ OD (A)) ×X/G X → X].

Since
P(OD ⊕ OD (A)) ×X/G X ∼ P( π ∗ (OD ⊕ OD (A)) ) ∼ P(OD∗ ⊕ OD∗ (A∗ )),
=
=
43

equations (3) and (4) are equivalent. This finishes the first half of the proof : well-definedness
of Ψ.
It remains to show the well-definedness of the inverse Ψ−1 . By Proposition 3.19, it is
well-defined at the level of MG (X)+ . It remains to show that for a given GDPR setup
φ : Y → X with divisors A1 , . . . , An , B1 , . . . , Bm on Y and corresponding homomorphism
G,
Ψ−1 ◦ G(GX ) = Ψ−1 ◦ G(GY )
n,m
m,n
as elements in ω(X/G).
First of all, Y is in D (by Proposition 3.18) implies that Y /G is in Sm and is equidimensional. In addition, for all i, the G-invariant divisor Ai is in D. So, Ai /G is in Sm.
Moreover, dim Ai /G = dim Ai − dim G implies that Ai /G is a smooth divisor on Y /G. By
similar arguments,
A1 /G + · · · + An /G + B1 /G + · · · + Bm /G
is a reduced strict normal crossing divisor on Y /G. On the other hand, by definition, there
exists f ∈ H0 (Y, K∗ )G such that
A1 + · · · + An − B1 − · · · − Bm = div f.
By the fact that H0 (Y, K∗ )G ∼ H0 (Y /G, K∗ ), we can consider f as an element in H0 (Y /G, K∗ )
=
and deduce that
A1 /G + · · · + An /G − B1 /G − · · · − Bm /G = div f.
By Proposition 3.19, φ/G : Y /G → X/G is projective. Hence, we obtain a GDPR setup
over X/G given by φ/G : Y /G → X/G with divisors A1 /G, . . . , An /G, B1 /G, . . . , Bm /G on
Y /G. Let G be the corresponding homomorphism. By Corollary 3.10,
G (GX ) = G (GY )
n,m
m,n
in ω(X/G). So, it will be enough to show G = Ψ−1 ◦ G. We will need the following claim
first.

44

Claim : For morphisms Z → X and Z → X with X, Z, Z ∈ D, we have the following
isomorphism :
(Z ×X Z )/G ∼ Z/G ×X/G Z /G.
=
Notice that
(Z/G ×X/G Z /G) ×X/G X ∼ Z/G ×X/G (Z /G ×X/G X)
=
∼ Z/G ×
=
X/G Z
(by Proposition 3.19)
∼ Z/G ×
=
X/G X ×X Z
∼ Z× Z
=
X
(by Proposition 3.19).
Again, by Proposition 3.19, we get
Z/G ×X/G Z /G ∼ ((Z/G ×X/G Z /G) ×X/G X) /G ∼ (Z ×X Z )/G.
=
=
The proves the claim.
p

Consider a general term Xi · · · Uk · · · in R. On one hand,
p

p

G (Xi · · · Uk · · · ) = [Ai /G ×Y /G · · · ×Y /G (Pk ) ×Y /G · · · → X/G]
p

where (Pk ) is the corresponding tower defined by {Ai /G}.
On the other hand,

p

p

Ψ−1 ◦ G(Xi · · · Uk · · · ) = Ψ−1 [Ai ×Y · · · ×Y Pk ×Y · · · → X]
p

where Pk is the corresponding tower defined by {Ai }
p

= [(Ai ×Y · · · ×Y Pk ×Y · · · )/G → X/G]
p

= [Ai /G ×Y /G · · · ×Y /G Pk /G ×Y /G · · · → X/G]
(by the claim).
45

p
p
Thus, it remains to show (Pk ) ∼ Pk /G. Consider the case when p = 1. Let D be the
=

divisor A1 + · · · + Ak . Then, we have
1
(Pk ) ×Y /G Y

∼ P( π ∗ (O
=
Y /G ⊕ OY /G (D/G)) )
∼ P(O ⊕ O (D))
=
Y
Y
1
= Pk .

p
p
1 = 1
By Proposition 3.19, we have (Pk ) ∼ Pk /G. Similarly, (Pk ) ∼ Pk /G for p = 2, 3.
=

When X is an object in D, there are some natural formulas relating the push-forward,
pull-back and external product with their non-equivariant versions.

Proposition 3.21. Suppose f : X → X is a morphism in D.
(1) If f is projective, then f /G is projective, we have push-forward
(f /G)∗ : ω(X /G) → ω(X/G)
and
f∗ = Ψ ◦ (f /G)∗ ◦ Ψ−1
as morphisms from UG (X ) to UG (X).
(2) If f is smooth and X, X are both equidimensional, then f /G is smooth,
we have pull-back
(f /G)∗ : ω(X/G) → ω(X /G)
and
f ∗ = Ψ ◦ (f /G)∗ ◦ Ψ−1
as morphisms from UG (X) to UG (X ).

Proof. (1) First of all, f /G is projective by Proposition 3.19. Also, X/G, X /G are both
in Sm. Hence, the push-forward (f /G)∗ : ω(X /G) → ω(X/G) is well-defined. Moreover, by
46

definition,
Ψ ◦ (f /G)∗ ◦ Ψ−1 [Y → X ] = Ψ ◦ (f /G)∗ [Y /G → X /G]
= Ψ [Y /G → X/G]
= [Y /G ×X/G X → X]
= [Y → X].
(2) By the descent of smoothness (Proposition 2 of [EG]), the morphism f /G is smooth.
Also, X/G, X /G ∈ Sm are both equidimensional. Hence, the pull-back (f /G)∗ : ω(X/G) →
ω(X /G) is well-defined. Moreover,
Ψ ◦ (f /G)∗ ◦ Ψ−1 [Y → X] = Ψ ◦ (f /G)∗ [Y /G → X/G]
= Ψ [Y /G ×X/G X /G → X /G]
= [Y /G ×X/G X /G ×X /G X → X ]
= [Y /G ×X/G X → X ]
= [Y /G ×X/G X ×X X → X ]
= [Y ×X X → X ]
by Proposition 3.19.

There is a also similar formula for the external product, which is somewhat harder to
state. We need some trivial facts first.
Let γ : G → H be a group scheme homomorphism between the group schemes G, H.
Then, for all X ∈ H-Sm, it induces a natural abelian group homomorphism
Φγ : UH (X) → UG (X)
by sending [Y → X] with H-actions to [Y → X] with G-actions via γ. This homomorphism
obviously respects GDPR, so Φγ is well-defined.
47

Denote the ascending homomorphism corresponding to G-action as ΨG : ω(X/G) →
UG (X).
Proposition 3.22. Suppose X, X are two objects in D. Then, the external product
× : UG (X) × UG (X ) → UG (X × X )
of the element (a, b) ∈ UG (X) × UG (X ) can be given by
a × b = Φ∆ ◦ ΨG×G (Ψ−1 a × Ψ−1 b)
G
G
where ∆ : G → G × G is the diagonal morphism.
Proof. Follows from the definition.

48

4. The Chern class operator c(L)
Suppose X is an object in G-Sm and L is a G-linearized invertible sheaf over X. Our goal
in this section is to define an abelian group homomorphism
G
G
c(L) : U∗ (X) → U∗−1 (X)

which satisfies some natural properties.
Recall that in section 4 of [LeP], when L is a globally generated invertible sheaf over a
k-scheme X ∈ Sm, c(L) : ω∗ (X) → ω∗−1 (X) is defined as follow. Let [f : Y → X] be
an element in ω(X) such that Y is irreducible. Since f ∗ L is a globally generated invertible
sheaf over Y , there is a smooth divisor H on Y such that OY (H) ∼ f ∗ L. Then, we define
=
def

c(L)[f : Y → X] = [H → Y → X].
It is natural to try to give a similar version in our equivariant setting. However, since
there is no assumption on how the group G acts on the scheme X, there is no guarantee that
even a single non-zero invariant global section of L can be found. For example, if the action
on X is transitive, then no matter how nice a G-linearized invertible sheaf L over X is, there
is no invariant global section that cuts out an invariant divisor. Hence, c(L)[I : X → X] can
not be defined in a similar manner.
Moreover, even if there is an invariant section cutting out a smooth invariant divisor, it
def

def

may not be generic. For example, take G = GL(2) and X = P2 with action




a b



x





 
 ·  y  def
  =
 
c d
z

a b 0



 c d 0

0 0 1



x



 
 
 y 
 
z

Consider the case when L = O(1), which is naturally G-linearized. Then, there is only one
invariant section s ∈ H0 (X, L)G that cuts out an invariant divisor, namely s = z. In this
case, for a projective map f : Y → X, we can not define c(L)[f : Y → X] by f ∗ (s) because
there is no reason to believe that Hf ∗ s (the subscheme cut out by f ∗ s) will be smooth, or
even a divisor. So, it is important that the choice of section is generic. Indeed, we will

49

see later that this freedom of choice is essential for the well-definedness of our Chern class
operator.
4.1. First approach. As pointed out in the subsection 3.4, the theory UG works nicely in
the subcategory D. Hence, our first approach is to restrict to this subcategory and define
the Chern class operator. We first need a little lemma to ensure we stay inside the quasiprojective setup.
Lemma 4.1. If X is quasi-projective over k and π : E → X is a vector bundle, then E is
quasi-projective over k.
Proof. Consider P(E ∨ ⊕ OX ) where E is the locally free sheaf over X corresponding to E.
Since P(E ∨ ⊕ OX ) → X is projective, the scheme P(E ∨ ⊕ OX ) is quasi-projective. Then, E
can be considered as an open set inside P(E ∨ ⊕ OX ), hence is quasi-projective.
Here is the natural definition of c(L) when X is in D.
Definition 4.2. Suppose X is an object in D and L is a sheaf in PicG (X). We define the
G
G
Chern class operator c(L) : U∗ (X) → U∗−1 (X) by
def

c(L) = Ψ ◦ c(π∗ LG ) ◦ Ψ−1
where π : X → X/G is the quotient map and Ψ : ω(X/G) → UG (X) is the ascent isomor˜
phism defined in subsection 3.4.
Since X is in D, the sheaf π∗ LG over X/G is invertible. Hence, the abelian group homomorphism c(π∗ LG ) : ω∗ (X/G) → ω∗−1 (X/G) is well-defined (see sections 4 and 9 in [LeP]
for more detail).
Remark 4.3. For X ∈ D and L ∈ PicG (X) such that L is globally generated by invariant
sections, we can construct c(L)[f : Y → X] by following the definitions of Ψ and c(π∗ LG ).
First, descend Y → X to get Y /G → X/G. Then, (f /G)∗ (π∗ LG ) will be a globally generated invertible sheaf over Y /G (A G-linearized invertible sheaf L being globally generated
by invariant sections is equivalent to π∗ LG being globally generated). Pick a global section
50

s ∈ H0 (Y /G, (f /G)∗ (π∗ LG )) that cuts out a smooth divisor Hs on Y /G. Then, ascend
Hs → Y /G → X/G to obtain [Hs ×X/G X → X]. Thus,
c(L)[f : Y → X] = [Hs ×X/G X → X].
It can be seen that c(L)[f : Y → X] can also be obtained in the following way. Since L is
globally generated by invariant sections, f ∗ L is also globally generated by invariant sections.
Pick a section s ∈ H0 (Y, f ∗ L)G that cuts out an invariant smooth divisor Hs on Y . Then,
c(L)[f : Y → X] = [Hs → Y → X].
Because of the natural isomorphism between UG (X) and ω(X/G) when X is in D, we
can now easily show the equivariant versions of some properties of the Chern class operator
listed in [LeP], namely (A3)-(A5), (A8), (Dim), etc.
4.2. Second approach. Instead of imposing a restriction on X, we may impose a restriction
on L. Our second approach is to first define the notion of a “nice” G-linearized invertible
sheaf. Then, we define the Chern class operator for “nice” sheaves L and extend this definition to more general G-linearized invertible sheaves through the formal group law.
Before proceeding to describe this second approach, let us recall the definition of the formal
group law and some basic properties.
We denote the Lazard ring by L (see section 1.1 in [LeMo]). Let {aij } with i, j ≥ 0 and
(i, j) = (0, 0) be the standard set of generators of the Lazard ring, i.e. L = Z[aij ]. Then,
the formal group law F is the power series in L[[u, v]] :
aij ui v j = u + v +

F (u, v) =
i,j≥0

aij ui v j
i,j≥1

(see section 2.4.3 in [LeMo]). To help our intuition, we will think of the formal group law as
giving “addition”. By definition, we have
F (u, 0) = u.
F (u, v) = F (v, u).
F (u, F (v, w)) = F (F (u, v), w)
51

and the relations on aij are the ones imposed by these equalities.
Moreover, there is a power series χ(u) ∈ L[[u]] that satisfies
F (u, χ(u)) = 0.
The power series χ(u) can be regarded as giving the “inverse” of u. Hence, we can define
“subtraction” by
def

F − (u, v) = F (u, χ(v)).
For our purpose, we also need the notion of “multiplication by a positive integer” :
def

F n (u) = F (u, F (u, · · · F (u, u) · · · ))
(n − 1 times application of F )
Finally, we will need the notion “division by a positive integer”. For simplicity, denote
1
L ⊗Z Z[ n ] by Ln . The Lazard’s Theorem states that L is a polynomial algebra over integers

with infinitely many generators (see [L]). In particular, L has no torsion and L → Ln .
Lemma 4.4. For all n ≥ 1, there exists a power series in Ln [[u]], denoted by F 1/n (u), such
that
F 1/n (F n (u)) = F n (F 1/n (u)) = u.
def

Proof. Let F n (u) =

i
i≥1 ai u

for some ai ∈ L.

Claim : a1 = n.
We proceed by induction on n. Obviously, the claim is true for n = 1. Suppose the claim
is true for n − 1. Notice that we can always ignore terms with degree of u greater than 1.
Hence,
F n (u) = F (u, F n−1 (u))
= u + F n−1 (u) + higher degree terms
= u + (n − 1)(u) + · · ·
= nu + · · · .
52

That proves the claim.
def

Let F 1/n (u) =

i≥1 bi u

i

∈ Ln [[u]] with coefficients {bi } yet to be determined. The

equality we want is
u = F 1/n (F n (u)) = b1 (a1 u + a2 u2 + · · · ) + b2 (a1 u + a2 u2 + · · · )2 + · · · .
That gives us the following set of equations :
1 = b 1 a1
0 = b 1 a2 + b 2 a2
1
0 = b1 a3 + b2 2a1 a2 + b3 a3 , etc.
1
Thus, we have b1 = 1/a1 = 1/n ∈ Ln . After b1 , . . . , bi−1 are determined, we can define
bi ∈ Ln by the equation with respect to ui and the fact that the term corresponding to bi is
just bi ai = ni bi . That gives us a power series F 1/n (u) ∈ Ln [[u]] such that u = F 1/n (F n (u)).
1
def

To show the second equality F n (F 1/n (u)) = u, let F n (F 1/n (u)) =

i≥1 ci u

i.

Then,

b1 u + b2 u2 · · · = F 1/n (u)
= F 1/n (F n (F 1/n (u)))
= F 1/n (

ci u i )
i≥1

= b1 (c1 u + c2 u2 + · · · ) + b2 (c1 u + c2 u2 + · · · )2 + · · · .
By comparing the coefficients, we obtain the following set of equations :
b1 = b 1 c 1
b2 = b 1 c 2 + b 2 c 2
1
b3 = b1 c3 + b2 2c1 c2 + b3 c3 , etc.
1
1
Since b1 = n , the first equation implies c1 = 1. Substituting c1 = 1 into the second equation

implies that c2 = 0. Inductively, ci = 0 for all i ≥ 2. Hence, F n (F 1/n (u)) = u.

53

Remark 4.5. By examining the proof carefully, it can be shown that if F 1/n (u) =

i≥1

bi ui ,

then ni(i+1)/2 bi ∈ L.
As mentioned at the beginning of this subsection, we will start by defining the notion of
a nice G-equivariant invertible sheaf.
Definition 4.6. Suppose X is an object in G-Sm and L is a sheaf in PicG (X). We say that
L is nice if there exists a morphism in G-Sm, ψ : X → Pn (with trivial G-action on Pn ) such
that L ∼ ψ ∗ O(1).
=
Here are some basic properties.
Lemma 4.7. Suppose X is an object in G-Sm.
1. The structure sheaf OX is nice.
2. If the sheaves L, L ∈ PicG (X) are both nice, then L ⊗ L is also nice.
3. If f : X → Y is a morphism in G-Sm and L ∈ PicG (Y ) is nice, then f ∗ L is nice.
Proof. 1. By considering the map ψ : X → P0 ∼ Spec k.
=
2. Suppose we have two morphisms ψ : X → Pn and ψ : X → Pm such that ψ ∗ O(1) ∼ L
=
and ψ ∗ O(1) ∼ L . Let ψ be the following composition :
=
ψ×ψ

Segre

X − − → Pn × Pm − − → PN .
−−
−−
Then, ψ ∗ O(1) ∼ L ⊗ L .
=
3. By definition.
We will start with a definition of the Chern class operator which depends on ψ. Suppose
that L is a sheaf in PicG (X) and there is a map ψ : X → Pn such that ψ ∗ O(1) ∼ L. We
=
would like to define cψ (L)[f : Y → X] as [Y ×Pn H → Y → X] where H is a hyperplane in
Pn such that Y ×Pn H is a smooth invariant divisor on Y . Clearly, it is enough to consider
the case when Y is G-irreducible. In what follows, we will show that this is well-defined,
i.e. that such an H exists, that this element is independent of the choice of H and that the
construction respects GDPR.
Lemma 4.8. Denote the dual projective space P(H0 (Pn , O(1))) by (Pn )∗ . Then, there is
a non-empty open set U in (Pn )∗ such that for any section s in U , the closed subscheme
54

Y ×Pn H ⊆ Y , where H is the hyperplane in Pn cut out by the section s, is a smooth
invariant divisor on Y .
Proof. This is a variation of the Bertini’s Theorem when char k = 0. We have f : Y → X
and ψ : X → Pn as above. Let H be the analog of the universal Cartier divisor, i.e.
def

H = { (y, s) | s(ψ ◦ f (y)) = 0 } ⊆ Y × (Pn )∗ .
Claim : H is smooth and of dimension dim Y + n − 1.
Let Pn = Proj k[x0 , . . . , xn ] and (Pn )∗ = Proj k[c0 , . . . , cn ]. Let D(xi ) be the affine open
subscheme of Pn given by xi = 0 and similarly for D(ci ). Also, let Spec A be an affine
open subscheme of (ψ ◦ f )−1 (D(xi )). Then, ψ ◦ f is locally given by a map Spec A →
Spec k[x0 /xi , . . . , xn /xi ], which corresponds to sending the elements xj /xi to some elements
aj ∈ A. So, the universal Cartier divisor H is locally given by the equation

j=i (cj /ci )aj

=

0 inside Spec A × D(ci ). Hence, the claim is true.
Consider the projection H → (Pn )∗ . For a section s ∈ (Pn )∗ , the fiber is exactly Y ×Pn H
where H is the hyperplane cut out by s. Hence, the open set we want will be the set of
regular values of this projection map.
Lemma 4.9. Let s, s be two sections in (Pn )∗ , cutting out H, H respectively, such that
Y ×Pn H and Y ×Pn H are both smooth invariant divisors on Y . Then we have
[Y ×Pn H → X] = [Y ×Pn H → X]
as elements in UG (X).
Proof. Observe that H, H are equivariantly linearly equivalent divisors on Pn . Thus,
Y ×Pn H = (ψ ◦ f )∗ H ∼ (ψ ◦ f )∗ H = Y ×Pn H
as invariant divisors on Y . The result then follows from GDP R(1, 1).
Lemma 4.10. Sending [Y → X] to [Y ×Pn H → X] defines an abelian group homomorphism
G
G
from U∗ (X) to U∗−1 (X).

55

Proof. As before, let G be the map corresponding to a GDPR setup Y → X with divisors
A1 , . . . , An , B1 , . . . , Bm on Y . We need to show
cψ (L) ◦ G(GX ) = cψ (L) ◦ G(GY ).
n,m
m,n
For simplicity, we will denote X ×Pn H by XH . Consider the projective morphism YH →
XH . By the freedom of choice of H, we may assume XH is a smooth invariant divisor on X
and the same for YH . In particular, YH , XH are both in G-Sm and YH is equidimensional.
Similarly, we may assume the same property holds for AiH and Bj H and also,
A1H + · · · + An H + B1H + · · · + Bm H
is a reduced strict normal crossing divisor on YH . Since the divisors are given by pull-back
along YH → Y , we have
A1H + · · · + An H ∼ B1H + · · · + Bm H .
Thus, we can define a map G : R → UG (XH ) by the GDPR setup YH → XH with
A1H + · · · + An H ∼ B1H + · · · + Bm H . So, it is enough to show

cψ (L) ◦ G = i∗ ◦ G
where i : XH → X.
p

For a general term Xi · · · Uk · · · ,
p

p

cψ (L) ◦ G(Xi · · · Uk · · · ) = cψ (L)[Ai ×Y · · · ×Y Pk ×Y · · · → X]
p

= [(Ai ×Y · · · ×Y Pk ×Y · · · )H → X]
p

= [AiH ×Y · · · ×Y (Pk )H ×Y · · · → XH → X].
H
H
H
p

Hence, it is enough to show (Pk )H is the same as the corresponding tower given by invariant
divisors {AiH }. The p = 1 case follows from the fact that
P(OY ⊕ OY (D))H ∼ P(OY ⊕ OY (DH ))
=
H
H

56

and the p = 2, 3 cases can be proved similarly. That shows the well-definedness of the
G
G
homomorphism. The fact that it sends U∗ (X) to U∗−1 (X) is clear.

Hence, we have the following definition.
Definition 4.11. Suppose that L is a sheaf in PicG (X) such that there exists an equivariant
morphism ψ : X → Pn with ψ ∗ O(1) ∼ L. We define the Chern class operator cψ (L) :
=
G
G
U∗ (X) → U∗−1 (X) by
def

cψ (L)[f : Y → X] = [Y ×Pn H → Y → X]
where H is a hyperplane in Pn such that Y ×Pn H is an invariant smooth divisor on Y .
We definitely do not want the definition of the Chern class operator to depend on the
particular morphism ψ : X → Pn .
Lemma 4.12. cψ (L) is independent of ψ.
Proof. Suppose we have two equivariant morphisms ψ1 : X → Pn and ψ2 : X → Pm such
∗
that ψ1 O(1) ∼ L ∼ ψ2 O(1). Consider the pull-back of sections
= = ∗

∗
ψ1 : H0 (Pn , O(1)) → H0 (X, L).
∗
Then, the image of ψ1 will lie in H0 (X, L)G and the same for ψ2 . Let {s1i } be a k-basis for
∗
∗
H0 (Pn , O(1)) and {s2j } be a k-basis for H0 (Pm , O(1)). Then, k−span{ψ1 s1i , ψ2 s2j } will

be a finite dimensional vector space in H0 (X, L)G . In addition, it is base-point free. This
defines an equivariant morphism ψ3 : X → PN which can be factored as X → Pn → PN or
∗
X → Pm → PN . Also, ψ3 O(1) ∼ L. Thus, it is enough to show cψ1 (L) = cψ3 (L).
=
Consider an element [Y → X] in UG (X). Pick a hyperplane H ⊆ PN such that Pn ∩ H

is a hyperplane in Pn (this is equivalent to Pn ×PN H being a smooth divisor on Pn ) and
Y ×PN H is a smooth divisor on Y . Then,
cψ1 (L)[Y → X] = [Y ×Pn (Pn ∩ H)]
= [Y ×PN H]
= cψ3 (L)[Y → X].
57

Hence, for a nice G-linearized invertible sheaf L over X ∈ G-Sm, we have a natural
definition of the Chern class operator
G
G
c(L) : U∗ (X) → U∗−1 (X).

4.3. Special pull-back and the formal group law. Recall that in the ω∗ theory in [LeP],
we have the following property (Proposition 9.4 in [LeP]).
For any X ∈ Sm and invertible sheaves L, M over X, we have
c(L ⊗ M) = F (c(L), c(M))
as abelian group endomorphisms on ω(X) where F ∈ L[[u, v]] ∼ ω(Spec k)[[u, v]] is the formal
=
group law with ω(X) considered as a ω(Spec k)-module by the external product. Since the
Chern class operator always cuts down the dimension of the domain by one, F (c(L), c(M))
indeed acts as a finite sum on any given element in ω(X).
We will follow the notation in [LeP] and denote this property by (FGL). Our objective in
this subsection is to prove it holds in our equivariant setting, when all G-linearized invertible
sheaves involved are nice. First of all, we will need some basic facts.
Proposition 4.13. Suppose f : Y → X is a morphism in G-Sm. Then, there exists a
G-representation V and an equivariant immersion i : Y → P(V ) × X such that f = π2 ◦ i.
If we further assume f to be projective, then i will be a closed immersion.
Proof. First, assume that G is reductive and connected. Since Y is quasi-projective, there
def

exists an (not necessarily equivariant) immersion i0 : Y → Pn . Define L = i∗ O(1) as
0
an (not necessarily G-linearized) invertible sheaf over Y . By Theorem 1.6 in [Su], there
exists an integer m such that Lm is G-linearizable. Fix a G-linearization of Lm . Since we
have a G-linearized very ample invertible sheaf L over Y , by Proposition 1.7 in [MuFoKi],
there exists an equivariant immersion i1 : Y → P(V ) for some G-representation V such that
i∗ O(1) ∼ Lm . Then, the map i1 × f : Y → P(V ) × X will be the equivariant immersion we
=
1
want.
58

Now assume that G is finite. As above, L = i∗ O(1) is a very ample invertible sheaf over
0
Y . Then, ⊗α∈G α∗ L will be a G-linearized very ample invertible sheaf over Y , which gives
us the equivariant immersion i1 .
If f is projective, then the image of i = i1 × f will be a closed subscheme of P(V ) × X.
Suppose X is a scheme over k and U is a subscheme of X. We will denote the closure of
U in X by closX U . Also denote the singular locus of X by Sing(X).
Proposition 4.14. (Equivariant immersion with smooth closure)
(1) If Y is an object in G-Sm, then there exists a G-representation V where Y
can be equivariantly embedded into P(V ) such that its closure is smooth.
(2) Suppose X, Y are objects in G-Sm and U ⊆ X is an invariant open subscheme. If a morphism f : Y → U in G-Sm is equivariant and projective,
then there exists a G-representation V , an equivariant closed immersion
i : Y → U × P(V ) such that f = π1 ◦ i, and closX×P(V ) Y is smooth.
Proof. (1) By Proposition 4.13, we may assume there exists an equivariant immersion
Y → P(V ) for some G-representation V . By the canonical resolution of singularities (Theorem 1.6 in [BiMi]), for any variety Z over k (char k = 0), there exists a smooth variety Z res
and a morphism Z res → Z which is given by a series of blowups along canonically chosen
smooth centers. As pointed out in Remarks 4-1-1 in [M], since the blowups are canonical,
Z res has a natural G-action and Z res → Z will be G-equivariant. Apply this on our case
def

by setting Z = closP(V ) Y , then we have an equivariant morphism π : Z res → Z.
First of all, since Y is smooth, π is an isomorphism away from Sing(Z) ⊆ Z − Y . That
implies the equivariant immersion Y → Z lifts to an equivariant immersion Y → Z res
and closZ res Y = Z res . Moreover, Z res is projective because π is projective and Z is projective. By Proposition 4.13, Z res can be equivariantly embedded into P(V ) for some Grepresentation V . Hence, we have Y → Z res → P(V ) such that closP(V ) Y = Z res is
smooth.
(2) Since f : Y → U is projective, by Proposition 4.13, there exists an equivariant
immersion i : Y → U × P(V ) for some G-representation V such that f = π1 ◦ i . Consider
def

U × P(V ) as an invariant open subscheme in X × P(V ) and let Z = closX×P(V ) Y . By
59

canonical resolution of singularities as above, we have an equivariant projective morphism
Z res → Z. By considering Z res → Z → X × P(V ) → X, we know that Z res → X
is equivariant and projective. By Proposition 4.13, there exists an equivariant immersion
Z res → X × P(V ) for some G-representation V . Again, the equivariant immersion Y → Z
lifts to Y → Z res and we have Y → Z res → X × P(V ) where closX×P(V ) Y = Z res is
smooth. Consider the following commutative diagram :
Z res → X × P(V )
↓
→ X × P(V ) → X.

Z

Consider its restriction over U . Then, we obtain the following commutative diagram :
Z res |U ∼ Y
=

→ U × P(V )

↓
Z|U ∼ Y
=

→ U × P(V ) → U.

That gives us an equivariant closed immersion i : Y → U × P(V ) such that the closure
closX×P(V ) Y = Z res is smooth. Moreover, the composition π1 ◦ i is given by
Y → Z res |U → Z|U ∼ Y → U × P(V ) → U,
˜
˜
=
which is π1 ◦ i = f .
In order to prove the (FGL) property , we need some reduction of arguments, which
requires the following special type of pull-back.
Let ψ : X →

n
iP i

and the G-action on

be a G-equivariant morphism where X ∈ G-Sm is equidimensional
n
iP i

is trivial. We are going to define ψ ∗ : UG ( i Pni ) → UG (X).

Our proof is basically the equivariant version of Lemma 6.1 in [LeP]. Let Q be the group
scheme

i GL(ni

+ 1) which acts on

n
iP i

naturally. We consider Q as a variety with

trivial G-action, so Q is in G-Sm.

Lemma 4.15. Let f : Y →

n
iP i

be a projective morphism in G-Sm such that Y is

G-irreducible.
60

(1) There exists a non-empty open subscheme U (ψ, f ) ⊆ Q such that, for all
closed points β ∈ U (ψ, f ), the morphisms β · ψ and f are transverse.
(2) For any two closed points β, β ∈ U (ψ, f ), we have
[X ×β·ψ Y → X] = [X ×β ·ψ Y → X]
as elements in UG (X).
n
iP i

Proof. (1) First of all, β · ψ is G-equivariant because β :

→
˜

n
iP i

is trivially

G-equivariant. Define a map Q × X →

n
iP i

by (β, x) → β · ψ(x), which is clearly

G-equivariant. In addition, since Q acts on

n
iP i

transitively, the map
Pn i )

Tβ Q ⊕ Tx X = T(β,x) (Q × X) → Tβψ(x) (
i

is surjective (Tx X means the tangent space of X at x). Since the domain and codomain are
both smooth, by Proposition 10.4 in Ch. III in [Ha] (char k = 0), the map Q × X →
is smooth. That implies (Q × X) ×

n
iP i

n
iP i

Y is smooth.

Let (Q × X) ×

n
iP i

Y → Q be the projection. If a closed point β ∈ Q is a regular value,

then ((Q × X) ×

n
iP i

Y )β = X ×β·ψ Y is smooth and

dim X ×β·ψ Y

= dim((Q × X) ×
= dim(Q × X) ×

n
iP i
n
iP i

Y )β

Y − dim Q
Pni − dim Q

= dim Q × X + dim Y − dim
i

Pn i .

= dim X + dim Y − dim
i

In other words, f and β · ψ are transverse. Hence, the open set U (ψ, f ) we want is just the
set of regular values of (Q × X) ×

n
iP i

Y → Q.

(2) Consider the following commutative diagram :
−→
−→
Q
(Q × X) × Pni Y − − Q × X − −
i


(U × X) ×


n
iP i

AN

−−
−→


def


def

Y − − U × X − − U = Q ∩ A1 − − A1 = line through β, β
−→
−→
−→

where the group scheme Q =

i GL(ni

+ 1) is considered as an open subscheme of AN for

some large N (trivial G-action on AN ). Notice that U is a non-empty open subscheme of A1 .
61

All maps in the diagram are trivially G-equivariant. The morphism (Q × X) ×

n
iP i

Y →

Q × X is projective because it is an extension from f . By using a smaller U (as long as
U ⊆ U (ψ, f )), we can assume the projection map
(U × X) ×
to be smooth. Hence, (U × X) ×

n
iP i

((U × X) ×

n
iP i

Y →U

Y is smooth. Notice that the fibers are
n
iP i

Y )β = X ×β·ψ Y.

Denote the map
def

Z = (U × X) ×

n
iP i

Y →U ×X

by g. Then, g is a projective morphism in G-Sm. In addition, Z is equidimensional because U
is equidimensional and Z → U is smooth. By Proposition 4.14, there exists a G-equivariant
closed immersion i : Z → (U × X) × P(V ) for some G-representation V such that g = π1 ◦ i
and the closure of Z in (P1 × X) × P(V ) is smooth. Let us denote this closure by Z. Thus,
we obtain a projective morphism Z → P1 × X → X in G-Sm such that the fibers of Z over
β, β ∈ P1 agree with the fibers of Z over β, β , namely Z β = Zβ and Z β = Zβ . Since
β, β can be considered as G-invariant divisors on P1 and they are G-equivariantly linearly
equivalent, we have Z β ∼ Z β , as G-invariant divisors on Z. Hence, by GDP R(1, 1),
[X ×β·ψ Y → X] = [Z β → X] = [Z β → X] = [X ×β ·ψ Y → X].

∗
∗
We will define the special pull-back ψ ∗ : UG ( i Pni ) → UG (X) by sending the element

[f : Y →

n
i P i]

to [X ×β·ψ Y → X] with β ∈ U (ψ, f ). Its well-definedness is given by the

following Lemma.

Lemma 4.16. Sending [f : Y →

n
i P i]

to [X ×β·ψ Y → X] defines an abelian group

∗
∗
homomorphism from UG ( i Pni ) to UG (X).

62

Proof. This proof is roughly the same as the proof of the well-definedness of cψ (L). We need
to show it respects GDPR. This can be achieved by using the fact that the choice of β in
the group Q is generic which is similar to the generic choice of H in Pn in the other proof.
As before, let G be the map corresponding to a GDPR setup φ : Y →

n
iP i

with

G-invariant divisors A1 , . . . , An , B1 , . . . , Bm on Y . Consider the following commutative
diagram :
(β·ψ)

def

= Y × Pni X − − →
−−
 i

Y

Y


φ

φ

β·ψ

n
iP i

−−
−→

X

By picking β ∈ U (ψ, φ), we may assume that Y is smooth and of dimension
Pn i .

dim X + dim Y − dim
i

def

Similarly, there is a non-empty open subscheme U ⊆ Q such that Ai = (β · ψ) −1 (Ai ) is
a smooth invariant divisor on Y for all β ∈ U . By taking intersection with some more
open subschemes, we may assume A1 + · · · + An + B1 + · · · + Bm is a reduced strict normal
crossing divisor on Y for all β in some non-empty open subscheme U ⊆ Q. The divisors
are given by pull-back, so A1 + · · · + An ∼ B1 + · · · + Bm . Thus, φ : Y → X together with
A1 , . . . , An , B1 , . . . , Bm defines a GDPR setup over X. Denote its corresponding map by G .
p

For a general term Xi · · · Uk · · · ,
p

p

ψ ∗ ◦ G(Xi · · · Uk · · · ) = ψ ∗ [Ai ×Y · · · ×Y Pk ×Y · · · →

Pn i ]
i

p

= [X ×β·ψ (Ai ×Y · · · ×Y Pk ×Y · · · ) → X]
p

= [(X ×β·ψ Ai ) ×Y · · · ×Y (X ×β·ψ Pk ) ×Y · · · → X].
p

= [Ai ×Y · · · ×Y (X ×β·ψ Pk ) ×Y · · · → X].
On the other hand,
p

p

G (Xi · · · Uk · · · ) = [Ai ×Y · · · ×Y (Pk ) ×Y · · · → X].
p
p
p
Observe that X ×β·ψ Pk = Y ×Y Pk ∼ (Pk ) . Hence, ψ ∗ ◦ G = G .
=

63

Hence, for any G-equivariant morphism ψ : X →

n
iP i

such that X is equidimensional,

we obtain a special pull-back

∗
ψ ∗ : UG (

∗
Pni ) → UG (X)
i

which sends [f : Y →

i

Pn i ]

to [X ×β·ψ Y → X] where β is a closed point in Q such that

β · ψ and f are transverse.
Now we can proceed to the proof of (FGL). Here are a few simple properties we will need.
Lemma 4.17. Suppose ψ : X → Pn × Pm is a morphism in G-Sm such that X is equidi∗
∗
∗
∗
mensional. Denote the sheaves π1 OPn (1), π2 OPm (1) and π1 OPn (1) ⊗ π2 OPm (1) by O(1, 0),

O(0, 1) and O(1, 1) respectively.
(1) If L is either O(1, 0), O(0, 1) or O(1, 1), then L is nice and
ψ ∗ ◦ c(L) = c(ψ ∗ L) ◦ ψ ∗
as morphisms from UG (Pn × Pm ) to UG (X).
(2) The special pull-back ψ ∗ is a UG (Spec k)-module homomorphism.
Proof. (1) The sheaves O(1, 0), O(0, 1) and O(1, 1) are nice by definition. The equalities
follow immediately from our construction.
(2) Same reason as the usual smooth pull-back.
Lemma 4.18. Suppose f : X → X is a projective morphism in G-Sm and L ∈ PicG (X )
is a nice invertible sheaf, then
f∗ ◦ c(f ∗ L) = c(L) ◦ f∗
as morphisms from UG (X) to UG (X ).
Proof. Let [Y → X] be an element in UG (X) and ψ : X → Pn be a morphism in G-Sm such
that ψ ∗ O(1) ∼ L. Then,
=
c(L) ◦ f∗ [Y → X] = c(L)[Y → X ]
= [Y ×Pn H → X ]
(fiber product via the map Y → X → X → Pn ).
64

On the other hand,
f∗ ◦ c(f ∗ L)[Y → X] = f∗ [Y ×Pn H → X]
(fiber product via the map Y → X → X → Pn )
= [Y ×Pn H → X ].

We are now ready to prove the formal group law property (FGL) of the Chern class
operator for nice G-linearized invertible sheaves. As mentioned before, the formal group law
is the power series
aij ui v j ∈ L[[u, v]].

F (u, v) =
i,j≥0

G
For nice sheaves L, M ∈ PicG (X), we consider F (c(L), c(M)) as a morphism from U∗ (X)
G
to U∗−1 (X) given by

aij c(L)i ◦ c(M)j
i,j≥0

where aij are considered as elements in UG (Spec k) via the maps
Φγ

L ∼ ω(Spec k) ∼ U{1} (Spec k) −→ UG (Spec k)
=
=
where Φγ is induced by the group scheme homomorphism γ : G → {1} (See definition of
Φγ in subsection 3.4. We will see that this is a ring embedding in Corollary 7.4). As in the
non-equivariant theory, the Chern class operator decreases the homological grading by one.
Since we have UiG (X) = 0 when i < 0, the power series

i,j≥0 aij c(L)

i ◦ c(M)j

indeed acts

as a finite sum for any given element in U G (X).
Proposition 4.19. If X is an object in G-Sm and L, M ∈ PicG (X) are both nice, then
c(L ⊗ M) = F (c(L), c(M))
G
G
as morphisms from U∗ (X) to U∗−1 (X).

Proof. Since [f : Y → X] = f∗ [IY ], by Lemma 4.18, it is enough to prove the statement on
the element [IX ] such that X ∈ G-Sm is equidimensional.
65

∗
∗
Let ψ1 : X → Pn and ψ2 : X → Pm be the maps such that ψ1 O(1) ∼ L and ψ2 O(1) ∼ M.
=
=

Let ψ : X → Pn × Pm be the map defined by ψ1 and ψ2 . Then,
c(L)[IX ] = c(ψ ∗ O(1, 0)) ◦ ψ ∗ [IPn ×Pm ]
= ψ ∗ ◦ c(O(1, 0))[IPn ×Pm ]
(by Lemma 4.17).
The same holds for M and L ⊗ M. Hence, we have
c(L ⊗ M)[IX ] = ψ ∗ ◦ c(O(1, 1))[IPn ×Pm ]
and
F (c(L), c(M))[IX ] = ψ ∗ ◦ F (c(O(1, 0)), c(O(0, 1)))[IPn ×Pm ].
Thus, without loss of generality, we can assume X = Pn × Pm , L = O(1, 0) and M =
O(0, 1). Notice that the G-actions on X, L, M and IX are all trivial now. Let
Φγ : ω(Pn × Pm ) ∼ U{1} (Pn × Pm ) → UG (Pn × Pm )
=
be the abelian groups homomorphism induced by the group scheme homomorphism γ :
G → {1}. By Proposition 9.4 in [LeP], (FGL) holds in the non-equivariant theory ω∗ . In
particular,
(5)

c(O(1, 1))[IPn ×Pm ] = F (c(O(1, 0)), c(O(0, 1)))[IPn ×Pm ]

as elements in ω(Pn × Pm ). Observe that, for L = O(1, 0), O(0, 1) or O(1, 1), we have
Φγ ◦ c(L)[IPn ×Pm ] = [Hs → Pn × Pm ] = c(L)[IPn ×Pm ]
where s ∈ H0 (Pn × Pm , L) is a global section such that Hs is a smooth divisor on Pn × Pm .
By applying Φγ on equation (5), the same equality holds in UG (Pn × Pm ).
4.4. Extending the definition. In order to extend our definition to arbitrary G-linearized
invertible sheaves, we need to first consider the sheaf O(1) ∈ PicG (P(V )) for arbitrary Grepresentation V . In the case when G is a finite abelian group with exponent e, it turns
66

out the only way to define c(O(1)), so that the property (FGL) still holds, will force us to
invert the element e ∈ Z. Hence, we introduce the notation
def

UG (X)[1/e] = UG (X) ⊗Z Z[1/e].

Remarks 4.20. We will explain why we cannot expect a more general definition of c(L)
that satisfies the (FGL) without inverting the exponent of the group. Let us consider the
following example. Suppose G is a cyclic group of order p (prime) and the ground field k
def

contains a primitive p-th root of unity ξ. Let V = k−span{x, y} with action α · x = ξx and
def

α · y = y where α is a generator of G. Let X = P(V ).
Suppose we have defined c(O(1)) : UG (X) → UG (X) such that (FGL) holds. Then, we
will have
c(O(p))[IX ] = c(O(1)⊗p )[IX ] = F p (c(O(1)))[IX ].
Notice that
F (c(O(1)), c(O(1)))[IX ] = 2 c(O(1))[IX ] + a11 c(O(1))2 [IX ] + · · · .
G
For any i ≥ 2, the element c(O(1))i [IX ] lies in U1−i (X), which is zero because the dimension

of X is one. So, we have F (c(O(1)), c(O(1)))[IX ] = 2 c(O(1))[IX ]. Inductively, we get
F p (c(O(1)))[IX ] = p c(O(1))[IX ].
On the other hand, consider the G-equivariant map ψ : X → P1 (with trivial action on
P1 ) given by (x; y) → (xp ; y p ). Then, OX (p) ∼ ψ ∗ OP1 (1). Hence, OX (p) is nice. By the
=
definition of the Chern class operator for nice G-linearized invertible sheaves,
c(O(p))[IX ] = [Hp → P(V )] = [G → P(V )]
where Hp ∼ G (the k-scheme of p points with free G-action). Hence, by pushing down both
=
equalities to UG (Spec k), we obtain
(6)

[G] = p a
def

where a = πk∗ (c(O(1))[IX ]) and πk : X → Spec k.
67

Let [Z1 ] − [Z2 ] be a representative of a ∈ UG (Spec k). Consider the group scheme homomorphism {1} → G. It induces an abelian groups homomorphism
{1}

G
Φ : U0 (Spec k) → U0 (Spec k) ∼ ω0 (Spec k) ∼ Z.
=
=

That implies
p (Φ[Z1 ] − Φ[Z2 ]) = Φ(p a) = Φ[G] = p
as elements in ω0 (Spec k). Since there is no torsion in ω0 (Spec k) ∼ Z, we conclude that
=
Φ[Z1 ] − Φ[Z2 ] = 1. On the other hand, since the order of the group G is a prime and the
dimension of Z1 is zero, Z1 ∼ Spec At
=

Spec Af where the action on At is trivial and the

action on Af is free. Moreover, At can be written as the product of Kt,i , where Kt,i are
finite field extensions of k. Similarly, Z2 ∼ Spec Bt Spec Bf and Bt = j Lt,j . By Lemma
=
2.3.4 in [LeMo], we have [Spec K] = [K : k][ISpec k ] as elements in ω(Spec k), where [K : k]
denotes the degree of the field extension. Hence,
(7) 1 = Φ[Z1 ] − Φ[Z2 ] =

[Kt,i : k] + Φ[Spec Af ] −
i

[Lt,j : k] − Φ[Spec Bf ].
j

Let us consider an G-irreducible component W of Spec Af . It can either be Spec K with
free action, or the disjoint union of p copies of Spec K with G permuting them. In the first
case,
Φ[W ] = Φ[Spec K] = [K : k] = [K : K G ][K G : k] = p[K G : k].
In the second case,
Φ[W ] = Φ[Spec

p
i=1 K]

= p[K : k].

Either case, p divides Φ[W ]. Hence, p divides Φ[Spec Af ]. Similarly, p divides Φ[Spec Bf ].
Now, if we apply the fixed point map F : UG (Spec k) → ω(Spec k) on equation (6) (see
section 7 for details), we obtain

68

0 = F[G] = F(p ([Z1 ] − [Z2 ]))
= p (F([Z1 ]) − F([Z2 ]))
= p ([Spec At ] − [Spec Bt ])
[Kt,i : k] −

= p(
i

[Lt,j : k]).
j

That implies
(8)

[Kt,i : k] −

0=
i

[Lt,j : k].
j

Combining equations (7) and (8) and the fact that p divides Φ[Spec Af ] and Φ[Spec Bf ], we
get a contradiction.
Hence, it is impossible to define c(O(1)) as an operator on UG (X) such that (FGL)
holds. It can also be seen in this example that the natural definition of c(O(1))[IX ] should
be (1/p)[Hp → X], as an element in UG (X)[1/p].
In order to simplify the calculation, we need a condition on G and k such that any
irreducible G-representation will be of dimension 1.
Definition 4.21. We will say that the pair (G, k) is split, if the group G is finite abelian
with exponent e and the field k contains a primitive e-th root of unity.
Lemma 4.22. If the pair (G, k) is split, then any irreducible G-representation has dimension
one.
Proof. Recall that we are assuming char k = 0. We can easily see that when (G, k) is split,
we have k[G] ∼
=

k. The result then follows.

For the rest of this subsection, we assume that the pair (G, k) is split. In this case, we
can extend our definition of the Chern class operator to arbitrary G-linearized invertible
sheaves. In order to preserve the (FGL) property, we would like to define c(L) by the
following formula :
F 1/e (F − (c(Le ⊗ M), c(M)))
69

where M is in PicG (X) such that Le ⊗ M and M are both nice (recall that Le means
L⊗e and F 1/e (u) is the operation “division by e” in formal group law, see subsection 4.2 for
details). We need the following two Lemmas for its well-definedness.
Lemma 4.23. For any L ∈ PicG (X), there exists an invertible sheaf M ∈ PicG (X) such
that Le ⊗ M and M are both nice.
Proof. Let us first consider the case when X = P(V ) where V is a G-representation and
L = O(1). By Lemma 4.22, X ∼ Proj k[x0 , . . . , xn ] such that, for all i, k−span{xi } is a
=
def

1-dimensional G-representation. Let Y = Proj k[y0 , . . . , yn ] with trivial action and
ψ : X = Proj k[x0 , . . . , xn ] → Proj k[y0 , . . . , yn ] = Y
be the morphism corresponding to the k-algebra homomorphism k[y0 , . . . , yn ] → k[x0 , . . . , xn ]
defined by yi → xe . Since e is the exponent of G, the map ψ is G-equivariant. Observe that
i
this map can also be considered as an e-uple embedding followed by a linear projection on
some G-invariant open subscheme. Hence, we have ψ ∗ OY (1) ∼ OX (e). In other words, the
=
sheaf OX (e) is nice.
For general X ∈ G-Sm and L ∈ PicG (X), by Proposition 4.14, there exists an equivariant
immersion ψ : X → P(V ). For large enough m, the sheaf L ⊗ ψ ∗ O(m) will be very ample.
By embedding P(V ) into some larger P(V ), we can assume m = 1. Since L ⊗ ψ ∗ O(1) is
very ample and G-linearized, by Proposition 1.7 in [MuFoKi], there exists an equivariant
immersion ψ : X → P(V ) such that ψ ∗ O(1) ∼ L ⊗ ψ ∗ O(1). Hence, we have ψ ∗ O(e) ∼
=
=
Le ⊗ ψ ∗ O(e). Then, the result follows because ψ ∗ O(e) and ψ ∗ O(e) are both nice.
Lemma 4.24. For any two sheaves M, M ∈ PicG (X) such that M, M , Le ⊗ M and
Le ⊗ M are all nice, we have
F 1/e (F − (c(Le ⊗ M), c(M))) = F 1/e (F − (c(Le ⊗ M ), c(M )))
as homomorphisms from UG (X)[1/e] to UG (X)[1/e].
Proof. By the fact that all sheaves involved are nice and Proposition 4.19, we have
F (c(Le ⊗ M), c(M )) = c(Le ⊗ M ⊗ M ) = F (c(Le ⊗ M ), c(M)).
70

That implies
F − (c(Le ⊗ M), c(M)) = F − (c(Le ⊗ M ), c(M ))
F 1/e (F − (c(Le ⊗ M), c(M))) = F 1/e (F − (c(Le ⊗ M ), c(M ))).

Definition 4.25. Assume the pair (G, k) is split. Suppose X is in G-Sm and L is in
G
G
PicG (X). We define the abelian group homomorphism c(L) : U∗ (X)[1/e] → U∗−1 (X)[1/e]

by the following formula :
def

c(L) = F 1/e (F − (c(Le ⊗ M), c(M)))
where M is in PicG (X) such that Le ⊗ M, M are both nice.
Remark 4.26. Suppose L ∈ PicG (X) is nice. In this new definition, we can pick M to be
L. Then,
F 1/e (F − (c(Le ⊗ L), c(L))) = F 1/e (c(Le )) = c(L).
That means the new definition is indeed a generalization of the definition of the Chern class
operator for nice G-linearized invertible sheaves.
Suppose X is an object in D and L ∈ PicG (X). Then we have two definitions of the
Chern class operator (as operators on UG (X)[1/e]), given by the first and second approach.
The last part of this section is to show that they agree.
Lemma 4.27. Suppose X is an object in D and L, M are sheaves in PicG (X). Let
π : X → X/G be the quotient map. Then, we have
π∗ (L ⊗ M)G ∼ π∗ LG ⊗ π∗ MG .
=
For any two sheaves L, M ∈ Pic(X/G), we have
π ∗ (L ⊗ M) ∼ (π ∗ L) ⊗ (π ∗ M).
=
In other words, descent and ascent both commutes with tensor product.
71

Proof. The second statement follows from a basic property of pull-back. For descent, since
X → X/G is a principle G-bundle, there is a one-to-one correspondence between PicG (X)
and Pic(X/G) given by π ∗ and π∗ (−)G . Therefore,

π∗ LG ⊗ π∗ MG ∼ π∗ (π ∗ (π∗ LG ⊗ π∗ MG ))G
=
∼ π∗ ((π ∗ π∗ LG ) ⊗ (π ∗ π∗ MG ))G
=
∼ π∗ (L ⊗ M)G .
=

Suppose the pair (G, k) is split, X is an object in D and L ∈ PicG (X). Denote the
corresponding Chern class operator defined by the first approach by c (L), i.e.
c (L) = Ψ ◦ c(π∗ LG ) ◦ Ψ−1
from UG (X)[1/e] to UG (X)[1/e]. Also denote the corresponding Chern class operator defined
by the second approach by c (L), i.e.
c (L)[Y → X] = [Y ×Pn H → X]
when L is nice (see subsection 4.2 for details), and for general L ∈ PicG (X),
c (L) = F 1/e (F − (c (Le ⊗ M), c (M)))
from UG (X)[1/e] to UG (X)[1/e] where M is in PicG (X) such that Le ⊗ M, M are both
nice.

Proposition 4.28. For any X ∈ D and L ∈ PicG (X), we have
c (L) = c (L)
as group homomorphisms from UG (X)[1/e] to UG (X)[1/e].
72

Proof. If L ∈ PicG (X) is nice, then there is an equivariant morphism ψ : X → Pn such that
ψ ∗ O(1) ∼ L. By definition,
=
c (L)[f : Y → X] = [Y ×Pn H → X]
where H is a hyperplance in Pn such that Y ×Pn H is an invariant smooth divisor on Y .
Let s ∈ H0 (Pn , O(1)) be the global section that cuts out H. Then, Y ×Pn H is cut out
by the invariant global section (ψ ◦ f )∗ s ∈ H0 (Y, f ∗ L)G . On the other hand, by remark
4.3, c (L)[Y → X] can also be given by the divisor cut out by any invariant global section
s ∈ H0 (Y, f ∗ L)G as long as the divisor is smooth. Hence, c (L) = c (L) when L is nice.
def

For general L ∈ PicG (X), let F 1/e (u) =

i≥1 bi

def

ui and F − (u, v) =

j,k≥0 cjk

Then, we have
c (L) = F 1/e (F − (c (Le ⊗ M), c (M)))
cjk c (Le ⊗ M)j c (M)k )i

bi (

=
i

j,k

=

cjk Ψ ◦ c(π∗ (Le ⊗ M)G )j ◦ c(π∗ MG )k ◦ Ψ−1 )i

bi (
i

j,k

(the two definitions agree for nice sheaves)

= Ψ◦(

cjk c(π∗ (Le ⊗ M)G )j c(π∗ MG )k )i ) ◦ Ψ−1

bi (
i

j,k

= Ψ ◦ F 1/e (F − (c(π∗ (Le ⊗ M)G ), c(π∗ MG ))) ◦ Ψ−1
= Ψ ◦ F 1/e (F − (c((π∗ LG )e ⊗ π∗ MG ), c(π∗ MG ))) ◦ Ψ−1
(by Lemma 4.27)
= Ψ ◦ c(π∗ LG ) ◦ Ψ−1
( because (FGL) holds in ω(X/G) )
= c (L).

73

uj v k .

5. More properties for UG
In this section, we will state and prove some more basic properties in our equivariant
algebraic cobordism theory UG , equipped with the Chern class operator for nice G-linearized
invertible sheaves. Some properties are related to the Chern class operator. In that case,
def

we will also prove them in the theory UG [1/e] = UG ⊗Z Z[1/e] for arbitrary G-linearized
invertible sheaves assuming that the pair (G, k) is split (recall that e is the exponent of G).
The non-equivariant version of these properties can be found in [LeP].
At this stage, we have established projective push-forward (D1), smooth pull-back (D2),
Chern class operator (D3) and external product (D4). For convenience, we will briefly
recall here some of the properties already shown in section 3.
(A1) If f : X → X and g : X → X are both smooth and X, X , X are all equidimensional, then
(g ◦ f )∗ = f ∗ ◦ g ∗ .
Moreover, I∗ is the identity homomorphism.
(A2) If f : X → Z is projective and g : Y → Z is smooth such that X, Y , Z are all
equidimensional, then we have g ∗ f∗ = f∗ g ∗ in the pull-back square
g

X ×Z Y − − X
−→



f

f

g

Y
−− Z
−→
(A3) If f : X → X is projective and L ∈ PicG (X ) is nice, then
f∗ ◦ c(f ∗ L) = c(L) ◦ f∗
in the theory UG . Moreover, if the pair (G, k) is split, then the same statement holds in the
theory UG [1/e] for arbitrary L ∈ PicG (X ).
Proof. The first part of the statement follows from Lemma 4.18. For the second part,

74

f∗ ◦ c(f ∗ L) = f∗ ◦ F 1/e (F − (c(f ∗ Le ⊗ f ∗ M), c(f ∗ M)))
(for some M ∈ PicG (X ) such that Le ⊗ M, M are both nice)
= f∗ ◦

cjk c(f ∗ Le ⊗ f ∗ M)j c(f ∗ M)k )i

bi (
i

j,k

where bi , cjk are coefficients for F 1/e (u), F − (u, v) respectively
= (

cjk c(Le ⊗ M)j c(M)k )i ) ◦ f∗

bi (
i

j,k

(by Lemma 4.18 and the fact that Le ⊗ M, M are nice).
Hence,
f∗ ◦ c(f ∗ L) = F 1/e (F − (c(Le ⊗ M), c(M))) ◦ f∗ = c(L) ◦ f∗ .

(A4) If f : X → X is smooth, X, X are both equidimensional and L ∈ PicG (X ) is
nice, then
f ∗ ◦ c(L) = c(f ∗ L) ◦ f ∗
in the theory UG . Moreover, if the pair (G, k) is split, then the same statement holds in the
theory UG [1/e] for arbitrary L ∈ PicG (X ).
Proof. Suppose that ψ : X → Pn is a morphism in G-Sm such that L ∼ ψ ∗ O(1). Let
=
[Y → X ] be an element in UG (X ) and H be a hyperplane in Pn such that Y ×Pn H is a
smooth invariant divisor on Y . Then,
f ∗ ◦ c(L)[Y → X ] = f ∗ [Y ×Pn H → X ]
= [X ×X (Y ×Pn H) → X].
On the other hand,
c(f ∗ L) ◦ f ∗ [Y → X ] = c(f ∗ L)[X ×X Y → X]
= [(X ×X Y ) ×Pn H → X].
75

Hence, they agree. The proof for arbitrary L is similar to the proof of the similar statement
of (A3).
(A5) If L, L ∈ PicG (X) are both nice, then
c(L) ◦ c(L ) = c(L ) ◦ c(L)
in the theory UG . Moreover, if the pair (G, k) is split, then the same statement holds in the
theory UG [1/e] for arbitrary L, L ∈ PicG (X).
Proof. Suppose that L, L are nice and let ψ : X → Pn and ψ : X → Pm be the corresponding maps for L and L respectively. Then, for some appropriately chosen hyperplanes
H ⊆ Pn and H ⊆ Pm ,
c(L) ◦ c(L ) [Y → X] = c(L) [Y ×Pm H → X]
= [(Y ×Pm H ) ×Pn H → X]
= [(Y ×Pn H) ×Pm H → X]
= c(L ) ◦ c(L) [Y → X].
The statement for arbitrary L, L ∈ PicG (X) can be shown by a similar argument as
before.
(A6) If f, g are projective, then
× ◦ (f∗ × g∗ ) = (f × g)∗ ◦ ×.
(A7) If f, g are smooth with equidimensional domains and codomains, then
× ◦ (f ∗ × g ∗ ) = (f × g)∗ ◦ ×.
(A8) Let a, b be elements in UG (X), UG (X ) respectively and let L ∈ PicG (X) be a nice
invertible sheaf. Then we have
∗
c(L)(a) × b = c(π1 L)(a × b).

76

Moreover, if the pair (G, k) is split, then the same statement holds in the theory UG [1/e] for
arbitrary L ∈ PicG (X).

Proof. Suppose that L is nice. Without loss of generality, we can assume a = [Y → X] and
b = [Y → X ]. Let ψ : X → Pn be the map corresponding to L. Then, for some H ⊆ Pn ,
(c(L)[Y → X]) × [Y → X ] = [Y ×Pn H → X] × [Y → X ]
= [(Y ×Pn H) × Y → X × X ]
= [(Y × Y ) ×Pn H → X × X ]
(via the map Y × Y → X × X → X → Pn )
∗
= c(π1 L)[Y × Y → X × X ].

For arbitrary L ∈ PicG (X),
c(L)(a) × b = F 1/e (F − (c(Le ⊗ M), c(M)))(a) × b
cjk c(Le ⊗ M)j c(M)k )i (a)) × b

bi (

= (
i

j,k

djk c(Le ⊗ M)j c(M)k (a)) × b

= (
j,k

(expand the series out and denote the coefficients by djk )
∗
∗
∗
djk c(π1 Le ⊗ π1 M)j c(π1 M)k (a × b)

=
j,k

=

∗
∗
∗
cjk c(π1 Le ⊗ π1 M)j c(π1 M)k )i (a × b)

bi (
i

j,k

∗
∗
∗
= F 1/e (F − (c(π1 Le ⊗ π1 M), c(π1 M))) (a × b)
∗
= c(π1 L)(a × b).

(Dim) If L1 , L2 , . . . , Lr ∈ PicG (X) are nice invertible sheaves and r > dim X, then
c(L1 ) ◦ c(L2 ) ◦ · · · ◦ c(Lr )[IX ] = 0.
77

Moreover, if the pair (G, k) is split, then the same statement holds in the theory UG [1/e] for
arbitrary L1 , L2 , . . . , Lr ∈ PicG (X).
G
G
G
Proof. It follows from the fact that c(L) : U∗ (X) → U∗−1 (X) and U<0 (X) = 0.

(FGL) If L, L ∈ PicG (X) are nice invertible sheaves, then
c(L ⊗ L ) = F (c(L), c(L ))
in the theory UG . Moreover, if the pair (G, k) is split, then the same statement holds in the
theory UG [1/e] for arbitrary L, L ∈ PicG (X).
Proof. The statement for nice L, L was proved in section 4. For arbitrary L, L ,
F (c(L), c(L )) = F ( F 1/e (F − (c(Le ⊗ M), c(M))) , F 1/e (F − (c(L e ⊗ M ), c(M ))) )
= F 1/e (F ( F − (c(Le ⊗ M), c(M)) , F − (c(L e ⊗ M ), c(M )) ))
= F 1/e (F − ( F (c(Le ⊗ M), c(L e ⊗ M )) , F (c(M), c(M )) ))
= F 1/e (F − ( c(Le ⊗ M ⊗ L e ⊗ M ) , c(M ⊗ M ) ))
= c(L ⊗ L )
because (L ⊗ L )e ⊗ (M ⊗ M ) and M ⊗ M are both nice.

78

6. Generators for the equivariant algebraic cobordism ring
The main objective of this section is to prove Theorem 6.22, which gives a set of generators
of the equivariant algebraic cobordism ring UG (Spec k). To achieve this, we need to use a
different version of splitting principle. We will assume the pair (G, k) is split in this section.

6.1. Splitting principle by blowing up along invariant smooth centers. In this subsection, for a sheaf E over Y and a map f : X → Y , we will denote f ∗ E by EX if there is no
confusion. Suppose X is a scheme over k and Z is a closed subscheme of X. We will denote
the blow up of X along Z by BlowZ X.
The main result in this subsection is similar to the equivariant analog of Theorem 4.7 in
[Kl].
Let S ∈ G-Sm be a ground scheme. Suppose N is a G-linearized locally free sheaf of rank
N over S and A → N is a rank 1 G-linearized locally free subsheaf. Recall the definition in
section 2.1 [Kl].
The scheme σ1,n (A, N ) is defined as the closed subscheme of Grassmannian Grn (N )
satisfying the following. A point (s, H) ∈ Grn (N ) (i.e. s ∈ S and H is a n-quotient of N |s )
is inside σ1,n (A, N ) if the composition A|s → N |s → H is zero.
Also recall the following definition in section 3.1 in [Kl].
Suppose X is in G-Sm and N is a G-linearized locally free sheaf of rank N over Spec k.
An equivariant immersion X → Grr (N ) is called twisted if it is the Segre product of an
equivariant map X → Grr (N1 ) and an equivariant immersion X → P(N2 ) for some Glinearized locally free sheaves N1 , N2 over Spec k.
Proposition 6.1. Suppose X ∈ G-Sm is G-irreducible with dimension d and there is a
twisted equivariant immersion
def

X → Grr (N ) = Y
for some G-linearized locally free sheaf N of rank N over Spec k (1 ≤ r < N ). Moreover, there is a 1-dimensional character ψ such that the dimension of the ψ component
def

H0 (Spec k, N )ψ is greater than r. Let Z = GrN −1 (N ) and A be the universal subbundle
over Z (A → NZ with rank 1). Then, there exists a closed point z of the fixed point locus
79

Z G , with residue field k(z) ∼ k, such that the closed subscheme σ1,r (A|z , N ) ⊆ Y is smooth
=
with codimension r and the dimension of X ∩ σ1,r (A|z , N ) is d − r.
Proof. This statement is similar to Theorem 3.3 in [Kl]. First of all, notice that
X × Z → Y × Z = Grr (N ) × Z ∼ Grr (NZ ).
=
On the other hand, the subsheaf A → NZ induces σ1,r (A, NZ ), which is a closed subscheme
of Grr (NZ ). So, we will consider σ1,r (A|z , N ) and X ∩ σ1,r (A|z , N ) as fibers of
σ1,r (A, NZ ) → Y × Z → Z
and
(X × Z) ∩ σ1,r (A, NZ ) → Y × Z → Z
respectively.
Suppose the G-representation corresponding to N is given by a k-basis {e1 , e2 , . . . , eN }
such that each ei defines a 1-dimensional G-representation. Let UN be the invariant affine
open subscheme of Z corresponding to e1 , . . . , eN −1 . Then, UN = Spec k[s1 , . . . , sN −1 ].
Since Z = GrN −1 (N ) ∼ P(N ∨ ) and dim H0 (Spec k, N )ψ ≥ r + 1, without loss of generality,
=
we may assume G acts on the coordinates s1 , . . . , sr trivially. In addition, it can be shown
that A → NZ is defined by


def
f = 



N −1

si ei  − eN
i=1

over UN .
Let U1,2,...,r be the affine open subscheme of Y corresponding to e1 , . . . , er . Then, we have
U1,2,...,r = Spec k[ti,j ] where 1 ≤ i ≤ r and 1 ≤ j ≤ N − r. Let (N /G, z) = (ti,j , sk ) be a
closed point in
Spec k[ti,j , sk ] = U1,2,...,r × UN ⊆ Y × Z = Grr NZ .
Then, the map A|z → N → N /G at this point corresponds to
k−span{f } → ⊕N k−span{ei } → (⊕N k−span{ei }) / k−span{g1 , . . . , gN −r }
i=1
i=1
where
80


def
gi = 



r

tj,i ej  − er+i

j=1

for 1 ≤ i ≤ N − r. The composition being zero is equivalent to f ∈ k−span{g1 , . . . , gN −r },
which is equivalent to

def



N −r−1

sj+r ti,j  − ti,(N −r) = 0

hi = si + 
j=1

for 1 ≤ i ≤ r. So, σ1,r (A, NZ ) is cut out by the equations h1 , . . . , hr inside U1,2,...,r × UN .
Let z = (q1 , . . . , qN −1 ) be a closed point in UN . Then, when restricted on the fiber of
U1,2,...,r × UN → UN over z, the closed subscheme σ1,r (A|z , N ) ∩ U1,2,...,r is cut out by r
linear equations :




N −r−1

qj+r ti,j  − ti,(N −r) = 0,

hi = qi + 
j=1

where 1 ≤ i ≤ r. So, σ1,r (A|z , N ) ∩ U1,2,...,r is smooth and of codimension r. Moreover,
since X → Grr (N ) is a twisted immersion and σ1,r (A|z , N ) ∩ U1,2,...,r is given by r linear
equations {hi = 0}, the scheme X ∩ σ1,r (A|z , N ) ∩ U1,2,...,r is of dimension d − r (See the
proof of Theorem 3.3 in [Kl] for details).
Because of the symmetry of f , the only other affine open subscheme of Y we need to
consider is U1,...,r−1,N . In this case, the map A|z → N → N /G corresponds to
k−span{f } → ⊕N k−span{ei } → (⊕N k−span{ei }) / k−span{g1 , . . . , gN −r }
i=1
i=1
where


def
gi = 

r−1


tj,i ej  + tr,i eN − er+i

j=1

for 1 ≤ i ≤ N − r − 1 and

def
gN −r = 



r−1

tj,N −r ej  + tr,N −r eN − er .
j=1

81

Hence, the equations that cut σ1,r (A, NZ ) out are


N −r−1

def

hi = si + 

sj+r ti,j  + ti,N −r sr = 0
j=1

for 1 ≤ i ≤ r − 1 and

def



N −r−1

hr = −1 + 

sj+r tr,j  + tr,N −r sr = 0.
j=1

Let B be the closed subscheme of UN defined by the equations sr = sr+1 = · · · =
sN −1 = 0 and z = (q1 , . . . , qN −1 ) be a closed point in UN − B. Then, in the fiber of
U1,...,r−1,N × UN → UN over z, the closed subscheme σ1,r (A|z , N ) is cut out by r linear
equations




N −r−1

hi = qi + 

qj+r ti,j  + ti,N −r qr = 0
j=1

for 1 ≤ i ≤ r − 1 and




N −r−1

hr = −1 + 

qj+r tr,j  + tr,N −r qr = 0.
j=1

Since at least one of qr , . . . , qN −1 is non-zero, the linear equations {hi | 1 ≤ i ≤ r} are
linearly independent. Hence, by the same reason, σ1,r (A|z , N ) ∩ U1,...,r−1,N is smooth with
codimension r and X ∩ σ1,r (A|z , N ) ∩ U1,...,r−1,N is of dimension d − r.
For a different affine open subscheme Ui1 ,...,ir−1 ,N of Y , there is a corresponding “bad”
closed subscheme B of UN defined by the set of equations {sj = 0} where j ∈ {i1 , . . . , ir−1 }.
/
Hence, the result follows by picking z = (q1 , . . . , qr , 0, . . . , 0) such that q1 , . . . , qr are all
non-zero.

Suppose A → N are G-linearized locally free sheaves of rank 1, N respectively, over
def

Spec k. Let Y = Grr−1 (N /A) and QY be its universal quotient. Let K be the kernel
of the composition NY → (N /A)Y → QY . Define a map g : Gr1 (K) → Grr (N ) as the
following.
82

For a point (y, H) in Gr1 (K), we get an exact sequence
0 → G → K|y → H → 0
where the rank of G will be N − r. Since K|y → N |y = N , we can consider N /G, which is
of rank r. Thus, we define
def

g(y, H) = N /G.
Proposition 6.2. The map g : Gr1 (K) → Grr (N ) constructed above is equivariantly isomorphic to the map corresponding to the blow up of Grr (N ) along σ1,r (A, N ).
Proof. This is the analog of Theorem 4.4 in [Kl]. First of all, it is not hard to see that
def

g is equivariant. Let X = Grr (N ), Y

def

˜ def
= Gr1 (K) and X = Blowσ (A,N ) Grr (N ).
1,r

˜
Also denote the blow up map from X to X by π. By Theorem 4.4 in [Kl], there exists an
˜
isomorphism µ : X → Y such that g ◦ µ = π. So, it is enough to show µ is equivariant. Take
an invariant open subscheme U ⊆ X such that g|U and π|U are both isomorphisms. Since
g|U , π|U are both equivariant, the map µ|U = (g|U )−1 ◦ π|U is also equivariant. Now, a map
being equivariant is a closed condition. Hence, µ is equivariant.
Theorem 6.3. Suppose X ∈ G-Sm is G-irreducible and E is a G-linearized locally free
˜
sheaf of rank r over X. Then, there exists an equivariant morphism f : X → X, which is
the composition of a series of blow ups along invariant smooth centers, and a G-linearized
˜
invertible subsheaf L → f ∗ E over X such that the sequence
0 → L → f ∗ E → (f ∗ E)/L → 0
is exact and (f ∗ E)/L is locally free with rank r − 1.
Proof. Let d be the dimension of X. The result is trivially true if d = 0, so we may assume
d ≥ 1. By Proposition 4.13, we can embed X into P(N2 ) for some G-linearized locally free
sheaf N2 over Spec k. Denote E ⊗ OX (m) by E(m) for simplicity. Assume X is projective
first. Let N1 be the G-linearized locally free sheaf over Spec k corresponding to H0 (X, E(m)).
For a sufficiently large m, we can assume the induced map (N1 )X → E(m) is surjective and
defines an equivariant immersion X → Grr (N1 ), which sends x to E(m)|x . Then, we define a
83

def

twisted equivariant immersion i : X → Grr (N ) = Y as the Segre product of X → Grr (N1 )
and X → P(N2 ). In particular, N ∼ N1 ⊗ N2 .
=
By construction, i∗ QY ∼ E(m + 1) where QY is the universal quotient of Y . Since dim
=
H0 (X, E(m)) is a polynomial of m with degree d, we may assume there is a 1-dimensional
character ψ such that the ψ component H0 (X, E(m))ψ has dimension much larger than r.
If X is not projective, we can pick N1 to be a sheaf corresponding to some finite dimensional G-representation inside H0 (X, E(m)) and construct i : X → Y in the same manner.
Let A be the universal subbundle of GrN −1 (N ). Let V1 , V2 and V be the G-representations
corresponding to N1 , N2 and N respectively. Then, the dimension of the ψ component of
V1 is much larger than r by construction. Thus, there is a 1-dimensional character ψ such
that the dimension of the ψ component of V is much larger than r. Hence, by Proposition
6.1, there exists a closed point z of the fixed point locus of GrN −1 (N ), with residue field
k(z) ∼ k, such that σ1,r (A|z , N ) ⊆ Y is smooth with codimension r and X ∩ σ1,r (A|z , N )
=
has dimension d − r.
For such z, denote σ1,r (A|z , N ) by σ for simplicity. Then, we have smooth invariant
closed subschemes X, σ of Y with dimension d and dim Y − r respectively. Moreover, X ∩ σ
has dimension d − r. By applying the embedded desingularization theorem in [BiMi] on
X ∪ σ → Y , we obtain the following commutative diagram :
i
˜ −→
X −− Y



f

p

i

X −− Y
−→
where p : Y → Y is the composition of a series of blow ups along smooth invariant centers
˜
˜
and f : X → X is the map corresponding to the strict transform of X. In addition, X ∪ σ
(denote the strict transform by

) is smooth and if E is the sum of the exceptional divisors

˜
on Y , then X, σ and E will intersect transversely. Since X and σ are both smooth and
do not contain each other, according to the Theorem 1.6 in [BiMi], it is not hard to see that
each smooth invariant center is either a proper closed subscheme of the strict transform of
X, or a subscheme away from it. Hence, f is the composition of a series of blow ups along
smooth invariant centers.
84

˜
˜
Observe that X and σ are disjoint because X ∪ σ is smooth. In addition,
˜
˜
i −1 ◦ p−1 (σ) = i −1 ( σ ∪ E) = X ∩ ( σ ∪ E) = X ∩ E,
˜
which is an invariant divisor on X. By the universal property of blow up, there is a unique
˜
map j : X → Blowσ Y such that the following diagram commutes.

i

˜ −j →
X − − Blowσ Y


q

p

Y −−
−→

Y

Since z is a fixed point with k(z) ∼ k, the sheaf A|z is a G-linearized locally free sheaf of
=
rank 1 over Spec k and it is naturally embedded inside N . Following the construction before.
def

Let Y1 = Grr−1 (N /A|z ), QY1 be its universal quotient, K be the kernel of NY1 → QY1
˜ def
˜
and Y = Gr1 (K). By Proposition 6.2, the equivariant map g : Y → Y is equivariantly
isomorphic to q : Blowσ Y → Y . Moreover, as pointed out in (4.1) in [Kl], there is an exact
sequence
(9)

def

˜

0 → L = QY → g ∗ QY → (QY1 )Y → 0
˜

˜
of G-linearized locally free sheaves over Y where L is of rank 1.
Consider the following commutative diagram :
˜ −j →
X − − Blowσ Y



Blowσ Y


q

f

i

X −−
−→

µ

g

←−
−−

Y

˜
Y

On one hand, f ∗ i∗ QY ∼ f ∗ E(m + 1). On the other hand, if we pull back the exact sequence
=
˜
(9) by µ and then j. We got an exact sequence of G-linearized locally free sheaves over X
0 → j ∗ µ∗ L → f ∗ E(m + 1) ∼ j ∗ µ∗ g ∗ QY → j ∗ µ∗ (QY1 )Y → 0.
=
˜
The result then follows by twisting the whole sequence by f ∗ OX (−m − 1).

85

6.2. Basic structure of G-linearized invertible sheaves. In this subsection, we will
state and prove some results about the structure of G-linearized invertible sheaves over some
X ∈ G-Sm.
Lemma 6.4. For any X ∈ G-Sm, we have
∗
kernel {PicG (X) → Pic(X)} = πk PicG (Spec k)

where PicG (X) → Pic(X) is the forgetful map.
Proof. Finding the kernel of the forgetful map is the same as asking how many G-linearizations
can OX have. A G-linearization of OX can be described by a set of isomorphisms
{α∗ : OX → OX | α ∈ G}.
˜
Each isomorphism α∗ induces an isomorphism
α∗ : H0 (X, OX ) → H0 (X, OX )
˜
which sends 1 to some element aα ∈ H0 (X, OX ). Since ae = 1 (e is the exponent of G) and
α
the pair (G, k) is split, aα is in k ∗ . In other words, there exists a 1-dimensional character
χ such that α∗ (1) = χ(α) for all α ∈ G. Then, the result follows from the one to one
correspondence between the set of 1-dimensional characters and PicG (Spec k).
Proposition 6.5. Suppose X ∈ G-Sm is G-irreducible and L is a G-linearized invertible
sheaf over X. Then, there exist an invariant divisor D on X and a sheaf N ∈ PicG (Spec k)
such that
∗
L ∼ OX (D) ⊗ πk N .
=

Proof. Without loss of generality, we may assume the action on X is faithful. Let U be a
non-empty, invariant open subscheme of X such that the action on U is free. By Theorem 1
in section 7 of [Mu], the geometric quotient U/G exists as a variety over k and π : U → U/G
is an ´tale morphism. By picking a smaller U , we may further assume U/G to be smooth.
e
Let D1 , . . . , Dn be some invariant divisors on X such that Di ⊆ X − U for all i and the
codimension of X − U − ∪i Di in X is at least 2.
86

Claim 1 : The kernel of the restriction map PicG (X) → PicG (U ) is generated by {OX (Di )}
and PicG (Spec k).
Consider the following commutative diagram :
PicG (Spec k)

∗
πk

a

Zn − −
−→


I

a

Zn − −
−→

PicG (X)

b

Pic(X)

c

− − PicG (U )
−→

b

c

−−
− → Pic(U ) − − 0
−→

where a sends “1” in the i-th position to OX (Di ), b is the forgetful map and c is the restriction
map.
Clearly, the third row is exact. Moreover, by Lemma 6.4, the second column is also exact.
Then, the result follows from some diagram chasing.
Since the action on U is free, according to Proposition 2 in section 7 in [Mu], there is a
one-to-one correspondence between PicG (U ) and Pic(U/G). In particular, π∗ (L|U )G is an
invertible sheaf over U/G. Since U/G is smooth, there is a divisor D on U/G such that
π∗ (L|U )G ∼ OU/G (D ). Thus, we have
=
L|U ∼ π ∗ (π∗ (L|U )G )
=
∼ π∗O
=
U/G (D )
∼ O (π ∗ D )
=
U
(π : U → U/G is ´tale).
e
Consider the sheaf OX (D ) ∈ PicG (X) where D is the invariant divisor on X given by
the closure of π ∗ D in X. Hence, L ⊗ OX (−D ) will be in the kernel of the restriction map
PicG (X) → PicG (U ). By claim 1, there are integers {mi } and a sheaf N ∈ PicG (Spec k)
such that
∗
mi Di ) ⊗ πk N .

L ⊗ OX (−D ) ∼ OX (
=
i
def

The result then follows by defining D = D +

87

i mi Di .

6.3. Reduction of towers. Next, we will define the notion of quasi-admissible tower and
admissible tower and prove we can reduce an quasi-admissible tower into something much
simplier. This subsection is an analog of section 7 in [LeP].
Definition 6.6. Suppose Y is an object in G-Sm. A morphism P → Y in G-Sm is called a
quasi-admissible tower over Y with length n if it can be factored into
P = Pn → Pn−1 → · · · → P1 → P0 = Y
such that, for all 0 ≤ i ≤ n − 1, Pi+1 = P(Ei ) where Ei is the direct sum of sheaves which is
either the pull-back of a G-linearized locally free sheaves over Y , or the pull back of OPj (m)
for some integer m and 1 ≤ j ≤ i.
In this subsection, for an object Y ∈ G-Sm, an invariant divisor D on Y and a G-linearized
locally free sheaf E over Y , we will denote E ⊗ OY (D) by E(D) for simplicity. Moreover, if
P → Y is a quasi-admissible tower, then we will denote the pull-back of E as a sheaf over Pi
by E if there is no confusion.
Definition 6.7. Suppose Y is an object in G-Sm. We will call a sheaf L ∈ PicG (Y ) admissible if there exist invariant smooth divisors D1 , . . . , Dk on Y and a sheaf N ∈ PicG (Spec k)
such that
L ∼ OY (
=

k
∗
i=1 mi Di ) ⊗ πk N

for some integers {mi }. Denote the subgroup of PicG (Y ) generated by admissible invertible
sheaves by APicG (Y ). Also, define the group of admissible invertible sheaves over Pi by
def

APicG (Pi ) = APicG (Y ) + ZOP1 (1) + · · · + ZOPi (1).
Then, a quasi-admissible tower P → Y is called admissible if all sheaves involved in the
construction are admissible invertible sheaves.
Remark 6.8. If all the G-linearized locally free sheaves involved in the construction of a
tower P → Y are invertible, then it is a quasi-admissible tower.
Proof. Since
Pic(Pi ) = Pic(Y ) + ZOP1 (1) + · · · + ZOPi (1)
88

and, by Lemma 6.4, the kernel of the forgetful map PicG (Pi ) → Pic(Pi ) is given by
PicG (Spec k), we have
PicG (Pi ) = PicG (Y ) + ZOP1 (1) + · · · + ZOPi (1).
Then, P → Y is a quasi-admissible tower by definition.

Lemma 6.9. Suppose Y ∈ G-Sm is G-irreducible and L is a sheaf in PicG (Y ). Moreover, E
is the direct sum of a finite number of invertible sheaves in PicG (Y ) and D is an invariant
smooth divisor on Y . Let
A
B
C

def

=

def

=

def

=

def

P

=

P(E ⊕ L ⊕ L(D))|D ,
P(E ⊕ L),
P(E ⊕ L(D)),
P(E ⊕ L ⊕ L(D)).

Then A, B, C are invariant smooth divisors on P, the sum of them is a reduced strict normal
crossing divisor, A + B ∼ C and
OP (A) ∼ OP (π ∗ D)
=
OP (B) ∼ (π ∗ L(D))∨ ⊗ OP (1)
=
OP (C) ∼ (π ∗ L)∨ ⊗ OP (1)
=
where π is the projection P → Y .

Proof. The fact that A, B, C are smooth divisors on P and the sum of them is a reduced
strict normal crossing divisor was stated in section 7.2 in [LeP]. They are obviously invariant.
Since π is smooth, OP (A) ∼ OP (π ∗ D). Moreover, as in the proof of Lemma 7.1 in [LeP],
=
P(E ⊕ L) ⊆ P(E ⊕ L ⊕ L(D)) is given by the vanishing of the composition of equivariant
morphisms
π ∗ L(D) → π ∗ (E ⊕ L ⊕ L(D)) → OP (1).
89

Hence,
OP (B) = OP (P(E ⊕ L)) ∼ (π ∗ L(D))∨ ⊗ OP (1).
=
Similarly,
OP (C) = OP (P(E ⊕ L(D))) ∼ (π ∗ L)∨ ⊗ OP (1).
=
Then, we have
OP (A) ⊗ OP (B) ∼ OP (π ∗ D) ⊗ (π ∗ L(D))∨ ⊗ OP (1)
=
∼ π ∗ O (D) ⊗ π ∗ L(D)∨ ⊗ O (1)
=
Y
P
∼ π ∗ L∨ ⊗ O (1)
=
P
∼ O (C)
=
P
By remark 3.2, that implies A + B ∼ C.
Lemma 6.10. Suppose Y is G-irreducible and D is an invariant smooth divisor on Y . If
P → Y is an admissible tower with length n and Pi+1 = P(⊕r Lj ), then there exist an
j=1
admissible tower P → Y of length n and quasi-admissible towers Q0 , Q1 , Q2 , Q3 → D such
that
P = Pn → · · · → Pi+1 → Pi → · · · → P0 = Y
r−1
where Pi+1 = P((⊕j=1 Lj ) ⊕ Lr (D)) and we have the following equality in UG (Y ) :

[P → Y ] − [P → Y ] = [Q0 → D → Y ] − [Q1 → D → Y ] + [Q2 → D → Y ] − [Q3 → D → Y ].
def
ˆ
ˆ
ˆ
Proof. Let Pi+1 = P((⊕r Lj ) ⊕ Lr (D)). Then, we have Pi+1 → Pi and Pi+1 → Pi+1 .
j=1

We will first construct an admissible tower
ˆ ˆ
ˆ
P = Pn → · · · → Pi+1 → Pi → · · · → P0 = Y
ˆ
such that Pk = Pi+1 ×P
Pk for all k > i. Since
ˆ
i+1
APicG (Pi+1 ) = APicG (Y ) + ZOP1 (1) + · · · + ZOPi+1 (1)
ˆ
APicG (Pi+1 ) = APicG (Y ) + ZOP1 (1) + · · · + ZOPi (1) + ZOP
ˆ

i+1

90

(1)

ˆ
and the restriction map PicG (Pi+1 ) → PicG (Pi+1 ) sends OP (1) to OPi+1 (1), if we write
ˆ
i+1
def
G (P
ˆ
Pi+2 = P(⊕L ) for some L ∈ APic
i+1 ), then we can define Pi+2 = P(⊕L ) by
j

considering L

j

as in

j
G (P
ˆ
APic

i+1 ).

Similarly, for higher levels,

ˆ
APicG (P

k)

→

j
G (P
APic

k)

is

ˆ
surjective and Pk+1 can be constructed.
Next, we will construct the admissible tower P → Y and quasi-admissible tower Q0 → Y .
def

As in the statement, Pi+1 = P((⊕r−1 Lj )⊕Lr (D)), which can be naturally embedded inside
j=1
def
ˆ
ˆ
Pi+1 . Then, we define Pk = Pi+1 ×P
Pk for all k > i+1, which is clearly admissible. The
ˆ
i+1
def
ˆ
quasi-admissible tower Q0 are defined by pull-back, i.e. (Q0 ) = D ×Y Pj for all 0 ≤ j ≤ n.
j

ˆ
By Lemma 6.9, Pi+1 , Pi+1 and (Q0 )i+1 are all invariant smooth divisors on Pi+1 , the
sum of them is a reduced strict normal crossing divisor and (Q0 )i+1 + Pi+1 ∼ Pi+1 . Pull
ˆ
them back to the top level, we have Q0 + P ∼ P as invariant smooth divisors on P. By
GDP R(2, 1), we have
(10)

ˆ
ˆ
ˆ
[P → P] = [Q0 → P] + [P → P]
ˆ
− [(Q0 ∩ P) ×P P 1 → P]
ˆ
ˆ
+ [(Q0 ∩ P ∩ P ) ×P P 2 → P]
ˆ
ˆ
− [(Q0 ∩ P ∩ P ) ×P P 3 → P]
ˆ

ˆ
as elements in UG (P), where
P1

def

P2

def

P3

def

=

=

=

P(O ⊕ O(Q0 ))
P(O ⊕ O(1)) → P(O(−P) ⊕ O(−P ))
P(O ⊕ O(−P) ⊕ O(−P )).

We then denote (Q0 ∩ P) ×P P 1 , (Q0 ∩ P ∩ P ) ×P P 2 and (Q0 ∩ P ∩ P ) ×P P 3 by Q1 , Q2
ˆ
ˆ
ˆ
and Q3 respectively. They all clearly lie over D. Since the towers Q1 , Q2 , Q3 → D are all
constructed by G-linearized invertible sheaves, by Remark 6.8, they are all quasi-admissible
towers. Hence, the result follows by pushing down equality (10) to UG (Y ).

91

Remark 6.11. Notice that Pj = Pj for all j < i + 1. For j > i + 1, if we identify the
admissible invertible sheaves over Pj−1 that comes from Y to those over Pj−1 and also the
sheaves of the form O(m) for some integer m, then Pj is defined by the exact same set of
admissible invertible sheaves as Pj .
Lemma 6.12. Suppose Y is an object in G-Sm and E is a G-linearized locally free sheaf of
rank r over Y . Furthermore, there exists an exact sequence of G-linearized sheaves over Y
0 → L → E → E/L → 0
such that L and E/L are locally free of rank 1, r − 1 respectively. Then,
P(E) ∼ P((E/L) ⊕ L)
as invariant smooth divisors on P(E ⊕ L) and they intersect transversely.
Proof. Without loss of generality, we may assume Y is G-irreducible. P(E) and P((E/L) ⊕ L)
are obviously invariant smooth divisors on P(E ⊕ L) and their intersection is P(E/L). So,
we only need to prove they are equivariantly linearly equivalent.
Ignore the G-action first. Locally, over an affine open subscheme Ui , we have E ∼ Rei1 ⊕
=
def
· · · ⊕ Reir where R = O (Ui ). Similarly, L ∼ Rfi . Let φ : L → E be the embedding
=
Y

of sheaves as in the statement. For simplicity, denote P(E ⊕ L) by P, P(E) by A and
P((E/L) ⊕ L) by B. Locally, P = Proj R[ei1 , . . . , eir , fi ], A is defined by fi = 0 and B is
def

defined by φ(fi ) = 0. So, it is enough to show g = fi /φ(fi ) ∈ K(P)∗ is independent of i,
namely, fi /φ(fi ) = fj /φ(fj ).
def

On the intersection Ui ∩ Uj , we can consider the ratio fi /fj = σij ∈ O(Ui ∩ Uj )∗ , which
defines the transition function of L. On the other hand, since φ : L → E is a morphism
between sheaves, we can also consider the ratio φ(fi )/φ(fj ) and it should be σij too. That
means
fi
φ(fi )
= σij =
.
fj
φ(fj )
Hence, g is independent of i. Finally,
α·g =α·

fi
α · fi
α · fi
=
=
= g.
φ(fi )
α · φ(fi )
φ(α · fi )
92

The following result is an analog of Lemma 5.1 in [LeP].
Lemma 6.13. If X is in G-Sm and Z is an invariant smooth closed subscheme of X, then,
as elements in UG (X),
[BlowZ X → X] − [IX ] = −[P1 → Z → X] + [P2 → Z → X]
for some projective morphisms P1 , P2 → Z in G-Sm.
def

Proof. Without loss of generality, X is G-irreducible. Let Y = BlowZ×0 (X × P1 ) (trivial
action on P1 ). Consider the projective map Y → X × P1 . For any closed point ξ = 0 in
P1 , we have [Yξ → X] = [IX ], where Yξ denotes the fiber of Y over ξ as before. Consider
def

∨
the fiber of Y over 0, we have Y0 = A ∪ B where A = P(OZ ⊕ NZ →X ) (the exceptional
def

∨
divisor) and B = BlowZ X (the strict transform of X). In addition, A ∩ B = P(NZ →X ).

Hence, Yξ , A, B are all invariant smooth divisors on Y and A, B intersect transversely. In
other words, Y → X × P1 defines an equivariant DPR. By Proposition 3.16, we have
[IX ]
∨
= [P(OZ ⊕ NZ →X ) → X] + [BlowZ X → X] − [P(OA∩B ⊕ OA∩B (A)) → A ∩ B → X]
∨
= [P(OZ ⊕ NZ →X ) → Z → X] + [BlowZ X → X] − [P(OA∩B ⊕ OA∩B (A)) → Z → X].
def

def

∨
Then, the result follows from defining P1 = P(OZ ⊕ NZ →X ) and P2 = P(OA∩B ⊕
∨
OA∩B (A)) and the fact that A ∩ B = P(NZ →X ) is projective over Z.

Remark 6.14. We can express P2 in a different way. Consider the following commutative
diagram :
∨
P(OA∩B ⊕ OA∩B (A)) − −
− → P(NZ →X ) = A ∩ B


∨
− − P(OZ ⊕ NZ →X ) = A
−→
Since A is the exceptional divisor of the blowup Y → X × P1 , we have OA (A) ∼ OA (−1).
=

P(OA ⊕ OA (A))

Thus, OA∩B (A) ∼ OA∩B (−1). Hence,
=
∨
P1 = P(O ⊕ NZ →X ) → Z

93

∨
P2 = P(O ⊕ O(−1)) → P(NZ →X ) → Z.

Definition 6.15. Define UG (Spec k) to be the abelian subgroup of UG (Spec k) generated
by admissible towers over Spec k.
Remarks 6.16. If P → Spec k and P → Spec k are two admissible towers, then the product
P × P → P → Spec k is also an admissible tower over Spec k. In other words, UG (Spec k)
is a subring of UG (Spec k).
Proposition 6.17. For any quasi-admissible tower P → Y where Y is G-irreducible, there
exist elements ai ∈ UG (Spec k) and maps Yi → Y in G-Sm with dim Yi ≤ dim Y such that
[P → Y ] =

ai [Yi → Y ]
i

as elements in UG (Y ).
Proof. We will prove the statement by induction on dimension of Y . We will handle the
induction step first. Suppose dim Y ≥ 1. Let UG (Y ) be the subgroup of UG (Y ) generated
by elements of the form [P → Y → Y ] where Y ∈ G-Sm is G-irreducible with dimension
less than dim Y and P → Y is a quasi-admissible tower. So, elements in UG (Y ) will be
handled by the induction assumption. Let P → Y be a quasi-admissible tower. If the length
of the tower n is 0, then we are done. Suppose n ≥ 1.
Step 1 : Reduction to a quasi-admissible tower constructed only by G-linearized invertible
sheaves.
Define the integer “total rank” as the sum of ranks of all sheaves involved in all levels.
Also, define the integer “number of sheaves” as the number of sheaves in all levels. For
example, the tower P(E1 ) → P(E2 ⊕ E3 ) → Y has total rank = rank E1 + rank E2 + rank E3
and number of sheaves 3.
Assume that, for the tower P → Y , number of sheaves is less than total rank. Then,
there exists a sheaf E, which is used in the construction of some level Pi , has rank greater
than 1. Notice that E has to come from Y because the tower is quasi-admissible. Let
def
˜
Pi = P((⊕j Ej ) ⊕ E). By Theorem 6.3, there exists a map π : Y → Y , which is the

composition of a series of blow ups along invariant smooth centers with dimensions less than
94

˜
dim Y , and a G-linearized invertible sheaf L over Y such that the sequence of G-linearized
sheaves
0 → L → π ∗ E → (π ∗ E)/L → 0
is exact and (π ∗ E)/L is locally free with rank r − 1.
˜
˜
˜
˜
Define the tower P → Y by pulling back each level, namely Pi = Pi ×Y Y . Then, the
˜
sheaves in the construction at each level of P is exactly the same as P if we identify π ∗ M
˜
˜
and M. Thus, P → Y is a quasi-admissible tower with the same total rank and number of
sheaves.
˜
˜
Claim 1 : π∗ [P → Y ] − [P → Y ] lies in UG (Y ) .
First, assume π is given by a single blow up along some invariant smooth center Z ⊆ Y .
˜
Observe that P can be considered as the blow up of P along P|Z . By Lemma 6.13, we obtain
the equality
˜
[P → P] − [IP ] = −[Q1 → P|Z → P] + [Q2 → P|Z → P].
Pushing them down to Y , we get
˜
˜
π∗ [P → Y ] − [P → Y ] = −[Q1 → P|Z → Z → Y ] + [Q2 → P|Z → Z → Y ].
Notice that the tower P|Z → Z is trivially quasi-admissible and, by Remark 6.14, the
sheaves involved in the construction of Q1 , Q2 → P|Z are either of the form O(m) or
∨
NP| →P ∼ NZ →Y in our notation. That implies Q1 → Z and Q2 → Z are both quasi= ∨
Z

admissible towers. The result then follows from the fact that dim Z < dim Y . The general
˜
case with more blow ups follows easily from the fact that π∗ UG (Y ) ⊆ UG (Y ) .
Hence, without loss of generality, we may assume the splitting
0 → L → E → E/L → 0
ˆ
happens in the original tower P → Y . Next, we will construct towers P, P → Y in a similar
manner as in the proof of Lemma 6.10. Define
ˆ def
Pi = P((⊕j Ej ) ⊕ E ⊕ L) and

95

def

Pi = P((⊕j Ej ) ⊕ (E/L) ⊕ L).

Then, by Lemma 6.12, Pi and Pi are equivariantly linearly equivalent invariant smooth
ˆ
ˆ
divisors on Pi and they intersect transversely. For each level k > i, we construct Pk by the
same set of sheaves used in Pk to form a tower
ˆ def ˆ
ˆ
P = Pn → · · · → Pi → Pi−1 → · · · → Y.
def ˆ
Also, for each level k > i, we construct Pk by fiber product, namely, Pk = Pk ×P Pi to
ˆ
i
form another tower
def

P = Pn → · · · → Pi → Pi−1 → · · · → Y.
ˆ
In this case, P ∼ P as invariant smooth divisors on P and they intersect transversely. By
ˆ
ˆ
GDP R(1, 1), we have [P → P] = [P → P] and hence,
[P → Y ] = [P → Y ]
as elements in UG (Y ). Observe that, for each level k = i, the set of sheaves involved in
the construction of Pk is exactly the same as those of Pk in our notation. For level i, by
definition, Pi = P((⊕j Ej ) ⊕ (E/L) ⊕ L). Hence, P → Y is a quasi-admissible tower with the
same total rank as P → Y and one higher number of sheaves. By repeating this procedure,
we will obtain the highest number of sheaves possible : the number of sheaves is equal to
the total rank. That means all sheaves involved in the construction of the quasi-admissible
tower are G-linearized invertible sheaves.
Step 2 : Reduction to an admissible tower.
By step 1, we may assume P → Y is a quasi-admissible tower constructed by G-linearized
invertible sheaves only. For each L ∈ PicG (Y ) used in the construction, there is an invariant
divisor DL on Y and a sheaf NL ∈ PicG (Spec k) such that
∗
L ∼ OY (DL ) ⊗ πk NL
=

by Proposition 6.5. We can then represent such a (Weil) divisor as a linear combination of
prime divisors {DL,k } on Y . Let
def

{D1 , . . . , DN } = {DL,k where L is used in the construction of P → Y }.

96

Consider ∪N Dk as a reduced closed subscheme of Y . Apply the embedded desingulark=1
˜
ization Theorem in [BiMi] on ∪N Dk → Y , we obtain a map π : Y → Y , which is the
k=1
composition of a series of blow ups along invariant smooth centers such that
is smooth. Let {El } be the set of exceptional divisors. Since

∪N Dk
k=1

∪N Dk
k=1

= ∪N Dk is
k=1

˜
smooth, the strict transforms { Dk } are disjoint invariant smooth divisors on Y . Moreover,
we have
π ∗ OY (Dk ) ∼ OY ( Dk +
= ˜

ml El )
l

for some integers ml and all invariant divisors involved are smooth. Hence, π ∗ L are all
˜
˜ def
˜
˜
admissible and the tower P → Y defined by Pi = Pi ×Y Y becomes admissible. By claim
1, we reduce to the case when P → Y is an admissible tower.
∗
Step 3 : Reduction to an admissible tower with P1 = P(πk E1 ) where E1 is a G-linearized

locally free sheaf over Spec k.
By step 2, we may assume P → Y is an admissible tower. Consider the first level P1 =
P(⊕r Lj ). Since the sheaves Lj are admissible, we have Lj ∼ OY (
=
j=1

∗
k ±Djk ) ⊗ πk Nj

for

some invariant smooth divisors Djk on Y and some Nj ∈ PicG (Spec k). By lemma 6.10, we
can twist P → Y to P → Y so that P1 = P((⊕j=p Lj ) ⊕ Lp (D)) and the difference will be
given by quasi-admissible towers Q → D. Notice that
[Qi → Di → Y ]

[Q → D → Y ] =
i

def

where {Di } are the G-components of D and Qi = Q×D Di defines a quasi-admissible tower
over Di . So, [P → Y ] − [P → Y ] lie in UG (Y ) . Hence, by twisting each Lj by suitable
choices of D, we may assume there exists a sheaf L ∈ APicG (Y ) such that
∗
L j ∼ L ⊗ πk N j
=

for all j. In other words,
∗
P1 = P(L ⊗ πk E1 )
def

∗
where E1 = ⊕j Nj is a G-linearized locally free sheaf over Spec k. Notice that P(L ⊗πk E1 ) is
def

∗
∗
isomorphic to P(πk E1 ) as equivariant projective bundles over Y . If we define P1 = P(πk E1 )

97

def

and Pi = Pi ×P1 P1 for all 2 ≤ i ≤ n, then we obtain a tower P → Y which is isomorphic
to P → Y . Since all the sheaves involved in the construction of P are invertible, by Remark
6.8, P → Y is a quasi-admissible tower. By applying step 2 on P → Y , we obtain an
˜
˜
admissible tower P → Y . Then, the result follows from claim 1 and the fact that
∗
˜
˜ =
P1 = P1 ×Y Y ∼ P(πk E1 ).

Step 4 : Finish the induction step.
By step 3, it is enough to prove the statement in the case when P → Y is an admissible
∗
tower with P1 = P(πk E1 ). Consider the second level P2 = P(⊕r Lj ). Since the sheaves Lj
j=1

are admissible and
APicG (P1 ) = APicG (Y ) + ZOP1 (1),
by the same trick as in step 3, we can twist P → Y until there exists a sheaf L ∈ APicG (Y )
such that
∗
Lj ∼ L ⊗ OP1 (mj ) ⊗ πk Nj
=

for all j. By Remark 6.11, the twisting will not affect P1 . By defining
def

E2 = ⊕j (OP(E ) (mj ) ⊗ Nj )
1
∗
and p1 : P1 = P(πk E1 ) → P(E1 ), we obtain an isomorphism

P2 = P(L ⊗ p∗ E2 ) ∼ P(p∗ E2 ).
=
1
1
Simiarly, we get an isomorphic quasi-admissible tower P → Y and then, an admissible tower
˜
˜
P → Y by blow ups. Thus, we have the following commutative diagram :
∗
˜
P(E2 ) ← − P(p∗ E2 ) = P2 ← − P(q1 p∗ E2 ) = P2
−−
−−
1
1



p
q
∗
∗
˜
P(E1 ) ← 1 − P(πk E1 ) = P1 ← 1 − P(πk E1 ) = P1
−
−
−
−




Spec k ← −
−−

←−
−−

Y

˜
Y

That handles the second level. By repeating the process until level n, we obtain an admissible
tower
98

Q = Qn = P(En ) → · · · → P(E1 ) → Q0 = Spec k
such that
[P → Y ] = [Y × Q → Y ] = [Q → Spec k][Y → Y ].
Step 5 : dim Y = 0 case.
In this case, any G-linearized locally free sheaf E over Y splits into the direct sum of
G-linearized invertible sheaves (by direct calculation or Theorem 6.3). Moreover, if L is a
∗
sheaf in PicG (Y ), then, by Proposition 6.5, we have L ∼ OY (D) ⊗ πk N ∼ πk N . That means
=
= ∗
∗
P1 = P(⊕Lj ) = P(⊕πk Nj ) = Q1 × Y
def

where Q1 = P(⊕Nj ). The same argument applies to higher levels. Hence,
[P → Y ] = [Q → Spec k][Y → Y ]
with admissible tower Q → Spec k.

6.4. Generators for UG (Spec k). We are now in position to prove the generators Theorem.
First of all, we will prove that any two birational objects Y , Y ∈ G-Sm agree in some
truncated theory.

Definition 6.18. For any X ∈ G-Sm, we define the abelian group U G (X) as the quotient
of U G (X) by the subgroup generated by elements of the form [Z][Y → X] where [Z] is in
G
U≥1 (Spec k) and [Y → X] is in U G (X), i.e.
def

G
U G (X) = U G (X) / U≥1 (Spec k) U G (X).

Remark 6.19. UG can be considered as a theory on G-Sm with projective push-forward,
smooth pull-back, Chern class operator (for nice invertible sheaves) and external product.
In this truncated theory, the formal group law becomes additive, i.e.
c(L ⊗ M) = c(L) + c(M).
99

Proof. In section 7.3 in [LeP], the abelian group ω(Spec k) is defined as the subgroup of
ω(Spec k) generated by admissible towers (Without group action, the notions of “admissible
tower” in [LeP] and in our paper are equivalent). By Corollary 7.5 and equation 8.1 in [LeP],
the coefficients aij used in the formal group law in the theory ω are all inside ω≥1 (Spec k) .
Then, the result follows from the fact that the formal group law in the theory ω and the
formal group law in our theory UG share the same set of coefficients aij if we consider
ω(Spec k) → UG (Spec k).
Proposition 6.20. Suppose Y , Y ∈ G-Sm are both projective and G-irreducible. If they
are equivariantly birational, then [Y ] = [Y ] as elements in UG (Spec k).
Proof. By the equivariant weak factorization theorem (Theorem 0.3.1) in [AKMW], there
exists a sequence of blowups and blowdowns along smooth invariant centers to go from Y to
Y . So, it is enough to consider a single blowup. By Lemma 6.13,
[BlZ Y → Y ] − [IY ] = −[P1 → Z → Y ] + [P2 → Z → Y ]
as elements in UG (Y ). Pushing them down to UG (Spec k) gives
[BlZ Y ] − [Y ] = −[P1 → Z → Spec k] + [P2 → Z → Spec k]
as elements in UG (Spec k). For simplicity, assume Z is G-irreducible. By Remark 6.14, P1 ,
P2 → Z are both quasi-admissible towers. By Proposition 6.17, [Pi → Z] =

a [Z → Z] for

some a ∈ UG (Spec k) and Z ∈ G-Sm such that dim Z ≤ dim Z. Since dim Pi = dim Y >
G
dim Z, the elements {a} are all in U≥1 (Spec k) . Hence, the element [Pi → Z → Spec k]

vanishes in UG (Spec k).
Finally, we are ready to prove our main Theorem. The generators of our equivariant
algebraic cobordims ring UG (Spec k), as a L-algebra, will be admissible towers over Spec k
and some “exceptional objects”. For an integer n ≥ 0 and a pair of subgroups G ⊇ H ⊇ H ,
since G is abelian, we can write
H/H ∼ H1 × · · · × Ha
=
100

def m
where Hi is a cyclic group of order Mi = pi i for some prime pi . Let αi be a generator of

Hi . Define a (H/H )-action on Proj k[x0 , . . . , xn , v1 , . . . , va ] as the following. First, H/H
acts on x0 , . . . , xn trivially. Then, for all i, the subgroup Hi acts on vi by αi · vi = ξi vi for
some primitive Mi -th root of unity ξi . For all j = i, the subgroup Hj acts on vi trivially.

Lemma 6.21. There exist homogeneous polynomials g1 , . . . , ga ∈ k[x0 , . . . , xn ] with degrees
M1 , . . . , Ma respectively, such that the projective variety
M
M
Proj k[x0 , . . . , xn , v1 , . . . , va ] / (v1 1 − g1 , . . . , va a − ga ),

is smooth and has dimension n.

Proof. Let U be the open subscheme ∪n D(xi ) of Proj k[x0 , . . . , xn , v1 , . . . , va ]. For 1 ≤
i=0
i ≤ a, let ψi : U → PNi be the (H/H )-equivariant map sending (x0 ; . . . ; xn ; v1 ; . . . ; va ) to
M
M −1
M
M
(x0 i ; x0 i x1 ; . . . ; xn i ; vi i )

(the first Ni − 1 coordinates run though all degree Mi monomials given by x0 , . . . , xn ). By
Lemma 4.8, there exist hyperplanes Hi ⊆ PNi such that U × N1 H1 × N2 H2 × N3 · · · ×PNa
P
P
P
Ha is smooth and has dimension n. The result then follows by observing each Hi defines a
homogeneous polynomial gi with degree Mi and
M
M
U × N1 H1 × N2 H2 × N3 · · ·×PNa Ha = Proj k[x0 , . . . , xn , v1 , . . . , va ] / (v1 1 −g1 , . . . , va a −ga ).
P
P
P

Pick g1 , . . . , ga as in Lemma 6.21. Let X be the projective variety
M
M
Proj k[x0 , . . . , xn , v1 , . . . , va ] / (v1 1 − g1 , . . . , va a − ga ).

Then, X is in (H/H )-Sm. Fix a set of representatives {βj } of G/H. The exceptional object
En,H,H is defined as G/H × X such that for all α ∈ G and (βj , y) ∈ En,H,H ,
def

α · (βj , y) = (βk , γ · y)
101

where βk ∈ G/H and γ ∈ H are uniquely determined by the equality αβj = βk γ. We will
see that the element [En,H,H ] ∈ UG (Spec k) is independent of the choice of {gi }.
Theorem 6.22. If the pair (G, k) is split, then UG (Spec k) is generated by the set of exceptional objects {En,H,H | n ≥ 0 and G ⊇ H ⊇ H } and the set of admissible towers over
Spec k as a L-algebra.
def

Proof. Let S be the set of generators mentioned in the statement, i.e. S = {[En,H,H ], [P]}.
Consider the following diagram of abelian groups :
G
Un (Spec k)

↓
G
Un (Spec k)

Pn
Pn

U G (Spec k) / L[S]

Our goal is to prove S gives a set of generator of UG (Spec k) as L-algebra. It is obviously
enough to show that Pn = 0 for all n. Suppose we have shown that P0 = P1 = · · · = Pn−1 =
0. Then, since U G (Spec k) is a subgroup of L[S] and
n
G
(U≥1 (Spec k)

G
U G (Spec k)) ∩ Un (Spec k)

G
UiG (Spec k) Un−i (Spec k),

=
i=1

the homomorphism Pn is well-defined and the diagram is commutative. In addition, Pn = 0
will imply Pn = 0 and P0 , P0 agree. So, it is enough to show that Pn = 0 for all n.
G
Suppose n ≥ 0 and [Y ] ∈ Un (Spec k) is G-irreducible. Assume Y is irreducible and the Gdef m
action is faithful first. Let G = G1 × · · · × Ga where Gi is a cyclic group of order Mi = pi i

for some prime pi and αi be a generator of Gi .
Claim 1 :
M
M
k(Y ) ∼ k(x1 , . . . , xn )[xn+1 , v1 , . . . , va ] / (f, v1 1 − g1 , . . . , va a − ga )
=

for some f , gi ∈ k[x1 , . . . , xn+1 ] such that the G-action on xi is trivial, Gj acts on vi trivially
if i = j and αi · vi = ξi vi where ξ ∈ k is a primitive Mi -th root of unity.
Denote the function field k(Y ) by K. Since the G-action on Y is faithful, the degree of the
extension K/K G is equal to the order of G and it is a Galois extension (separable because
char k = 0). Let Ki be the subfield of K consists of elements fixed by
102

j=i Gj .

Then,

K = K1 · · · Ka , the intersection of any Ki = Kj is K G and the extension Ki /K G is Galois
with Gal(Ki /K G ) ∼ Gi .
=
Since the dimension of the scheme Y /G is n, the field k(Y /G) ∼ K G has transcen=
dence degree n over k. So, there is an element f ∈ k[x1 , . . . , xn+1 ] such that K G ∼
=
k(x1 , . . . , xn )[xn+1 ]/(f ). Consider Ki as a Gi -representation over K G . Since the pair
(Gi , K G ) is split, Ki can be written as direct sum of 1-dimensional Gi -representations over
K G . Then, the action of αi on at least one of the Gi -representations is given by ξi . Let bi
M
be a generator of such representation. Since bi i is fixed by Gi , it is in K G . Denote it by

gi . Without loss of generality, gi ∈ k[x1 , . . . , xn+1 ]. Consider the polynomial
Mi −1

v Mi

(v − ξ j bi ) ∈ K G [v].

− gi =
j=0

j
j
It is irreducible because if j < Mi , then α does not fix bi , hence bi ∈ K G . Since v Mi − gi
/

has degree Mi , the field Ki has to be generated by bi . In other words,
M
Ki ∼ k(x1 , . . . , xn )[xn+1 , vi ] / (f, vi i − gi ).
=

Also, the G-action on vi corresponds to the G-action on bi , which is exactly as the one
described in the statement.
Let
Y

def

M
M
= Proj k[x0 , . . . , xn+1 , v1 , . . . , va ] / (f, v1 1 − g1 , . . . , va a − ga )

and
def

P = Proj k[x0 , . . . , xn+1 , v1 , . . . , va ]
where f , gi ∈ k[x0 , . . . , xn+1 ] are homogeneous polynomials with degree d and Mi respectively, the G-action on xi is trivial and the G-action on vj are the one described in claim
1. For simplicity, we will denote P simply as Proj k[x, v]. By claim 1, Y is equivariantly
birational to Y . By applying the embedded desingularization theorem [BiMi] on Y → P ,
there is a commutative diagram
Y


−− P
−→


Y

−− P
−→
103

is smooth and equivariantly birational to Y ,

where Y , P are both in G-Sm. Since Y
by Proposition 6.20, we may assume Y

= Y . Moreover, since P → P is projective, by

Proposition 4.13, there exist free variables y0 , . . . , ym with G-actions and a set of polynomials
{h} ⊆ k[x, v, y0 , . . . , ym ] which are bihomogeneous with respect to (x, v) and y such that
P ∼ BiProj k[x, v][y]/(h)
=
and
M
M
Y = BiProj k[x, v][y] / (h) + (f, v1 1 − g1 , . . . , va a − ga ).

Define a set of indices
def

{monomial in k[x] with degree Mi }.

J = {monomial in k[x] with degree d}
i
def

Let C = k[{cj | j ∈ J}] be the polynomial ring generated by free variables indexed by J.
Then, f (x) can be considered as f (c0 , x) for some c0 ∈ Spec C and similarly for gi . Let
def
M
M
T = Proj C[x, v] / (f (c, x), v1 1 − g1 (c, x), . . . , va a − ga (c, x)).

If we assign a trivial G-action to Spec C, then there is an equivariant, projective, surjective
map φ : T → Spec C with fiber Tc0 ∼ Y and T is a closed subscheme of Spec C × P ∼
=
=
Proj C[x, v]. Also let
def
M
M
T = BiProj C[x, v][y] / (h) + (f (c, x), v1 1 − g1 (c, x), . . . , va a − ga (c, x)).

Similarly, there is an equivariant, projective, surjective map φ : T → Spec C with fiber
Tc0 ∼ Y and T is a closed subscheme of Spec C × P ∼ BiProj C[x, v][y]/(h).
=
=
Claim 2 : T is in G-Sm and has dimension dim Spec C + n.
Without loss of generality, k is algebraically closed. Notice that T is cut out from Spec C ×
P , which is smooth and has relative dimension n+a+1 over Spec C, by the equations f (c, x)
M
M
and vi i − gi (c, x). We will show that the gradients { f (c, x), (vi i − gi (c, x))} are linearly

independent and they are also linearly independent to any

104

h at any closed point in T . Since

h is in k[x, v][y], φ∗ h = 0. So, it will be enough to show that the vectors
M
{φ∗ f (c, x), φ∗ (vi i − gi (c, x))}

are linearly independent. Over D(xj ), if we denote xk /xj by tk , then we have
d−1
φ∗ f (c, x) = (td , t0 t1 , . . . , td , 0, . . . , 0)
n+1
0

and
M
M M −1
Mi
φ∗ (vi i − gi (c, x)) = −(0, . . . , 0, t0 i , t0 i t1 , . . . , tn+1 , 0, . . . , 0)

(zero except coordinates corresponding to the coefficients of gi ). Moreover, over D(vj ), if
we denote xk /vj by tk , then we obtain the same equations for the vectors φ∗ f (c, x) and
M
φ∗ (vi i − gi (c, x)). Thus, they are linearly independent as long as (x0 ; · · · ; xn+1 ) = 0.

Suppose x = (x0 ; · · · ; xn+1 ) = 0 for a certain closed point in T . Then, v1 , . . . , va are all zero
too. So, the coordinate of this point is (c, 0; 0, y) ∈ T ⊆ C × P . We then get a contradiction
by realizing that the map C × P → P → P will send (c, 0; 0, y) to (0; 0) ∈ P .
The same argument also shows that T is in G-Sm and has dimension dim Spec C + n.
Notice that T and Spec C are both smooth, the map φ is projective, surjective and has
relative dimension n and the fiber Tc0 is smooth with dimension n. So, the map φ is
smooth if restricted in an open neighborhood of c0 (because the point c0 is not in the
image of {critical point}, which is closed). Call such a neighborhood U0 . Pick a point
c1 = (c1j ) ∈ Spec C such that the fiber Tc1 is in G-Sm with dimension n (such point exists
by Lemma 6.21). Similarly, the map φ : T → Spec C is smooth if restricted in an open
neighborhood of c1 . Call such a neighborhood U1 .
Claim 3 : There exists an equivariant, projective, birational map µ : T → T of schemes
over Spec C.
The map µ is given by the restriction of the map Spec C × P → Spec C × P which sends
(c, x; v, y) to (c, x; v). So, it is clearly equivariant and projective. Notice that P → P is
birational. That means if η1 is the generic point of P , then Pη1 → Spec η1 is an isomorphism,
i.e. Proj k(x∗ , v∗ )[y]/(h∗ ) → Spec k(x∗ , v∗ ) where x∗ , v∗ and h∗ are the dehomogenizations
˜
of x, v and h with respect to x0 , respectively. Let η2 be the generic point of T , as scheme
105

over Spec C. Then,
M
Tη2 ∼ Proj C(x∗ , v∗ )[y] / (h∗ ) + (f∗ , v1 ∗ 1 − g1 ∗ , . . . , va Ma − ga ∗ )
=
∗

∼ Spec C(x∗ , v∗ ) / (f∗ , v1 M1 − g1 , . . . , va Ma − ga )
=
∗
∗
∗
∗
∼ Spec η2 .
=
That means µ is birational, as a morphism of schemes over Spec C.
Denote the open subscheme U0 ∩U1 ⊆ Spec C by U . Then, φ : T |U → U and φ : T |U → U
are both smooth and µ : T |U → T |U has birational fibers (over U ). Also, denote the affine
line in Spec C connecting c0 and c1 by L and pick a closed point c2 ∈ U ∩ L. Consider the
equivariant, projective map φ : T |L → L. It is smooth over U0 ∩ L. That means Sing(T |L ) is
disjoint from the fibers Tc0 and Tc2 . By resolution of singularities (Theorem 1.6 in [BiMi]),
we can assume T |L is smooth (The blow ups will not affect the two fibers). Now, T |L has
fibers Tc0 and Tc2 which are both smooth invariant divisors. By Proposition 4.14, we can
extend T |L → L to some equivariant, projective map T → P1 where T is in G-Sm. Then,
GDP R(1, 1) will imply
[Tc0 → T ] = [Tc2 → T ]
as elements in UG (T ). Push them down to Spec k, we got [Tc0 ] = [Tc2 ]. By applying the
same argument on φ : T |L → L, we got [Tc1 ] = [Tc2 ]. Hence,
[Y ] = [Tc0 ] = [Tc2 ] = [Tc2 ] = [Tc1 ]
as elements in UG (Spec k) by Proposition 6.20 and the fact that Tc2 , Tc2 are birational and
are both smooth.
Because of the freedom of choice of c1 = (c1j ), we can assume
M
M
Y ∼ Proj k[x, v] / (f, v1 1 − g1 , . . . , va a − ga )
=

for any choice of f (x), gi (x) as long as the degrees of f , gi are d, Mi respectively and Y is
smooth. Consider the equivariant map

106

def
M
M
ψ : W = Proj k[x, v] / (v1 1 − g1 , . . . , va a − ga ) → Proj k[x] ∼ Pn+1 .
=

Then, Y can be considered as the preimage of a generic degree d hypersurface. More precisely,
as elements in UG (W ),
[Y → W ] = c(ψ ∗ O(d))[IW ] = d c(ψ ∗ O(1))[IW ]
because ψ ∗ O(1) is nice and formal group law becomes additive by Remark 6.19. In other
words, it is enough to consider the case when d = 1. Without loss of generality, we may
assume f (x) = xn+1 . Hence, we have
M
M
Y ∼ Proj k[x0 , . . . , xn , v1 , . . . , va ] / (v1 1 − g1 (x), . . . , va a − ga (x)),
=

which is the exceptional object En,G,{1} . So, Pn [Y ] = 0. That proves the case when Y is
irreducible with faithful G-action.
If the G-action on Y ∈ G-Sm is faithful, but Y is reducible, then Y ∼ G/H × X for some
=
subgroup H ⊆ G and some irreducible X ∈ H-Sm. By applying claim 1 on X with H-action,
we can define G/H × X , G/H × X, G/H × P and G/H × P in the same manner to obtain
the following commutative diagram :
G/H × X − − G/H × P
−→


G/H × X − − G/H × P
−→
We can also define the polynomial ring C and the C-schemes G/H ×T and G/H ×T . We will
also have G-equivariant, projective maps φ : G/H ×T → Spec C and φ : G/H ×T → Spec C
such that φ is smooth around c0 and φ is smooth around some c1 = (c1j ). Similarly, the
natural map µ : G/H × T → G/H × T will also be G-equivariant, projective and has
birational fibers over Spec C. Hence, as before,
[Y ] = [G/H × X] = [(G/H × T )c0 ] = [(G/H × T )c2 ] = [(G/H × T )c2 ] = [(G/H × T )c1 ].
In other words, we may assume
M
M
X ∼ Proj k[x, v] / (f, v1 1 − g1 , . . . , va a − ga ).
=

107

Define ψ : G/H × W → Pn+1 similarly to get the same reduction on f . We may further
assume
M
M
X ∼ Proj k[x, v] / (v1 1 − g1 , . . . , va a − ga ).
=

Hence, we have Y ∼ G/H × X ∼ En,H,{1} .
=
=
In general, if we have subgroups G ⊇ H ⊇ H such that the (G/H )-action on Y is faithful
and Y ∼ G/H × X for some irreducible X ∈ (H/H’)-Sm, then we may assume
=
M
M
X ∼ Proj k[x0 , . . . , xn , v1 , . . . , va ] / (v1 1 − g1 (x), . . . , va a − ga (x))
=

for some generic g1 , . . . , ga where v1 , . . . , va are given by H/H . Hence, Y ∼ En,H,H . That
=
finishes the proof.
Remark 6.23. Notice that we did not use the full power of the generalized double point
relation in our proof of Theorem 6.22. More precisely, if we define our equivariant algebraic
cobordism theory by imposing the extended double point relation GDP R(2, 1) alone, the
same set of generators will still generate the equivariant algebraic cobordism ring. But, with
the aid of the generalized double point relation, we can actually simplify the exceptional
objects further.
Suppose the dimension of an exceptional object En,H,H is greater than the order of the
group H/H . Let
Ma−1
def
M
W = G/H × Proj k[x0 , . . . , xn , v1 , . . . , va ] / (v1 1 − g1 , . . . , va−1 − ga−1 ).
M
Then, the invariant smooth divisor G/H × {va a = ga } = En,H,H is equivariantly linearly

equivalent to the sum of invariant smooth divisors G/H × {xi = 0} where i runs from 0 to
Ma − 1. Moreover, by the freedom of choice of {gi }, we can assume
Ma −1

G/H × {xi = 0}

En,H,H +
i=0

is a reduced strict normal crossing divisor. Thus, by the generalized double point relation
GDP R(Ma , 1),
Ma −1

[En,H,H → W ] =

[G/H × {xi = 0} → W ]
i=0

108

as elements in UG (W ) (“extra terms” are always of the form [P → Z → W ] where P → Z is
a quasi-admissible tower and dim Z < dim P = n). In other words, it is enough to consider
objects of the form
Ma−1
M
G/H × Proj k[x0 , . . . , xn−1 , v1 , . . . , va ] / (v1 1 − g1 , . . . , va−1 − ga−1 )

instead. Similarly, we can apply the same argument to reduce En,H,H into
G/H × Proj k[x0 , . . . , xn−a , v1 , . . . , va ] = G/H × P(V )
for some (H/H )-representation V . In particular, if the group G is a cyclic group of prime
order, then
[G/H × P(V )] = [G/H] [P(V )]
where V is some G-representation and H can be either G or the trivial group. Notice that
E0,{1},{1} ∼ G and P(V ) is an admissible tower over Spec k. Hence, only finite number of
=
exceptional objects are needed to generate UG (Spec k) in this case.

109

7. Fixed point map
In this section, we will prove the well-definedness of the canonical fixed point map
F : UG (X) → ω(X G ),
which is an analogue of the fixed point map in topology. Recall the following definition
of fixed point locus from [Fo]. If X is a scheme over a field k, let XG be the G-scheme X
equipped with trivial G-action. If Y is a G-scheme over k, let hG (X) be the set of morphisms
Y
from XG to Y in the category of G-schemes over k. Then, hG (−) is a cotravariant functor
X
from the category of schemes over k to category of sets. By Theorem 2.3 (for schemes of
finite type over a field k), hG (−) is represented by a closed subscheme of X with trivial
X
G-action. We refer to this closed subscheme as the fixed point locus of X and denote it by
X G.
In order to show that the fixed point map is well-defined, we need to first make sure the
fixed point locus of any object in the category G-Sm stays inside the category Sm in our
basic setup (char k = 0 and G is either a reductive connected group or a finite group).

Proposition 7.1. For any object X ∈ G-Sm, the fixed point locus X G is smooth. Moreover,
if x ∈ X G is a closed point, then there is no non-zero conormal vector in N ∨ G
|x which
X →X
is fixed by the natural G-action.

Proof. By Proposition 3.4 in [Ed], the fixed point locus X G is smooth if G is finite. In
the case when G is linearly reductive, let x ∈ X G be a closed point and C(X, x) be the
tangent cone of X at x. Since X is smooth at x, the tangent cone C(X, x) is isomorphic
to Spec k(x)[t1 , . . . , td ] where d is the dimension of OX,x and t1 , . . . , td are independent
indeterminates corresponding to a system of parameters of OX,x . Moreover, G acts on
k(x) trivially and the G-action on t1 , . . . , td is linear. By Theorem 5.2 in [Fo], we have
C(X, x)G = C(X G , x). Therefore, C(X G , x) is a linear subspace of C(X, x), i.e. Ad
for
k(x)
some d . But then d = dim C(X G , x) = dim X G . Hence, the fixed point locus X G is smooth
at x. That shows the first part of the statement.
110

For the second part, when G is finite, we have T X G |x ∼ (T X|x )G by Proposition 3.2 in
=
[Ed]. Moreover, the following exact sequence of G-representations splits :
0 → T X G |x → T X|x → NX G →X |x → 0.
Hence, there is no non-zero normal vector of X G which is fixed by G, and the same holds
for conormal.
When G is reductive,
T X G |x ∼ T C(X G , x)|0 ∼ T C(X, x)G |0 ∼ (T C(X, x)|0 )G ∼ (T X|x )G .
=
=
=
=
Then the result follows similarly.

Theorem 7.2. Suppose X is an object in G-Sm and {Z} is the set of irreducible components
of its fixed point locus X G . Then, sending [Y → X] to

Z [Y

G

×X G Z → Z] defines an

abelian group homomorphism :
F : UG (X) →

ω(Z).
Z

Before going into the proof, let us illustrate how this fixed point map respects the generalized double point relation by the following example. We would like to thank Professor
P. Brosnan for inspiration.
Example : Suppose C is the ground field and G is a cyclic group of order 3. Let X(3) be
the fine moduli space for generalized elliptic curves with Γ(3)-structure and E → X(3) be its
corresponding universal family (see [DR]). By the Γ(3)-structure, there are two sections s,
s : X(3) → E such that, for each closed point µ ∈ X(3), s(µ) and s (µ) is a set of generators
of the 3-torsion Eµ [3]. As in section 1.2 in [DR], the universal family can be given explicitly
by
E = {ν(x3 + y 3 + z 3 ) = 3µxyz} ⊆ Proj C[x, y, z] × Proj C[µ, ν]
projecting down to
X(3) = Proj C[µ, ν] = P1 .
111

Notice that the fiber over ∞ :
E∞ = {0 = 3xyz} ⊆ Proj C[x, y, z]
is a N´ron 3-gon. Denote {x = 0}, {y = 0}, {z = 0} by A, B, C respectively and A ∩ B,
e
A ∩ C, B ∩ C by P , Q, R respectively. Then,
E∞ − {P, Q, R} ∼ Z/3Z × Gm
=
and, without loss of generality, the element s(∞) ∈ E∞ corresponds to an element (0, ξ3 ) ∈
Z/3Z × Gm , where ξ3 is a primitive cubic root of unity. In other words, if we define a
G-action on E by translation by s, then
(1) φ : E → X(3) ∼ P1 is a projective morphism in G-Sm (trivial G-action on
=
X(3)).
(2) The fiber E0 is an elliptic curve with free G-action.
(3) E∞ = A ∪ B ∪ C and A, B, C are all G-invariant.
(4) The G-actions on A ∼ B ∼ C ∼ P1 are non-trivial and their fixed point
=
=
=
loci are AG = {P, Q}, B G = {P, R} and C G = {Q, R}.
Now, consider the GDP R(3, 1) setup given by πC : E → Spec C with G-invariant divisors
E0 , A, B, C on E such that E0 ∼ A + B + C and E0 + A + B + C is a reduced strict normal
crossing divisor. Then, as elements in U G (Spec C), we have
[A] + [B] + [C] − [P1 ] − [P2 ] − [P3 ] = [E0 ]
where P1 = P(O ⊕ O(A)) → P , P2 = P(O ⊕ O(A + B)) → Q and P3 = P(O ⊕ O(A + B)) →
R. But since
O(A)|P ∼ NA →E |P ∼ T B|P ,
=
=
O(A + B)|Q ∼ O(A)|Q ∼ NA →E |Q ∼ T C|Q ,
=
=
=
O(A + B)|R ∼ O(B)|R ∼ NB →E |R ∼ T C|R ,
=
=
=
we have P1 ∼ P2 ∼ P3 ∼ A ∼ B ∼ C. Hence, [E0 ] = 0 in U G (Spec C). In this case, the
=
=
= =
=
fixed point map will take both sides to zero because the G-action on E0 is free.
112

Furthermore, if we consider P as an irreducible component of E G , sending [Y → E] to
[Y G |P → P ] will define a map from U G (E) to ω(P ). So, if we consider the GDP R(3, 1)
setup given by IE : E → E with the same set of divisors, we will have
[A → E] + [B → E] + [C → E] − [P1 → E] − [P2 → E] − [P3 → E] = [E0 → E],
as elements in U G (E). In this case, the fixed point map (restricted over P ) will send the
right hand side to zero and the left hand side to
[IP ] + [IP ] + 0 − [ 0 ∪ ∞ → P ] − 0 − 0,
which is also zero.

Proof of Theorem 7.2. By Proposition 7.1, Z is smooth and sending [Y → X] to [Y G ×X G
Z → Z] is well-defined at the level of MG (X)+ → M (Z)+ . If X G is the empty set, then
⊕Z ω(Z) = 0 and there is nothing to prove. So, we can assume X G is non-empty. The
strategy of this proof is very similar to that of the Proposition 3.9.
First of all, it is clearly enough to show the well-definedness of F with respect to one
fixed component Z, i.e. FZ [Y → X] = [Y G ×X G Z → Z]. Consider a generalized double
point relation setup given by φ : Y → X with A1 + · · · + An ∼ B1 + · · · + Bm on Y . Let
G : R → MG (X)+ be the corresponding map. What we need to show is
FZ ◦ G(GX ) = FZ ◦ G(GY )
n,m
m,n
as elements in ω(Z).
p

For a general term Xi · · · Uk · · · in R,
p

p

FZ ◦ G(Xi · · · Uk · · · ) = FZ [Ai ×Y · · · ×Y Pk ×Y · · · → Y → X]
p

= [(Ai ×Y · · · ×Y Pk ×Y · · · )G ×X G Z → Y G ×X G Z → Z].
If Y G ×X G Z is empty, then FZ ◦ G(GX ) = FZ ◦ G(GY ) = 0. So, we may assume
n,m
m,n
Y G ×X G Z is non-empty. Let {W } be the set of irreducible components of Y G ×X G Z
and πW : W → Z be the natural projective map. Let G : R → MG (Y )+ be the map
113

corresponding to the GDPR setup given by I : Y → Y with the same set of divisors on Y .
Then,

p

p

πW ∗ ◦ FW ◦ G (Xi · · · Uk · · · ) = πW ∗ ◦ FW [Ai ×Y · · · ×Y Pk ×Y · · · → Y ]
p

= πW ∗ [(Ai ×Y · · · ×Y Pk ×Y · · · )G ×Y G W → W ]
p

= [(Ai ×Y · · · ×Y Pk ×Y · · · )G ×Y G W → W → Z].
Hence,
FZ ◦ G =

πW ∗ ◦ F W ◦ G .
W

That means it is enough to prove
FW ◦ G (GX ) = FW ◦ G (GY )
n,m
m,n
as elements in ω(W ). In other words, we may assume φ = IX . In particular, X is equidimensional. For simplicity, we will denote FZ by F.
Within this proof, we will call a G-linearized invertible sheaf L over X “good” if L|Z
has trivial G-action. Otherwise, we will call it “bad”. We will also call an invariant divisor D on X “good” (“bad”) if the corresponding G-linearized invertible sheaf OX (D) is
“good”(“bad”).
For a set of invariant divisors A1 , . . . , An , B1 , . . . , Bm on X such that A1 + · · · + An ∼
B1 + · · · + Bm , we define a ring homomorphism F from R to End (ω(Z)) by the following
rules :


 c(O(A ))
i
Xi →

1

114

if Ai is good
if Ai is bad


 (p1 ) (p1 )∗
D ∗ D
1 →
Uk

2

def

if D = A1 + · · · + Ak is good
if D is bad

where p1 : P(O ⊕ O(D)) → Z
D

2
Uk

→


















∗

(p2 )∗ (p2 )
k
k

if D, Ak , D + Ak are all good
def

where D = A1 + · · · + Ak−1
2(p1 )∗ (p1 )
D
D

∗

if D is good but Ak , D + Ak are bad

∗


2 + (p1 ) (p1 )

Ak ∗ Ak




∗
1
 2 + (p1


D+Ak )∗ (pD+Ak )





4

if Ak is good but D, D + Ak are bad
if D + Ak is good but Ak , D are bad
if D, Ak , D + Ak are all bad

where p2 : P(O ⊕ O(1)) → P(O(−Ak ) ⊕ O(−D − Ak )) → Z
k

3
Uk →


















∗

(p3 )∗ (p3 )
k
k

if D, Ak , D + Ak are all good
def

where D = A1 + · · · + Ak−1
1 + (p1 )∗ (p1 )
D
D

∗

∗


1 + (p1 ) (p1 )

Ak ∗ Ak




∗
1
 1 + (p1


D+Ak )∗ (pD+Ak )





3

if D is good but Ak , D + Ak are bad
if Ak is good but D, D + Ak are bad
if D + Ak is good but Ak , D are bad
if D, Ak , D + Ak are all bad

where p3 : P(O ⊕ O(−Ak ) ⊕ O(−D − Ak )) → Z
k
q

if 1 ≤ i, k ≤ n. Otherwise, send it to zero. Define F (Yj ), F (Vl ) similarly by replacing
“A” by “B”. As shown in the proof of Proposition 3.9, c(O(D)) and p∗ p∗ commutes with
2
3
each other. Hence, F is well-defined. Notice that since, in the Uk , Uk cases, we have

D + Ak ∼ (A1 + · · · + Ak ), it is impossible to have only one of Ak , D, D + Ak being bad.
Thus, the definition covers all possibilities.
Claim 1 : F (GX ) = F (GY ) as elements in End (ω(Z)).
n,m
m,n

115

By a similar symbolic cancelation as in the proof of Proposition 3.9, it is enough to show
the claim in the case when A + B ∼ C. In this case,
3
2
1
GX = X1 + X2 − X1 X2 U1 + Y1 X1 X2 (U2 − U2 )
2,1

and
GY = Y1 .
1,2
We will prove the claim case by case.
Case 1 : A, B, C are all good.
In this case,
1
2
3
F (GX ) = F (X1 + X2 − X1 X2 U1 + Y1 X1 X2 (U2 − U2 ))
2,1

= c(O(A)) + c(O(B)) − c(O(A))c(O(B))p1 p1
A∗ A
∗

∗

∗

+ c(O(C))c(O(A))c(O(B))( p2 p2 − p3 p3 )
2∗ 2
2∗ 2
where p1 : P(O ⊕ O(A)) → Z
A
p2 : P(O ⊕ O(1)) → P(O(−B) ⊕ O(−C)) → Z
2
p3 : P(O ⊕ O(−B) ⊕ O(−C)) → Z.
2
On the other hand,
F (GY ) = F (Y1 )
1,2
= c(O(C)).
Thus, the difference F (GX ) − F (GY ) is exactly what we defined to be H(O(A), O(B))
2,1
1,2
in the proof of Proposition 3.9, which was proved to be zero.

116

Case 2 : A is good but B, C are bad.
F (GX ) = c(O(A)) + 1 − c(O(A))p1 p1
2,1
A∗ A

∗

∗

∗

+ c(O(A))( 2p1 p1 − 1 − p1 p1 )
A∗ A
A∗ A
= 1
= F (GY ).
1,2
Case 3 : B is good but A, C are bad.
F (GX ) = 1 + c(O(B)) − c(O(B))(2)
2,1
∗

∗

+ c(O(B))( 2 + p1 ∗ p1 − 1 − p1 ∗ p1 )
B B
B B
= 1
= F (GY ).
1,2
Case 4 : C is good but A, B are bad.
∗

∗

F (GX ) = 1 + 1 − (1)(2) + c(O(C))( 2 + p1 ∗ p1 − 1 − p1 ∗ p1 )
2,1
C C
C C
= c(O(C))
= F (GY ).
1,2
Case 5 : A, B, C are all bad.
F (GX ) = 1 + 1 − (1)(2) + (1)(4 − 3)
2,1
= 1
= F (GY ).
1,2
That proves the claim.
The next step is to verify the correspondence between F and F . To be more precise, let
G : R → MG (X)+ be the map corresponding to a GDPR setup given by A1 + · · · + An ∼
B1 + · · · + Bm on X such that A1 + · · · + An + B1 + · · · + Bm is a reduced strict normal
crossing divisor and let F : R → End (ω(Z)) be the map we just defined corresponding to
117

this setup. Consider the fixed point map F as a map from MG (X)+ to ω(Z). The equation
we are going to prove is
F ◦ G(s) = F (s)[IZ ]

(11)

p

q

for any element s ∈ Z{Xi · · · Yj · · · Uk · · · Vl · · · | power of any Xi , Yj ≤ 1}.
Suppose equation (11) is true. Then,
F ◦ G(GX ) = F (GX )[IZ ]
n,m
n,m
= F (GY )[IZ ]
m,n
(by claim 1)
= F ◦ G(GY ),
m,n
which is what we want. That means it is enough to verify equation (11). First of all, we
need to understand the meaning of an invariant divisor being “good”.
Claim 2 : Suppose D is a smooth invariant divisor on X. Then, D is good if and only if
D ∩ Z is a smooth divisor on Z. Also, D is bad if and only if D ∩ Z = Z.
First of all, observe that
D ∩ Z = D ×X Z = D ×X X G ×X G Z = DG ×X G Z,
which is always smooth. If D ∩ Z = ∅, then OZ (D) ∼ OZ . That means it is good and D ∩ Z
=
is the zero divisor.
Suppose D ∩ Z is non-empty. Take a closed point x ∈ D ∩ Z. Notice that since the action
on Z is trivial and Z is irreducible, the action OZ (D) is trivial if and only if the action
on OZ (D)|x is trivial. Moreover, OZ (D)|x ∼ OD (D)|x ∼ ND →X |x . Hence, the action on
=
=
ND →X |x is trivial if and only if D is good.
Suppose the action on ND →X |x is trivial and D ∩ Z is not a divisor on Z. That means
∨
∨
D ∩ Z = Z, i.e. Z ⊆ D. Thus, we have a natural injective map ND →X |x → NZ →X |x . It
∨
contradicts with the fact that there is no non-zero vector in NZ →X |x fixed by G (Proposition

7.1).

118

Suppose D ∩ Z is a divisor on Z. Then D and Z intersect transversely. That means
T X|x = T D|x + T Z|x and T D|x ∩ T Z|x = T (D ∩ Z)|x . Therefore, we have ND →X |x →
T Z|x / T (D ∩ Z)|x and hence, the G-action on ND →X |x is trivial.
Suppose the smooth invariant divisor Ai is good. Then, we have
F ◦ G(Xi ) = F[Ai → X]
= [AG ×X G Z → Z]
i
= [Ai ∩ Z → Z]
= c(O(Ai ))[IZ ]
(by claim 2 and (Sect) axiom in the theory ω)
= F (Xi )[IZ ].
On the other hand, if Ai is bad, then we have Ai ∩ Z = Z by claim 2. In this case,
F ◦ G(Xi ) = [AG ×X G Z → Z] = [Z → Z] = F (Xi )[IZ ].
i
Hence, equation (11) holds for Xi and Yj .
def

1
For Uk , if D = A1 + · · · + Ak is good, then P(OZ ⊕ OZ (D)) has trivial action. Thus,
1
F ◦ G(Uk ) = [P(O ⊕ O(D))G ×X G Z → Z]

= [P(OZ ⊕ OZ (D)) → Z]
∗

= p1 p1 [IZ ]
D∗ D
where p1 : P(O ⊕ O(D)) → Z
D
1
= F (Uk )[IZ ].

If D is bad, then P(OZ ⊕ OZ (D)) has non-trivial fiberwise action. That implies
P(OX ⊕ OX (D))G |Z = P(OZ ⊕ OZ (D))G = P(OZ (D))

119

P(OZ ).

Thus,
1
F ◦ G(Uk ) = [P(O ⊕ O(D))G ×X G Z → Z]

= [P(OZ (D))

P(OZ ) → Z]

1
= 2[IZ ] = F (Uk )[IZ ].
1
Hence, equation (11) holds for Uk and Vl1 .
def

2
For Uk , let D = A1 + · · · + Ak−1 as in the definition of F . There are five different cases

to consider.
Case 1 (Divisors D, Ak , D + Ak are all good) :
The action on the projective bundle P(OZ (−Ak ) ⊕ OZ (−D − Ak )) will be trivial and so
is the projective bundle P(O ⊕ O(1)) above it. Thus,
2
F ◦ G(Uk ) = [P(O ⊕ O(1))G ×X G Z → Z]

= [P(O ⊕ O(1)) → P(OZ (−Ak ) ⊕ OZ (−D − Ak )) → Z]
∗

= p2 p2 [IZ ]
k∗ k
(p2 as in the definition of F )
k
2
= F (Uk )[IZ ].

Case 2 (Divisor D is good but Ak , D + Ak are bad) :
In this case,
P(OZ (−Ak ) ⊕ OZ (−D − Ak )) ∼ P(OZ (D) ⊕ OZ ),
=
which has trivial action. Moreover, this isomorphism takes O(1) to O(1) ⊗ OZ (−D − Ak ).
Hence, the tower
P(O ⊕ O(1)) → P(OZ (−Ak ) ⊕ OZ (−D − Ak )) → Z
is isomorphic to
P(O ⊕ (O(1) ⊗ OZ (−D − Ak ))) → P(OZ (D) ⊕ OZ ) → Z.

120

Hence,
2
F ◦ G(Uk ) = 2 [P(O(D) ⊕ O) → Z]
∗

= 2 p1 p1 [IZ ]
D∗ D
2
= F (Uk )[IZ ].

Case 3 (Divisor Ak is good but D, D + Ak are bad) :
Since D is bad, P(OZ (−Ak ) ⊕ OZ (−D − Ak )) ∼ P(OZ (D) ⊕ OZ ) has fixed point locus
=
P(OZ (−Ak ))

P(OZ (−D − Ak )). Moreover, the tower
P(O ⊕ O(1)) → P(OZ (−Ak )) → Z

is isomorphic to
P(O ⊕ (O(1) ⊗ OZ (−Ak ))) → P(OZ ) → Z,
which is simply P(O ⊕ OZ (−Ak )) → Z and also, the tower
P(O ⊕ O(1)) → P(OZ (−D − Ak )) → Z
is isomorphic to
P(O ⊕ (O(1) ⊗ OZ (−D − Ak ))) → P(OZ ) → Z.
Hence,
2
F ◦ G(Uk ) = [P(O(Ak ) ⊕ O) → Z] + 2[IZ ]
∗

= (p1 p1 + 2)[IZ ]
Ak ∗ Ak
2
= F (Uk )[IZ ].

Case 4 (Divisor D + Ak is good but D, Ak are bad) :
Similarly, the fixed point locus of P(OZ (−Ak ) ⊕ OZ (−D − Ak )) is the disjoint union of
P(OZ (−Ak )) and P(OZ (−D − Ak )), and the corresponding towers are the same as in case

121

3. Hence,
2
F ◦ G(Uk ) = 2[IZ ] + [P(O(D + Ak ) ⊕ O) → Z]

= (2 + p1
D+A

k∗

p1
D+A

∗
k

)[IZ ]

2
= F (Uk )[IZ ].

Case 5 (Divisors D, Ak , D + Ak are all bad) :
The fixed point locus of P(OZ (−Ak ) ⊕ OZ (−D − Ak )) is again the disjoint union of
P(OZ (−Ak )) and P(OZ (−D − Ak )), and the corresponding towers are the same. Hence,
2
F ◦ G(Uk ) = 2[IZ ] + 2[IZ ]
2
= F (Uk )[IZ ].
2
That proves equation (11) holds for Uk and similarly for Vl2 .
def

3
For Uk , similarly, let D = A1 + · · · + Ak−1 . In case 1,
∗

3
3
F ◦ G(Uk ) = [P(O ⊕ O(−Ak ) ⊕ O(−D − Ak )) → Z] = p3 p3 [IZ ] = F (Uk )[IZ ].
k∗ k

In case 2,
3
F ◦ G(Uk ) = [P(O)

P(O(−Ak ) ⊕ O(−D − Ak )) → Z]

= [IZ ] + [P(O(D) ⊕ O) → Z]
∗

= (1 + p1 p1 )[IZ ]
D∗ D
3
= F (Uk )[IZ ].

In case 3,
3
F ◦ G(Uk ) = [IZ ] + [P(O(Ak ) ⊕ O) → Z] = (1 + p1
A

k∗

p1
A

∗
k

3
)[IZ ] = F (Uk )[IZ ].

In case 4,
3
F ◦ G(Uk ) = [IZ ] + [P(O(D + Ak ) ⊕ O) → Z] = (1 + p1
D+A

122

k∗

p1
D+A

∗
k

3
)[IZ ] = F (Uk )[IZ ].

In case 5,
3
F ◦ G(Uk ) = [P(O)

P(O(−Ak ))

3
P(O(−D − Ak )) → Z] = 3[IZ ] = F (Uk )[IZ ].

3
That proves equation (11) holds for Uk and similarly for Vl3 .

Let s, t be two terms in
def

p

q

R = Z{Xi · · · Yj · · · Uk · · · Vl · · · | power of any Xi , Yj ≤ 1}.
By definition, the domain of G(st) = the domain of G(s) ×X the domain of G(t). For simplicity, we will focus on domains. By abuse of notation, we will still call it G. Observe
that

F[Y1 ×X Y2 → X] = [(Y1 ×X Y2 )G ×X G Z → Z]
= [Y1G ×X G Y2G ×X G Z → Z]
= [(Y1G ×X G Z) ×Z (Y2G ×X G Z) → Z].
def

Hence, F(Y1 ×X Y2 ) = F(Y1 ) ×Z F(Y2 ), by abuse of notation again. Suppose s = Xi , Yj ,
p

q

Uk or Vl and t is a term in R such that st is also in R. By induction, we assume equation
(11) holds for s and t. In that case,
F ◦ G(st) = F[G(st) → X]
= F[G(s) ×X G(t) → X]
= [F(G(s) ×X G(t)) → Z]
= [F ◦ G(s) ×Z F ◦ G(t) → Z].
On the other hand,
F (st)[IZ ] = F (s) ◦ F (t)[IZ ] = F (s)[F ◦ G(t) → Z]

123

by induction assumption. Denote F ◦ G(t) by Y and Y → Z by f . By the above calculation,
[F ◦ G(s) → Z] = m1 [IZ ] + m2 [P → Z] + m3 [D ∩ Z → Z]
for some non-negative integers m1 , m2 , m3 , tower P and good, smooth, invariant divisor D
on X.
Claim 3 : The map F ◦ G(s) → Z is transverse to f : Y → Z.
The claim is clearly true for [IZ ] and [P → Z]. So, we only need to consider the map
[D ∩ Z → Z] where D is a good, smooth, invariant divisor on X. Recall that
p

p

Y = F ◦ G(Xi · · · Uk · · · ) = F(Ai ) ×Z · · · ×Z F(Pk ) ×Z · · · .
p

Since F(Pk ) is the sum of towers and F(Ai ) = Z when Ai is bad, we may assume it only
involves good divisors, i.e.
F(Ai1 ) ×Z · · · ×Z F(Bj1 ) ×Z · · · = Ai1 ∩ · · · ∩ Bj1 ∩ · · · ∩ Z.
Notice that since st is in R, the divisor D and the set of divisors {Ai1 , · · · , Bj1 , · · · } are
all distinct. For simplicity, we will only show the transversality involving good divisors D,
D . More precisely, we will show if D, D are good, smooth, invariant divisors on X such
that D + D is a reduced strict normal crossing divisor, then D ∩ Z + D ∩ Z is a reduced
strict normal crossing divisor on Z.
Since X is equidimensional, D is equidimensional. Let W be an irreducible component of
Z ∩ D. Then, D ∈ G-Sm is equidimensional, D ∩ D is an invariant smooth divisor on D
and W is an irreducible component of the fixed point locus of D. Notice that
OW (D ∩ D ) ∼ OX (D )|W ∼ OZ (D )|W .
=
=
Thus, D ∩ D is a good divisor on D with respect to W , for all W , because D is a good
divisor on X with respect to Z. By applying claim 2 with X, D, Z replaced by D, D ∩ D ,
W respectively, D ∩ D ∩ W is a smooth divisor on W . So, D ∩ D ∩ Z is a smooth divisor
on D ∩ Z. Hence, D ∩ Z and D ∩ Z intersect transversely inside Z.

124

Let {Yi } be the irreducible components of Y . Notice that f is projective. So, the pushforward f∗ : ω(Y ) → ω(Z) is well-defined. Since Y , Z are both smooth and quasi-projective,
the map f is a local complete intersection morphism (See section 5.1.1 in [LeMo]). In
addition, the algebraic cobordism theories ω and Ω are canonically isomorphic (Theorem 1
in [LeP]) and, for any local complete intersection morphism g : X → X with equidimensional
domain and codomain, the pull-back g ∗ : Ω(X ) → Ω(X) is well-defined (see definition 6.5.10
in [LeMo]). Hence, f ∗ : ω(Z) → ⊕i ω(Yi ) ∼ ω(Y ) is also well-defined.
=
Suppose we have shown that
(12)

F (s)[f : Y → Z] = f∗ f ∗ F (s)[IZ ].

Then, we have
F (st)[IZ ] = F (s)[f : Y → Z]
= f∗ f ∗ F (s)[IZ ]
= f∗ f ∗ [F ◦ G(s) → Z]
(by induction assumption)
= [(F ◦ G(s)) ×Z (F ◦ G(t)) → Z]
(by claim 3 and Theorem 6.5.12 in [LeMo])
= F ◦ G(st).
That means equation (11) holds for st ∈ R. Hence, it remains to show equation (12).
By the previous calculation,
F (s) = m1 + m2 p∗ p∗ + m3 c(OZ (D))
for some non-negative integers m1 , m2 , m3 , smooth, projective map p : P → Z and good,
smooth, invariant divisor D on X. The equation obviously holds for the identity operator.

125

For c(OZ (D)),
c(OZ (D))[Y → Z] = [(D ∩ Z) ×Z Y → Z]
(by claim 3 and (Sect) axiom in ω)
= f∗ f ∗ [D ∩ Z → Z]
(by claim 3 and the Theorem 6.5.12 in [LeMo])
= f∗ f ∗ c(OZ (D)) [IZ ].
For p∗ p∗ ,
p∗ p∗ [Y → Z] = [P ×Z Y → Z]
= f∗ f ∗ [P → Z]
(by Theorem 6.5.12 in [LeMo])
= f∗ f ∗ p∗ p∗ [IZ ].
That proves equation (12) and hence finishes the proof of the Theorem.

Corollary 7.3. If X is an object in G-Sm, then sending [Y → X] to [Y G → X G ] defines
an abelian group homomorphism
F : UG (X) → ω(X G ).
Proof. Let {Z} be the set of irreducible components of the fixed point locus X G . By Theorem 7.2, sending [Y → X] to

Z

[Y G ×X G Z → Z] defines an abelian group homomorphism

UG (X) → ⊕Z ω(Z). Then, the map F : UG (X) → ω(X G ) can be considered as the composition
UG (X) → ⊕Z ω(Z) → ⊕Z ω(X G ) → ω(X G )
defined by sending
126

[Y G ×X G Z → Z]

[Y → X] →
Z

[Y G ×X G Z → Z → X G ]

→
Z

[Y G ×X G Z → Z → X G ] = [Y G → X G ].

→
Z

Corollary 7.4. Suppose X is an object in G-Sm with trivial G-action. Then, the abelian
group ω(X) ∼ U{1} (X) is a direct summand of UG (X) via the homomorphism
=
Φγ : U{1} (X) → UG (X)
induced by the group homomorphism γ : G → {1}. In particular, the Lazard ring L is
naturally a subring of the equivariant algebraic cobordism ring UG (Spec k).
Proof. The fixed point map
F : UG (X) → ω(X G ) = ω(X) ∼ U{1} (X)
=
is a left inverse of the homomorphism U{1} (X) → UG (X). Also,
Φγ : L ∼ U{1} (Spec k) → UG (Spec k)
=
is a ring homomorphism.

127

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Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
E-mail address : liuchun@math.msu.edu

131