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THE FINE MODULI SPACE OF REPRESENTATIONS OF
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EMRE COSKUN

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5/08 KIIProleoc8-PresICIRCIDataDuehdd

THE FINE MODULI SPACE OF REPRESENTATIONS OF CLIFFORD
ALGEBRAS

By

Emre Coskun

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Mathematics

2009

ABSTRACT

THE FINE MODULI SPACE OF REPRESENTATIONS OF CLIFFORD
ALGEBRAS

By

Emre Coskun

Given a fixed binary form f (u, v) of degree d over a field k, the associated Clifford
algebra is the k-algebra Cf = k{u,v} / I , where I is the two-sided ideal generated
by elements of the form (au + fl'v)d — f (a, ,3) with a and fl arbitrary elements in
It. All representations of Cf have dimensions that are multiples of d, and occur
in families. In this article we construct fine moduli spaces U = U f9, for the rd-
dimensional representations of C f for each 1‘ Z 2. Our construction starts with the

projective curve C C P% defined by the equation ti)“

= f(u, v), and produces Uf.r as
a quasiprojective variety in the moduli space M(T, (tr) of stable vector bundles over

C with rank 7‘ and degree dr = 7‘(d + g — 1), where 9 denotes the genus of C.

 

ACKNOWLEDGMENT

This project would not have come to completion were it not for the help of several
people. First and foremost, I would like to thank my advisor, Rajesh Kulkarni, for
his help in this project.
I would like to thank my family for the support they have given me over the years.
Finally, two more people deserve special mention. They gave me invaluable sup-
port through some difficult moments. And I don’t think I could have finished this

dissertation without their help. Phillip and Whitney, thanks for everything.

iii

TABLE OF CONTENTS

1 Introduction ........................... 1
2 Background and Preliminaries .................. 6
2.1 Conventions ................................ 6 '
2.1.1 Notations ........................ - ..... 6
2.2 Vector Bundles over Curves ....................... 8
2.3 Geometric Invariant Theory ....................... 13
2.4 The Moduli Space M(r. D) ....................... 15
2.5 The Clifford Algebra and its Representations .............. 19
2.6 Grothendieck Topologies ......................... 21
2.7 Azumaya Algebras ............... ‘ ............. 23
3 Construction of the Universal Representation .......... 27
4 The Moduli Problem ...................... 41
Bibliography .......................... 54

iv

Chapter 1

Introduction

Let f be a binary form of degree d over a field k. The Clifford algebra C f associated
to f is the quotient of the tensor algebra on two variables by the two-sided ideal
generated by {(0121 + [310d — f (a, ,3) | (1,6 6 k}. We will be mostly interested in the
case of degree d > 3. We also assume that f (a, v) is nondegenerate, that is, f has no

repeated roots over the algebraic closure of k.

The structure and representations of Clifford algebras has been a subject of study
in many recent papers. The degree 2 case is classical, for an overview of the subject,
see [14] and [20]. The degree 3 case was examined by Haile in [9]. Assuming that
the characteristic of the base field is not 2 or 3, and that f (u, v) is nondegenerate,
he proved that Cf is an Azumaya algebra (see Section 2.7) over its center. He also
proved that the center is isomorphic to the coordinate ring of an affine elliptic curve
J. The curve J is the Jacobian of 1123 = f (u, v) and the affine elliptic curve is the
complement of the identity point in J.

Next we describe what is known for d > 3. Let C be the curve over k defined
by the equation wd = f (u, v) in P2, and let 9 denote its genus. Since f is assumed
to be nondegenerate, the curve C is nonsingular, and g = (d — 1)(d — 2) /2. Haile

proved in [9] that the dimensions of representations of C f are divisible by d. Van

1

den Bergh proved in [24] (assuming that the base field It is algebraically closed of
characteristic 0) that the equivalence classes of rd-dimensional representations of the
Clifford algebra Cf are in one-to—one correspondence with vector bundles E over
C having rank 7‘, degree r(d + g — 1) such that H0(C, E(—1)) = 0. These vector
bundles are always semistable and the stable bundles correspond to the irreducible
representations. Then, assuming that (1 does not divide the characteristic of k, Haile
and Tesser proved in [10] that C f = Cf/ n 77, where 17 runs over the kernels of the
dimension d representations, is Azumaya over its center. The center of C f is then
the fine moduli space of d—dimensional representations (hence r = 1) of C f. Kulkarni
then proved in [13] that this center is the affine coordinate ring of the complement of

a O-divisor in the space P’lCEhjf1 of degree d + g - 1 line bundles over C.

This article generalizes Kulkarni’s work to the higher rank case, 7‘ 2 2; under the
assumptions that k is algebraically closed and the characteristic of It does not divide
d. We begin with Van den Bergh’s correspondence between equivalence classes of
rd—dimensional representations of Cf and semistable vector bundles over C of rank
r and degree r(d + g — 1), which can be described as follows. Give a representation
(b : Cf —* Mrdlkl- Set au = q5(u) and av = ¢(v). Then we define a map from
S = k[u,v,w]/(wd — f(u, 11)) to Mrd(k[u,v]) by sending it to ulrd, v to vlrd, and
w to uau + vav. This makes $711 k[u, 12] into a graded S—module. It is easy to see
that the corresponding coherent sheaf is a vector bundle. In this way we get a rank
r vector bundle 8 over C such that qsg 9’ 01;? where q : C -> P}: is the map defined
by the inclusion k[u, v] —» S. The condition H 0(C, E(—1)) = 0 is equivalent to the

condition that q...£ 3-“ 01;?
The moduli problem we solve in this dissertation can be stated as follows: Define

a contravariant functor Reprd( C f, —) from the category of k-schemes to the category
as sets by sending a k-scheme S to the set of equivalence classes of rd-dimensional

irreducible S-representations of C f. (For the definition of an S-representation of C f,

2

see Definition 2.7.4.) Procesi proved that this functor is representable by a scheme
U (see Theorem 1.8 in Chapter 4 of [21].) Unfortunately, Procesi’s method gives no
geometric description of U. In this dissertation, we prove that U is isomorphic to an
open subset of the (coarse) moduli space M(r, r(d+ g- 1)) of stable vector bundles of
rank r and degree r(d + g - 1) over C. Using this geometric description, we construct
the universal representation A of C f over U.

More explicitly, the universal representation of If of a given dimension rd is a
k—algebra homomorphism 11) : C f —+ H 0(A), where A is a sheaf of Azumaya algebras
of rank (rd)2 defined over U. The base variety U is the open subset of M(r, r(d +
g — 1)) consisting of stable vector bundles B such that H 0(C, E( —1)) = 0. The sheaf
A is constructed as follows. There is a Quot scheme Q (see Theorem 2.4.2), that
parametrizes the quotients of the trivial vector bundle of large enough rank N over
C, having rank r and degree r(d + g — 1), and there is a universal bundle 8 over
C x Q. We take the open subset Q of Q consisting of stable vector bundles E with
H 0(C, E(—1)) = 0. We prove in Lemma 3.0.11 that the pushforward of 8 to Q under
the projection map 7r : C x Q —+ Q is a rank rd vector bundle. The algebraic group
CL( N) acts on Q and also on mg. The stabilizer of a point in Q under this action
is the group of scalar matrices, so the action of GL(N) on Q descends to an action
of PCL(N). But the scalar matrices act as scalar multiplication on mg. So we get
a PGL(N)-action on 8nd(7r...8). The resulting Geometric Invariant Theory quotient
is the variety U together with a sheaf of algebras A on it. We then construct the
homomorphism 1,11 : C f —) H 0(A).

The main theorem of this dissertation is as follows:

Theorem 1.0.1. (Main Theorem) Let k be an algebraically closed base field.

Reprd(Cf, —) is represented by the pair (t,b,A) described above.

For the proof, we construct an isomorphism between (t/J,A) and the universal

3

representation in Procesi’s theorem. As mentioned above, Procesi showed that the
functor Reprd(Cf, —) is representable, that is, there is a universal representation
(‘11,3) consisting of a scheme T, a sheaf of Azumaya algebras 8 over T, and a k-
algebra homomorphism \Il : C f —i H 0(13). Since we have an irreducible representation
(1b,A) over U, we obtain a map a : U ——+ T such that oz*(\II,B) ’=" (¢,A). The idea
of the proof is to construct an inverse to a. To do this, we first consider irreducible
representations of the type ((3‘), £n.d(£5)), where 83 is a vector bundle of rank rd over
S. We construct an associated vector bundle in Lemma 4.0.20. By the coarse moduli
property of M(r,r(d + g — 1)), this gives us a map f : S -—) U. We use this to
construct a morphism )3 : T -—> U. We then prove that a and [3 are inverses, and that
(2/2, A) represents Reprd(Cf, —). This finishes the proof of the theorem.

We now outline an alternative way to prove that a is an isomorphism. The idea

is to first prove that the morphism 0 satisfies the following properties:

0 (r is a bijection on closed points,
0 the morphism induced by a on the tangent spaces of closed points is injective,

o a is a proper morphism.

Then it follows from Zariski’s Main Theorem that a must be an isomorphism. For
the case r = 1, this approach was used by Kulkarni in his Ph.D. dissertation. ([12])

Further questions can be asked about this universal representation. First, it can
be proved that this construction works for any base field k subject to the restriction
that the characteristic of k not divide d. We use Galois descent theory to prove this,
and the results will be included in a forthcoming article.

Secondly, A gives a class in the Brauer group Br(U). Since the Brauer group is
torsion, it is natural to ask what the period and index of this class (as defined in
Section 2.7) are. It is known that the period always divides the index, and the set

of primes dividing both of them is the same. Hence the index divides a power of the

4

period. The period-index problem is the problem of computing this power. This is
also part of an ongoing project.

Thirdly, it is an interesting question to examine the representations of Clifford
algebras of ternary forms. By the results of Van den Bergh in [24], these correspond
to vector bundles over a surface X in P3 defined by the equation zd = f (u,v,w),
whose direct images under the natural projection map X -—) P2 is a trivial vector

bundle. These will be studied in future articles.

Chapter 2

Background and Preliminaries

2. 1 Conventions

Let k be an algebraically closed field with characteristic 0 or not dividing d. In this
dissertation, a variety means a separated scheme of finite type over k, and points
mean closed points. A variety of dimension 1 is called a curve. X will always denote

a smooth projective curve of genus g.

2.1 .1 Notations

o All rings have an identity element.

All schemes are locally Noetherian over k and all morphisms are locally of finite

type.

The terms line bundle and invertible sheaf are used interchangeably.

M(r, d) denotes the coarse moduli space of stable vector bundles of rank r and

degree d over a curve C.

o For any vector bundle E, x(£) denotes the Euler characteristic of 5.

6

h component X,- is denoted pi,

o The projection of a fiber product onto the it
713-, p X,- or ”X2" When no subscript is indicated, 7r is the canonical map from

C x Y ——> Y for a variety Y.

o For technical reasons, we assume throughout the article that the binary form
f (1:, y) has no repeated factors over an algebraic closure of the base field It and

that the characteristic of A: does not divide d.

o q denotes the canonical map from C to P1, where C is the curve wd = f (u, v)
in P2. qS denotes the map q x ids : C x S -—> P1 x S for any scheme S, and

p5 : P1 x S ——> S is the natural projection.

o For any coherent sheaf .7: over a scheme X, we denote the ith cohomolo ’ of .7:
g}

by H 2)}- ) whenever X is clear from the context.

0 For a closed point y in a scheme S, and for a vector bundle 8 over C x S, 83)
denotes the pull-back of 8 under the canonical map id x 2'3) : C x Spec k(y) ——>
C x S.

The following lemma will be useful in proving the main theorem.

Lemma 2.1.1. Let Y be a quasi-projective variety. Let f : Y ——+ Y be a map that is

the identity on closed points. Then f is the identity map.

Proof. First we want to prove that f is the identity on all points. Let g be a non-
closed point. Assume that f (E) = 77 sé 5. Then E yé 7), where E and fi denote the
closures of g and 7), respectively. There are two cases to analyze:

Case 1: E Q fi. Pick a closed point y1 E E \ '77. Then f _1('17) is a closed set
that contains 5, and hence it contains E and in particular y, a contradiction. Case
2: E C h. Since this is a proper inclusion, the dimension of 17 is strictly greater than
the dimension of 5. But since 13(7)) is a subfield of 19(6), this is a contradiction for

dimension reasons.

Now since any variety can be covered by open affine subsets, we are reduced to the
case where Y is the spectrum of an (reduced) affine ring A(Y) = k[T1, . . . ,Tn]/I(Y).
Let i : Y —-> A" be the inclusion map. Assume that f is induced by the morphism of
rings cf) : .4(Y) -—) A(Y). From elementary algebraic geometry, the image 45(7)) is the
regular function on Y that is the ith coordinate function on Y. Since Y is reduced,

this is the same as Ti. This proves that (b, and hence f, is the identity map. E]

2.2 Vector Bundles over Curves

A vector bundle E of rank n over a variety X is a locally free sheaf of rank n on
X. This is equivalent to the classical definition of a vector bundle; the equivalence
is established by taking a vector bundle 7r : 8 —> X to its sheaf of sections. (See
Exercise 5.18, Chapter 2, [11].) A line bundle is a vector bundle of rank 1, also called
an invertible sheaf. For simplicity, we will assume that X is a nonsingular projective
curve from now on. This implies that every torsion—free coherent sheaf on X is locally
free. (See Lemma 5.2.1, [15].)

There are two invariants associated to a vector bundle 8 over X:
o The rank rk(£) = dinik(x)€ <8) [C(17) for any a: E X;

o The degree deg(£), which is the degree of the determinant line bundle del(8) of
8. Here, det(8) 2 Ar 8, where r is the rank of 5.

Consider a connected smooth projective variety V of dimension n. Then we have
the sheaf of differentials fly on V. This is a locally free sheaf on V. Set wv = Ar fly.
This is called the canonical sheaf of V. Serre duality tells us that for any vector bundle

8 over X, we have an isomorphism:

How) 2 H1(wv a 8*).

8

where 8* denotes the dual of E.

A very useful theorem for vector bundles is the following:

Theorem 2.2.1. (Riemann-Roch) Suppose X has genus g. Let 5 be a vector
bundle over X. We have:

us) = We — m) = (1 - awe) + deg<£>

Yet another important invariant of vector bundles is the Hilbert polynomial. Let
8 be a vector bundle over X. Let OX(H) denote an ample line bundle on X. We

denote E (n) = 8 ® (9X(nH). Then the Hilbert polynomial of 6 is given by:

By Riemann-Roch, we have:

FEM) = X(E(n)) = (1 - 9)Tk(E(n)) + d69(E(n))

= (1— g)rk(E) + deg(E) + n deg(H)rk(E).

Thus the Hilbert polynomial of E depends only on the rank and degree of E.

For a vector bundle E of rank r and degree d, we define the slope of E to be the
number ,u.(E) = d/ r. We say that E is semi-stable if for every non—zero subbundle F
of E, we have ,u(F) S p( E) If strict inequality holds, we say that E is stable.

The concepts of stability and semi-stability can also be defined in terms of quo-
tients. The bundle E is semi-stable if and only if, for every quotient bundle E” =
E / E’ of non-zero rank, we have ja(E”) 2 ME). E is stable if and only if, the same
inequality holds strictly.

The following proposition will be used later:

9

Proposition 2.2.2. (Proposition 5. 3. 3, [15/) Let E and F be two semi-stable bundles
on the curve X. If Hom(E, F) 75 0 then we have [1(E) g ,u(F). Furthermore, if E
and F are stable and of the same slope then every non-zero map f : E —+ F is an

isomorphism.

Corollary 2.2.3. Let E be a stable bundle. The only endomorphisms of E are scalar

multiplications.

Proof. Let f be an endomorphism of E. Consider the commutative algebra k[ f]
generated by f. This is a field by the above proposition. Then it must be a finite
algebraic extension of k and so it must be equal to k. This implies that f is a scalar

multiple of the identity endomorphism. El

We now prove a lemma that will be crucial in the construction of the moduli

space:

Lemma 2.2.4. (Remark, pg. 14, [7]) Let E be a semistable vector bundle over C of

rank r and degree d, and suppose that d > r(2g - 1). Then
1. 111(52): 0
2. E is generated by its global sections.

Proof. We make use of the short exact sequence 0 —) E(—:z:) ——) E —> Ex —> 0 and

the associated long exact sequence in cohomology:
0 —> H0(E(—x)) —» HO(E) —» H0(E$) —>

(PC is a line bundle of degree 2g -— 2. Hence it is stable with slope u = 2g — 2. Let
V be any semistable vector bundle over C with ,u(V) > 2g - 2. Then Hom(V, 9%)) =
0. By Serre duality, H1(V) = 0. Let E be as in the statement and let x E C.
Then u(E(—:r)) = u(E) — 1 > 29 — 2, and hence H1(E(—:z:)) = 0. Therefore we

10

have H0(E) —» H0(E/E(—;r.)) = ET. This proves that E is generated by global

sections. C]

There is a similar theorem, which does not require semi-stability. We mention it

here for reference:

Theorem 2.2.5. (Theorem 2.3.2, [15]) Let F be a coherent sheaf on a projective
variety X. For all n sufiiciently large

(A) the sheaf F(n) is generated by its global sections,

(B) we have Hq(X, F(n)) = 0 forq > 0.

Theorem 2.2.6. (Grothendieck) (Lemma 4.4.1, [15]) Let E be a vector bundle on

P1. Then E is a direct sum of line bundles.

Proof. We use induction on the rank r of E. The case r = 1 is obvious. For all i
sufficiently large, then by Serre duality and Theorem 2.2.5, E (—i) has no non-zero
sections. Consider the largest integer i such that E(—i) has a non-zero section. This
section can never vanish, so there exists a vector bundle F of rank r — 1 with an exact

sequence

0—>O(i)—iE—)F—i0.

Then, by induction, F = $j O(j)rj with j S i if rj # 0 because otherwise E(—i — 1)
would have a non-zero section. To see this, consider the twist of the above short exact

sequence by O(-—i — 1):
0 —+ O(—1)—> E(—i — 1) -—> F(—i — 1) —> 0.
Now consider the associated long exact sequence in cohomology:

o —» HOW-1)) = 0 —» H0(E(—i — 1)) e H0<F<—z' — 1)) —> H1(0(-1)) -—— o

11

Hence, we have

Emu 1510(2)) 2: H1(’Hom(F,O(i)))

a GE H1(Hom(ov>. 0(a))”
J‘

9:: EB H1<0v — an"?
3'

'5 EBH0(0<-2 4+»)?
J'

=0

which implies that the short exact sequence above splits. This proves the theorem. CI

Consider a semi-stable bundle E of slope )1. Such a bundle has a sub—bundle E1
of slope p which is stable: If not, we can take a descending sequence of semi-stable
sub-bundles of slope ,u in which the ranks strictly decrease. But this is impossible.
So E / E1 is also semi-stable of slope u and we can continue the construction in this

manner. We obtain a strictly increasing sequence of sub—bundles
0CE1CE2C---CEk=E

consisting of semi-stable bundles of slope it such that the successive quotients E,- /E,~_1
is a stable bundle of slope p. This is called a Jordan-Holder filtration of E; the integer
k is called the length of the filtration and the sum ®i E,- /E,-_1 is called the associated

grading.

Proposition 2.2.7. Let E be a semi-stable bundle of slope p. All the Jordan-Holder
filtrations of E have the same length k. Moreover, if (E) and (E;) are two such
Jordan-He'lder filtrations with associated gradings ®i Ei/EZ-_1 and @2- E;/E£_1 re-

spectively then there exist a permutation o E Sk such that Ei/Ez-_1 ”5 E; (0/ E; (2.)_1.

12

In particular, the gradings are isomorphic.

2.3 Geometric Invariant Theory

In this section we wish to briefly go over constructions involving quotients of varieties.
In this section, C will be a linear algebraic group, i.e. a closed subgroup of CLn for

an integer n 2 1.

Definition 2.3.1. An action of C on a variety Y is a morphism o : C x Y —> Y such
that:
( 1) for all g E C, the morphism org : Y —> Y induced from o by g is an automorphism
of Y,

(2) the map C —> Aut(Y) defined by taking g to 09 is a group homomorphism.

We denote o(g, y) by g.y. For a given y E Y, the set of g.y for all g E C is called
the orbit of y. The set of all g E C with g.y = y is called the stabilizer of y. We also

have the graph morphism w : G x Y —> Y x Y.
Definition 2.3.2. Let G and Y be as before. The action of G on Y is called:

0 closed if all of the orbits of the action are closed,
0 proper if it) is a proper morphism,
0 free if if; is a closed immersion.

Definition 2.3.3. Let Y and Z be two varieties with G-actions. A morphism d :

Y —-> Z is called C-equivariant if the following diagram commutes:

GxY—‘UY

l l

G X Z —0—+ Z
In other words, we have ¢>(g.y) = g.cb(y) for all g E C and y E Y.

13

We now proceed to describe quotients. In the following, Y is a variety with a

C-action.

Definition 2.3.4. A categorical quotient of Y by G is a pair (Z, a) given by a variety
Z with the trivial C-action and a C—invariant morphism ¢ : Y —-> Z having the
following universal property: Any C-invariant morphism w : Y —> Z ' factors through

(i) uniquely, i.e. there exists a unique f : Z -—) Z’ with w = fqb.

The universal property implies that the categorical quotient is unique if it exists.

Definition 2.3.5. A good quotient of Y by G is a pair (Z, a) given by a variety Z
with the trivial C-action and a C-invariant morphism (,2) : Y ——) Z such that:

(1) d is affine and surjective,

(2) the image of a closed C-invariant subspace of Y is closed in Z; and if Y1 and Y2
are two closed disjoint G-invariant subspaces of Y then their images in Z are disjoint,

(3) the structure sheaf 0 Z of Z is the sheaf of invariant sections of the structure sheaf

of Y.

By Proposition 6.1.7, [15], a good quotient is also a categorical quotient, and if
(Z, ()5) is a good quotient of Y by G, then Z is equipped with the quotient of the
Zariski topology of Y. Also, the closed points of Z are in one—to—one correspondence
with the closed orbits of G in Y.

Our last definition in this section is the definition of quotients we will be using

later:

Definition 2.3.6. A good quotient (Z, qb) of Y by G is called geometric if all the

orbits of C in Y are closed.

14

2.4 The Moduli Space M(r, D)

In this section, we describe the moduli problem of vector bundles on curves. This
moduli problem has a solution in terms of coarse moduli spaces. We describe the
relevant machinery below.

Consider the contravariant functor F which associates to an algebraic variety S
the set F (S ) of isomorphism classes of vector bundles E of rank r and degree D on
X x S such that the fiber E3 = E 69 k(s) is semi-stable of degree D for all s E S. The
moduli problem is the problem of describing this functor as the functor of points of a
variety. If such a variety exists, it is called a fine moduli space of the moduli problem.

A coarse moduli space for this functor is a variety M (r, D) with a morphism of
functors ¢ : F —i ll om Sch /k(—, M (r, D)) that is universal in the following sense: For
any other variety N with a morphism of functors w : F -> H 0mSch. /k(—, N) there
exists a unique morphism f : M (r. D) —> N such that w = f 0d), where f also denotes
the induced morphism HomSch/k(—, M(r, D)) —> Hom.SCh/k(—, N). This condition

determines the pair (M (r, D), (,b) uniquely.

Theorem 2.4.1. There exist coarse moduli spaces M(r, D) and /‘\/l88 (r, D) for sta-
ble and semistable bundles of rank r and degree D over any nonsingular irreducible
projective curve X of genus g 2 2. M(r, D) is a nonsingular quasiprojective variety
that is contained in M38 (r, D), which is a projective variety. M33 (r, D) is normal,
and its singular locus is given by M33 (r, D)\M(r, D). The dimension of these moduli
spaces is equal to r2(g - 1) + 1. Moreover, M (T, D) is a fine moduli space if and only
ifgcrl(1', D) = I.

The construction of the moduli space proceeds through an intermediate construc-
tion of the Hilbert scheme. This is a fundamental object in the study of vector bundles
on curves. We now proceed to describe this.

Let P be a degree 1 polynomial, and let E be a coherent sheaf on the variety

15

X. For any variety S, let E 5 denote the pull-back of E under the projection map
X x S —+ X and let H ile (E, S ) denote the set of coherent quotient sheaves G =
E 5 / F which are flat over S and such that for each s E S, the Hilbert polynomial of
C(s) is P. If f : S’ -» S is a morphism, then for all C E Hile(E, S), the pullback
(id X x f )*(C) is a coherent sheaf over X x S’ which is flat over S’ and which defines
a map from Hile(E, S) to Hile(E,S’) by taking G to (id X x f)*(G). We then
obtain a contravariant functor from the category of algebraic varieties to the category
of sets.

Let the curve C be as defined in the Introduction, P a linear polynomial with
integer coefficients and let IE be a coherent sheaf on C. Then the family of all coherent
sheaves F on C such that F is a quotient of E and the Hilbert polynomial of IF is P,
is denoted QUE, P). The following theorem, which is due to Grothendieck, ensures

that QUE, P) has the structure of an algebraic variety:

Theorem 2.4.2. (Grothendieck) ( Theorem 6.1, [23]) There is a unique projective
algebraic variety Q = Q(lE, P) and a surjective homomorphism 0 : p’f(lE) —> 8 of
coherent sheaves on C x Q (p1 is the canonical projection C x Q —> C) such that:

1: 8 is flat over Q;

2: the restriction of the homomorphism 0 : p’fUE) —i 8 to C x q E“ C, q 6 Q
corresponds to the element of Q(lE, P) represented by q;

3: given a surjective homomorphism ct : p]‘(lE) —> C of coherent sheaves on C x T,
where T is an algebraic scheme such that C is flat over T, and the Hilbert polynomial
of the restriction ofG' to C x t g C is P, there exists a unique morphism f : T —. Q

such that (b : p’]‘(lE) -—> C is the inverse image ofd : p’l“(lE) —i 8 by the morphism f.

Now consider the coarse moduli problem of stable vector bundles E over C with
given rank r and degree D. Let us assume that D > r(2g- 1) as required in the lemma.

The rank r and the degree D determine the Hilbert polynomial of the vector bundle

16

uniquely by Riemann-Roch, Theorem 2.2.1. Since H1(E) = 0 for these bundles, we

can compute the dimension of the space of global sections h0(E) as well:
N = h0(E) = (1 — 9)?“ + D.

Since a stable vector bundle E over C with rank r and degree D is generated by its
global sections, it can be considered as a quotient of the trivial vector bundle IE of
rank N.

Note that the group GL(N) may be identified with the group of automorphisms
of IE, hence GL(N) acts on Q, and also on the sheaf 8. The action of CL(N) on Q
induces an action of PGL(N) on Q, but does not induce a corresponding PGL(N)-
action on E. The scalar matrices act as scalar multiplication on E. (This is the reason
why the moduli space for vector bundles over curves is not fine in general.)

Let R3 be the subset of Q consisting of those :1: in Q for which the bundle 81; is
stable. This is an open subset of Q on which PGL(N) acts freely and hence has a
quotient. This quotient is the moduli space M(r, D).

Now we want to discuss some properties of 8. We will assume that the rank r
and the degree D are given, and that the vector bundles can be twisted by 00(m) to
make their degrees D larger than r(2g — 1), as required in Lemma 2.2.4. By Theorem
5.3, [19], the bundle 8 has the local universal property for families of bundles of rank

r and degree D which satisfy conditions (1) and (2) in Lemma 2.2.4:

Definition 2.4.3. Let .77 be a family of vector bundles over C with rank r, degree D
and satisfying (1) and (2); parametrized by a scheme T and flat over T. Then we can
cover T with open subsets T,- so that there exist maps fz- : T,- —> Q with f isomorphic

to (idC x fi)*£. We do not require the maps fi to be unique.

We want to be able to use the ideas in the preceding paragraph regardless of the

degree. So let the rank r, the degree D and the twist m be such that D + rdm >

17

r(2g — 1). We construct the Quot scheme Q and the bundle 8 as before. Let .7:
be a family of stable vector bundles over C with rank r and degree d; parametrized
by T. Consider the vector bundle f 8 pEOC(m). This gives us a family of stable
vector bundles over C with rank r and degree D. By the local universal property of
8 mentioned above, we cover T with open subsets T; and we get maps f,- as before.
This gives us

fly}. ® [)EOCUn) E” (idC X fi)*£,

Now consider 5 8a pEOCl-m). We have:

('idc >< ft)*(5 ®PZ~OC(-7'l)) ’5’ (MC >< ft)”: ® (idc >< ft)*P*COC(-m)
E (idc >< ft)? ®pZ~OC(-m)
E“ TIT,- 8‘ PEOCW) ® PEOd-m)

iii-HT,-

This proves that the bundle 8 <8) pEOC(—-m) has the local universal property for

families of stable bundles with rank r and degree D.

Note that for a closed point y E R5, its orbit under PCL(N) corresponds to a
stable bundle F of rank r and degree D. The bundle 6' has the property that its
restriction to C x Spec k(y) is isomorphic to F. When we twist E by pEOC(—m),
we get a similar correspondence between PGL( N )-orbits of closed points y E R3 and

stable bundles of rank r and degree D.

Let U denote the subset of M(r, r(d + g — 1)) consisting of vector bundles E over
C such that H 0(E (—1)) = 0. We want to prove that this is a nonempty open subset.
The fact that it is nonempty in case of curves was proven in [24], Theorem 2.4. To

prove that it is open, consider the subset Q of R3 consisting of vector bundles E over

18

C such that H0(E(—1)) = 0. Then PGL(N) acts freely on n by Lemma 8.3.3, [15]

and we can take the geometric quotient to construct U.
Lemma 2.4.4. 9 is open in Rs.

Proof. Note that the second projection C x R5 —> R3 is a projective morphism. Since
any affine subset of R3 is Noetherian, we can restrict to an affine open subset after
choosing an affine open cover. We also note that 5 ® pEOC(—m — 1) is a coherent
sheaf on C x R3 and is flat over Rs, by Grothendieck’s theorem. It now follows from
the Semicontinuity Theorem, Theorem 12.8, Chapter 3, [11], that the set D is open
in R3. CI

2.5 The Clifford Algebra and its Representations

Let f (u, v) be a binary form of degree at over It. We define the Clifford algebra of f,
denoted C f to be the associative k-algebra k{u, v} / I , where I is the two-sided ideal
generated by elements of the form (on + 5v)d — f (a, B), where a and [3 are arbitrary
elements of k. A representation of C f is an algebra homomorphism (b : C f —+ Mm(F),
where F is a field extension of k. The integer m is called the rank of the representation.

Let X be the curve in P2 defined by the equation wd = f (u, v), where u, v and
w are the projective coordinates. We assume that the binary form f (u, v) does not
have any repeated factors over an algebraic closure of k and that the characteristic

of I: does not divide d. With these assumptions, we have the following lemma:
Lemma 2.5.1. X is a smooth curve of degree d.

Proof. It is obvious that X is a curve of degree d. To prove that it is smooth, consider

the partial derivatives of the defining equation wd - f (u, v):

19

aurwd — f(u. v» = dwd‘l
cutie — f(u,v)) = —o.,f(u, h)

6.1qu - f(u, v» = -8vf(u. v)

It is now obvious that for a point [u : v : w] E X to be singular, f (u, v) and both
its partial derivatives have to vanish on it. But since we assumed that f (u, v) has no

repeated factors, this is not possible. El

From now on, X will denote this curve. We note that the genus of X is g =
(d — 1)(d — 2)/2. Assuming that d 2 4, we have 9 2 2. We also note that the map
[uzvzw]l—» [uzv] definesadegreedmapsz—ill’l.

We now prove that the rank of a representation of C f is divisible by d:

Proposition 2.5.2. Let f be a binary form of degree (1 over an infinite field I: with no
repeated factors over an algebraic closure of k. If 45 is a representation of the Clifford

algebra Cf, then the degree d of f divides the rank of ()5.

Proof. ([10], Proposition 1.1) Let ab : C f —> Mm( K) be a representation of C f of rank
m, where K is a field extension of k. Let Bu = ¢(u) and BU = qb(v). We have the
following relations:

Na. :3) = (a8. + .6130)“l

for all 0,3 E K. Taking determinants, we obtain (f(a',f3))m = (det(orBu + flBt.))d
for all 0,,3 E K. Since K is an infinite field, we infer that f(u,v)m = g(u,v)d,
where g is the polynomial with coefficients in K given by g(u, v) = det(uBu + va).
Each coefficient is a polynomial in the entries of the matrices Bu and By. Let f =
T17‘2 - - -rn, where each form rz- is irreducible over K. The r,- are distinct by hypothesis.

Since fm = gd, the polynomials f and 9 have the same irreducible factors. So we

20

can write g = 7.1711 r322 - - - ran". We conclude that mid = m for all i. Hence d divides

m. C]

We now want to describe representations of C f in more detail. Let a) : Cf —+
.llvlm(k) be a representation. Let R = k[u, v] have the standard grading and let
S = k[u,v,w]/(urd — f(u.v)). Note that X = Proj S, and the map p is induced by
the inclusion R —> S. Let au 2 ¢(u) and 0., = (15(v). These two matrices define a
map of graded algebras ¢f : S —» Mm(R) by sending u to uIm, v to va and w to
ua-u + vav. Conversely, if we have such a map, we can define a representation of f
by taking (,bf(u1m) and ¢f(v1m). Via this map, Rm becomes a graded S-module. In
this way, we get a vector bundle E over X such that piE is trivial of rank m. This

vector bundle E also satisfies H 0(E (—1)) = 0.

2.6 Grothendieck Topologies

We now give a brief discussion about sites and Grothendieck topologies. (A good
overview can be found in Section 2, [25].) An Open cover in a topological space X
can be seen as a collection of maps f,- : U,- —) X; where each Uz- is an open subset
of X and f,- is the corresponding inclusion, and the images of the maps f,- cover X.
Suppose we have a category with fiber products. A Grothendieck topology on this
category is basically a choice of a class of morphisms, that play the role of open sets.

This class of morphisms E must satisfy the following properties:

0 All isomorphisms are in E.
o A composite of two morphisms in E is in E.

0 Any base change of a morphism in E is in E.

A category with a Grothendieck topology is called a site. In the examples relevant

to us, the category will be the category of schemes over a base scheme S. Typical

21

examples of such classes of morphisms include:
0 Open immersions. This gives rise to the classical Zariski topology.
0 Etale morphisms.
0 Flat morphisms that are locally of finite type.

A covering of a scheme X over S will be a family of morphisms (fi : U1: —-> X )ie I
in E, whose images cover X. Two topologies on (Sch / S) are especially important:

Flat topology: Let X be a scheme over S. Then a covering of X is a collection
of morphisms (fz- : U,- —+ X lie I such that each fz- is a finitely presented flat morphism
(for locally Noetherian schemes, this is equivalent to flat and locally of finite type),
and X is the set-theoretic union of the images of the morphisms fi. We can consider
the fz- as giving a single map HUI: —> X, and this map is faithfully flat and of finite
presentation.

Etale topology: Same as above, but replacing fiat by étale. Recall that an étale
morphism is defined to be a morphism of schemes that is flat and unramified. This
can also be stated as follows. Let f : X —-) Y be a morphism. For each y E Y the
fiber X3) = X Xy k(y) as a k(y)-scheme is isomorphic to the disjoint union of finitely
many points that are spectra of finite separable field extensions of k(y).

A presheaf ’P on a site (Sch/S) is a contravariant functor (Sch/S) —+ (Set).
For each morphism, f : V —> U in (Sch/S), ’P gives us a ”restriction map” ’P( f ) :
’P(U) —> ’P(V). For 3 E ’P(U), we denote P(f)(s) by slv. A morphism of presheaves
is simply a natural transformation of the corresponding functors.

A presheaf ’P is a sheaf if it satisfies

0 If s,t E ’P(U) and there is a covering (U,- —+ UheI of U such that SlU, = tle.

for all i E I, then s = t.

22

o If (U1: —i Uliel is a covering and the family (302-61, 32- E P(Ui) is such that
Sile'XUUj = SleixUUj for all i,j E I then there is an s E P(U) such that
s|Ui = s,- for all i E I.

A presheaf can be shcafificd:

Theorem 2.6.1. (Theorem 2.1.1, [25]) For any presheafP on (Sch/S) there are a
sheaf P on (Sch/S) and a morphism cp‘ : P —> P such that any other morphism w

from P into a sheaf]: factors uniquely through 45. P is called the sheafification of P.

The sheafification of P is constructed as follows. For any U in (Sch/S) and
s,s’ E P(U), define 3 ~ 3’ if Sle‘ = SlUj for some covering {U¢}|i E I of U. The
elements of P are pairs ((si)|,-61,(U,- —+ Ulielli where SilUixUUj SleiXUUj' Two
pairs (i.e.-haw.- -—» Uher) end «es-mete;- —» Uljell ii the eevreinsg (U.- —»
Uliel and (U1,- —i U)jEJ have a common refinement (V)c —+ UlkeK such that the

families obtained by restricting s, and s;- to (I'k —> U) kE K are equal.

2.7 Azumaya Algebras

We follow the discussions in [22] and [16] for this review. Let A be an algebra over a
commutative ring R. We assume that R is the center of A. Then A is called Azumaya
over R if A is faithful, finitely generated and projective as an R-module and the map

(bA : A ®R .40 -i EndR(A) defined by
(M(Z 7‘2“ ‘8 800‘) = 2 7‘17'82'
i i
is an isomorphism.
The following proposition will be useful later:

Proposition 2.7.1. Let A and B be Azumaya algebras over R. Then A ®R B is

Azumaya over R.

23

By a construction similar to obtaining a quasicoherent sheaf over Spec R using an
R-module M, given an Azumaya algebra A over R, we can obtain a sheaf of algebras

over Spec R. We can use this as the motivation for the definition:

Definition 2.7.2. Let X be a scheme. An OX-algebra A is called an Azumaya
algebra over X if it is coherent as an OX-module and if, for all closed points a: of X,

AI is an Azumaya algebra over the local ring OX4:-

The conditions in Definition 2.7.2 imply that A is locally free of finite rank as an
OX-module.
Instead of using the definition to prove that an OX-algebra is Azumaya, we will

make use of the following proposition:

Proposition 2.7.3. Let A be an (OX-algebra that is of finite type as an OX-module.
Then A is an Azumaya algebra over X if and only if there is a flat covering (Uz- —+ X)

of X such that for each i, A 920x OUz‘ ’5 Mrz.(OUi) for some ri.

We can now define the Brauer group of X. We say that two Azumaya algebras
A and A' over X are similar if there exist locally free OX-modules E and E’, of
finite rank over 0X, such that A ®0X £nd0X(E) '.-‘=’ A’ ®0X €nd0X(E’). This
defines an equivalence relation on the collection of Azumaya algebras over X, because
8nd(E) @oX £nd(E') g £nd(E ®OX E’). The tensor product of two Azumaya
algebras is an Azumaya algebra, and it is compatible with the similarity relation.
We define a group operation on the set of similarity classes of Azumaya algebras on
X with [A] [A'] = [A 8 A’]. The identity element of this operation is [OX] and the
inverse of an element [A] is given by [A0]. This group is called the Brauer group of
X. It is known that the Brauer group is torsion when X has finitely many connected
components (See Proposition 2.7, Chapter 4, [16]). It is also known that if X is
integral and smooth, then there is an injective map i : Br(X) —i Br(K), where K is
the function field of X (Corollary 2.6, Chapter 4, [16]). This can be used to define

24

the index of a class in Br(X). To do this, simply consider this class as an element
of Br(K) using the injective map i defined above. Then the index of this class is the
smallest degree of a separable extension field K ’ of K that splits the class, i.e. the
pullback of the corresponding Azumaya algebra along Spec K’ —> Spec K is a matrix
algebra.

We apply the considerations about Grothendieck topologies in Section 2.6 to non-
commutative algebras. Let B be an associative k-algebra, finitely generated over

k.

Definition 2.7.4. If S is a k-scheme then an S-representation of degree n of B is
a pair (¢,0A), where O A is a sheaf of Azumaya algebras of rank n2 over S and
<25 : B ——i HO(S,OA) is a ring homomorphism. Two representations ((131,0A1) and
((152,0A2) are called equivalent if there is an isomorphism 6 : 0A1 —> 0A2 of sheaves
of rings such that $2 = H 0(S, 6) 0451. A representation of B is called irreducible if the
image of B generates O A locally. Let Repn(B, S) be the set of equivalence classes of

irreducible S—representations of degree n of B.

Theorem 2.7.5. ( Theorem 4.1, [25]) The functor Repn(B, —) is representable in
(Sch/k).

Sometimes it is more convenient and easier to work with representation into endo-
morphism sheaves of vector bundles. Let 9,,(3, —) be the subfunctor of Repn(B, S)
consisting of representations of endomorphism sheaves of vector bundles of rank n.
This is not a sheaf with respect to the flat topology. However, its sheafification with

respect to the flat topology (see Theorem 2.6.1) is well-known:
Lemma 2.7.6. (Lemma 4.2, [25]) Repn(B, —) E“ 3MB, —).

We want to end this section with a lemma that will be useful later. Let S be a
scheme, and 0 be a sheaf of algebras over S. A collection of global sections (a,- E

H0(S, Ollie I is said to generate 0 if the stalks (oz-)3 generate 03 for all s E S.

25

Lemma 2.7.7. Let S and (’2 be as above. If the 0,-(3) generate the fibers 0(5) for

closed points 5 E S, then the o,- generate 0.

Proof. The statement is local in S, so we will assume that S = Spec R for a ring R
and that O = A~ for an R-algebra A.

First, we claim that if the 0,-(s) generate 03 for closed points 3 as a k(s)-algebra,
then the (oz-)3 generate 03 as an RS-algebra. But this follows from Nakayama’s
lemma.

Second, we claim that if the (oz-)3 generate (93 for closed 3, then they generate Os
for all s E S. The 0,- correspond to elements a,- E A. We know that for all maximal
ideals in in R, (a,)m generate Am as an Rm-algebra.

Let B be the R—subalgebra of A generated by the a,-. Then Bm = Am. By
Corollary 2.9, [5], A = B and the lemma is proved. CI

26

 

Chapter 3

Construction of the Universal

Representation

In this section, we want to construct a sheaf of Azumaya algebras over the variety U
defined in the previous section. Recall that there is a coarse moduli space M(r, D) of
vector bundles over the curve C with rank r and degree D; this is a quasiprojective
variety. The variety U is the open subset of M(r, D) consisting of vector bundles E
over C such that H0(E(—1))= 0.

We first recall how to construct quotients of vector bundles. Let Y be an integral
algebraic variety and C a reductive algebraic group that Operates on Y. We will

assume that there exists a good quotient 7r : Y —+ M. We have the following definition:

Definition 3.0.8. (Definition 8.4.3, [15].) An algebraic vector C—bundle F ——i Y is
a vector bundle over Y, equipped with a G-action which is linear in each fiber and

such that the diagram
G X F ——> F

l l

GXY——>Y

commutes. In other words, for every y E Y, we have a linear map Fy —> g.y

F is said to descend to M if there is a vector bundle F’ over M such that the

27

algebraic vector G-bundles F and 7r*F’ are isomorphic.

This definition can also be stated in terms of sheaves. Let .7: denote the sheaf of
sections of the vector bundle F. For every 9 E C and every open subset V of Y, we
have a linear map .7: ( V) —i f(g.V). These maps are required to satisfy the obvious
compatibility conditions.

The following lemma gives a necessary and sufficient condition for an algebraic

vector G—bundle F to descend to M: (For the proof, consult [15] and [4].)

Lemma 3.0.9. Let C, Y and M be as before. Let F —i Y be an algebraic vector
G-bundle over Y. Then F descends to M if and only if for every closed point y of Y

such that the orbit of y is closed, the stabilizer of C at y acts trivially on Fy.

Recall that Q is the subset of the Quot-scheme Q consisting of vector bundles E
over the curve C such that H 0( E(—1)) = 0. We also have the bundle 8 parametrizing
quotients of the trivial vector bundle IE having the given Hilbert polynomial P. Recall
also that CL(N) acts on Q, and the stabilizers of points are the scalar matrices. Hence,
there is an induced action of PGL(N) on Q, and the good quotient is the variety U.

When we try to take the quotient of 8 by PGL(N), however; we are unable to
define an action of PGL(N) on 8 because of the fact that the scalar multiples of
identity in CL(N) do not act trivially.

We resolve this difficulty as follows. Consider the direct image of 8 under the
projection 7r : C x D ——i 9. Then consider the endomorphism bundle 8nd(7r...€) and
then define an action of PCL(N) on it. (We prove that 7r...8 is a vector bundle below in
Lemma 3.0.11.) Since CL(N) only acts on the second component of C x 9, this gives
a CL(N)-action on r48. Now let CL(N) act on 8nd(7r...8) by conjugation, see below
that the action of CL(N) descends to a PCL(N)-action: The action of the stabilizer
of a point on the fibers of 7r...£' is by scalar multiplication, but on £nd(7r...£), this action

becomes trivial. So the action of CL(N) on 8nd(7r...8) descends to PCL(N).

28

To be precise, let 9 E CL(N), f E F(V,8nd(rr...£)) for an open subset V C 9.
Then f is an endomorphism of mg over V, and we define g. f to be an endomorphism
of 7r...8 over g.V as follows. Let s be a section of rr...£ over an open subset W C V.
Then g. f is the endomorphism of r18 that sends s to g( f (g’ls)). It is obvious
that this defines a CL(N)-action on £nd(rr...8). Since a scalar matrix /\I acts as

multiplication by A on (7r*£');,;, it can easily be seen as acting trivially on £nd(7r...8)$:

(Al'f)(3)=()\1 o f 0(A1)“1)(s)
= Annie)

1
=AIon(s)

1

= Axflsl

= f(8)-

This gives us a PGL(N) action on £nd(rr...£). As seen above, we have a quotient
vector bundle, which we will denote as £nd(1r*8)PGL(N) over U.

Recall that there is a canonical map q : C —> P1, corresponding to the inclusion
k[u,v] —> k[u,v, w]/(wd — f(u,v)). Recall that for a closed point y in 0 8y denotes
the pull-back of 8 under the canonical map id x iy : C x Spec k(y) —+ C X D.

We want to prove the following proposition, which follows from a standard result

of Grothendieck. (See, for example, [24].)

Proposition 3.0.10. The coherent sheaf q...(£y) is isomorphic to the trivial bundle
$iil 01131 '

k(y)
Proof. Since ll”)c is a nonsingular projective curve, any torsion-free coherent sheaf is
a vector bundle. Since q... respects torsion-freeness, q...(£y) is a vector bundle on If”).

It has rank rd since 83) has rank r and the map q has degree d. Since any vector

bundle on If”)c is a sum of line bundles, we can write q...(8y) ’5 $221 OP] (71,-) for some

29

integers 71,-. We have x(q*(8y)) = x(£y). Then,

rd
may» = <rk<q.<a>>>(i — 9,4,) + deg(‘l*(5yll = rd + Z

i=1
X(5y)=(rk(£y))(1— g) + deg(£'y) = r(1— g) + r(d + g — 1) = rd.

So we have 2:11 Tli = 0.

We also have, by the projection formula:
h0(fi’1.qe(5y ®oC Oc(-1))) = h°<r1,<q.sy)<—1>>= how. 5.1—1» = 0.

So it follows that 1100114, (q.8y)(—1)) = 0. This is equal to h0(1P’1, 69173—1 0,,1(n,- — 1)).
It then follows that n,- — 1 < 0 for all i and hence n,- = 0 because the sum of the n,-

is equal to 0. D

We can use this result to prove the following result, which is due to Kulkarni. (See

Proposition 3.5, [13].)

Lemma 3.0.11. Let 7r : C x D —> O denote the projection onto the second factor.

Then mg is a locally free sheaf of rank rd.

Proof. For any point y in Q, let k(y) be the residue field of y. Denote by Q, the
vector bundle (id x iy)*£ on CW», where iy is the inclusion of the point y; that is,
Spec k(y) —i Q. We have to prove that dimk(y)H0(Ck(y),£y) is constant, and equal
to rd. Recall that Q is irreducible, and reduced. So, by Corollary 2 pp. 50, [17], it
will follow that 7r...8 is a locally free sheaf of rank rd.

If y is closed, then qsgy 2 $3,711 0P1 by Proposition 3.0.10. So h0(C,€y) =
h0(ll’1,q...£y) = rd. It is also well-known that the function y i—i dimk(y)H0(Ck(y), 8y)
is upper semicontinuous. (See, for example, Theorem 12.8 in Chapter 3, [11].) To-

gether with the fact that Q is a Jacobson scheme (since it is locally of finite type over

30

the spectrum of a field), it follows that dimk(y)H0(Ck(y),£y) is constant and equal

to rd for all y E O. This implies that ruff is a vector bundle of rank rd. [:1

Lemma 3.0.12. Let S be a scheme, and let f : S ——i Q be a morphism. Consider the
commutative diagram:

'd
st fl an

e1 1e

[Pl idpl xf

xS—elP’le.

Then the coherent sheaves (idPl x f)*(qQ)*8 and (q3)*(idC x f)*8 are isomorphic.

Proof. Since 8 is a coherent sheaf on C x Q = Proj(OQ[u,v,w]/(wd — f )), there
exists a sheaf of graded 09[u, v, w]/(wd — f )-modules M such that 8 is isomorphic
to M. (See [8], II, Sections 3.2 and 3.3.) Then we have the following isomorphisms,

as sheaves of 09hr, v]-modules:

(idles X f)*(q9).€ 2’ (Osluivl ®Ofl[u,v] omuijl'V.

and

(Qk(y))*(idC x fllk“: g (OSIui vi wl/(wd -‘ f(ui 71)) ®OQ[u,v,w]/(wd—f(u,v)) M)~'

05[u,v]

But the two graded k(y)[u, v]-modules on the right side of the equations above are

isomorphic. So the lemma is proved. E]

For the remainder of this section, we want to construct a map 7/) : C f —i H 0(A)
that is a universal representation for C f. The next theorem is crucial in this construc-
tion. It is proven in [11] that for any morphism g : X —> Y of schemes and a sheaf
g of (OX-modules, there is a natural morphism g*g...g —-> g. This is a consequence of
the adjointness of the functors g* and g... The proof of the following theorem follows

the discussion in [13] closely. (See, Proposition 3.8, [13].)

31

Theorem 3.0.13. Let f = (q9)*£. Then the natural morphism u : p;,(p9).f —-i f

is an isomorphism. (Recall that pg is the canonical projection P1 x Q —i Q.)

Proof. We know that (rims? = 77*8 is a locally free sheaf of rank rd by Lemma
3.0.11. So the sheaf pa(pn)*}“ is also a locally free sheaf of rank rd on I???

First we claim that f is a locally free sheaf of rank rd on 111%,. We prove that
dimHyfl: <83 k(y) is rd for any closed point y in P5. This will be sufficient by the
upper semicontinuity of the dimension function and by the fact that P}, is locally of

finite type over k.

For any closed point y in P1 x Q, consider the following commutative diagram:

Spec k(y) = Spec k(y)

We prove that dimk(y).7: ®Op1xg k(y) = dimk(y)j*f is rd. But dimk(y)j*}' =
dimk(y)i*(id x iy)*f'. From Lemma 3.0.12 and Proposition 3.0.10, it follows that
(id x iy)”‘.7-~ 3 (id x iy)*(qQ)...8 § q...(id X i)*£ 9—4 q...(£y) is a trivial vector bundle of
rank rd, so that dimk(y)i*(id x iy)*.7-' is rd.

Now 71 is a morphism of vector bundles of rank rd. To prove that it is an isomor-
phism, it is enough to show that ux : (p;2(pn)...f)$ -—i (7);; is bijective for all points
:1: in P1 X 0. But it is sufficient to prove this only for closed points ([8], I, Corollary

0.5.5.7).

By ([8], I, Corollary 0.5.5.6) it is sufficient to prove that

”y 81d: (sz(PQ)*f)y/my(Pf2(PQ)*fly —* fy/myfy

32

is surjective for all closed points y in P6. But this homomorphism is surjective if and
only if the morphism

1*" 1.136010%]: 41*}-

is surjective, which is the same as

i*(id X iy)*u : i*g(id X iy)*p§(p9)*f —i i*(id X iy)*.'F
being surjective. So it is sufficient to prove that

(id X iy)*u : (id X iy)*pfi(pg2)*}' —> (id X iy)*.7:
is an isomorphism. But we have the isomorphism
(7307 >< iy)*r§3(rn)ef E“ p;(y)i;(pn)e7-

Note that i;(pg)*f is a trivial vector bundle of rank rd, and so

Phi/rim)“ g Phi/)(EB OSpeckm) 9‘ ED 0,.1 -

rd rd My)

This proves that (id X iy)*pa(pg)*}7 is a trivial vector bundle. Recall that (id X
71y)“F is a trivial vector bundle of rank rd. So the morphism (id X iy)*u will be an

isomorphism if it is so on the global sections. But the morphism
r;(y)'i;(r)n)ef -* (id X is)?
on the global sections is an isomorphism if the natural morphism
(mm 20,, My) _. H°(rt(,,.n>

33

is an isomorphism. By the earlier part, and Corollary 2, p. 50, [17], this is the

case. Cl

Recall that we have the morphisms qg : C X 9 = C9 ——> P1 X (2 2 P19 and
pg : IF}, —i Q. Consider 09[u,v] and Og[u,v,w]/(wd — f). These are sheaves of
graded OQ-algebras generated by degree 1 elements. We can define their homogeneous
spectra Proj(09[u,v]) = P}, and Proj(09[u,v,w]/(wd - f)) = C X 0. Recall also
that we have the vector bundle .7: on P511. That this is a vector bundle follows from

Lemma 3.0.11 and Theorem 3.0.13.

Let g be a sheaf of 0P1 -modules. Define
S2

13(9) = 6909104900)-

nEZ

In particular, we have

E(Opbl = @(PQMOPSBWI
nE

I‘...(OIP1 ) is then a sheaf of graded 0P1 -algebras and I‘...(Q) becomes a sheaf of graded
Q Q

I“... (0P1 )-modules.
Q

Remark 3.0.14. These are particular cases of constructions carried out in [8], II, 3.3.

By Proposition 7.11, Chapter 2, [11], we have

P*(0 ) 9': OQ[U, vl

1
Pa
as sheaves of graded 09-modules.

In the next proposition, we use Theorem 3.0.13 to describe F*(f).
Proposition 3.0.15. The Og[u,v]-module I‘...(.7-') = ®n€Z(pQ)*(.P(n)) is isomor-
phic to (mg) @037 Opju, v] as a sheaf of graded (99hr, v]-modules.

34

Proof. By the theorem, it is enough to compute the graded module associated to the

coherent sheaf pa(pg)*f. By projection formula, we have

(pn)e(}'(n)) = (PQ)*(~7:) €909 (Pn)e(0ph(n))

By definition:

1': @((p9)*(}-) @00 (PQ)*(Opa(n)))
nE

g(p9)*(f)809(®(190l*(0p1 (71)»
7262 Q

= (mm) @909 may

2 (7138) 8’09 Ofllui Ul
as desired. [:1

Next we consider the question of the existence of the universal representation. This
representation will be an algebra homomorphism w : Cf —-i HO(U, A) that satisfies
a universal property that will be in the next chapter. Recall that A is the Azumaya
algebra obtained by taking the quotient of £nd(7r...£) by the action of PCL(N).

Theorem 3.0.16. There exists an algebra homomorphism 7b : Cf —+ HO(U, A).

Proof. We prove this theorem by showing the existence of elements in H 0(U , A) that
satisfy the relations of the Clifford algebra.

First we make a simple observation. Let n48 be as above, and let M be the graded
Og[u, v]-module 7r...£ <80n Og[u, v]. Note that M0 = 7r...8. mg is an OQ-mOdule and

the grading is determined by assigning u and v as degree 1 elements. Let M,- be the

35

ith graded piece of M. Then we have
HomOQ(M0, M1) = uE-ndoQ (M0) + vEndOQ (M0).

This follows from the fact that M1 = uMo EB vMO.

Consider the vector bundle 8 on C9. We have

11.16) = 69 man»

nEZ

= $(PQ)*(‘IQ)*(£("))

nEZ

= $(Pn)*((qn)e(5) 801d Opl ("ll
126% Q Q

= €B(pn)e(f(n))

nEZ

: RAJ?)

Hence, F..(.F), which we proved to be isomorphic to (NJ) (800 OQ[U, v], can also be
viewed as a sheaf of graded modules over F4009) E’ (99 [u, v, w] / (w‘ll — f). Now w
is a homogeneous element of degree one in this graded OQ-algebra, so we can view
717 (to be precise, the multiplication map by w) as an element of H onion (M0, M1).

By the comment above, there exist elements an and an in EndOQ(M0) such that we

have the following equality in H 077109 (M0, M1):

w = uau + vav.

Now we can consider the element wd

as an element of H 0mg)n (MO, Md). The relation
wd = f (u, v) holds in H omOQ(M0,Md) as well. This shows that the elements a“

and av in EndOQUl/IO) satisfy the relations of the Clifford algebra C f. So we get a

36

 

homomorphism:

X : Cf —i EndOQUVIo)

Finally, recall that PCL(N) acts on EndOQM/IO), as was shown in the beginning
of the chapter. Since w is invariant under this action, it follows that an and av are also
PGL(N)-invariant. Hence they give global sections of A that satisfy the relationships
of the Clifford algebra; and that means that we have a map if) : C f —+ H 0(A). C]

Now we want to prove the following proposition, which will be used in the proof

of the main theorem.

Proposition 3.0.17. Let :1: E U be a closed point, and ix : Spec k —i U the inclusion.
The pullback i;(7,b,A) is an rd-dimensional representation of C f that corresponds to

the vector bundle E over C defined by the point :7: under Van den Bergh’s correspon-

dence. (See, Lemma 2, [24].)

Proof. Note that any closed point :r E U can be lifted to a closed point y E Q. We

have the following diagram:

Spec k(y) = Spec k(:r)

il in

Q -——i U

Recall that q : (2 —i U is the good quotient map by the action of PGL(N). Hence,

we have
iii-A = (q 0 iy)*A
’5 i;q*A
E” iggnd(7r*£')

a: 8nd(i;7f*5) = 5nd((7i=i5lyl

37

Since £nd(7r...€) is an endomorphism bundle of dimension (rd)2, i;A also has dimen-
sion (rd)2. Note also that i;(au) and i;(av) satisfy the relations of the Clifford

algebra. This proves that i;A is an rd—dimensional representation of Cf.

We claim that this is the same representation that corresponds to the point :7:
under Van den Bergh’s correspondence. Recall the construction of a representation
of C f from a stable vector bundle E over C with rank r, degree r(d + g — 1), and
H 0(E(—1)) = 0: Consider the direct image q...(E). This is a trivial vector bundle
of rank rd on P1 by Proposition 3.0.10. Its associated graded module over k{u, v]
is $711 k{u,v], and this is also a graded module over k[u,v,w]/(wd - f(u,v)). The
action of w gives two matrices in Mrd( k) satisfying the relations of the Clifford algebra
and hence a map of algebras C f —> rd(k)- Conversely, let cf) be a representation of
C f. Consider the two matrices ¢(u) and ¢(v). The associated graded module of the
trivial vector bundle of rank rd on P1 over k{u, v] is ®rd k[u, ii]. We can define an
action of w given by the images of the generators of the Clifford algebra, i.e. w acts
as ugb(u) + v¢(v) on ®rd k{u, v]. This makes Q") k[u, v] into a graded module over
k[u, v, w] /(wd — f). In this way, we get a stable rank r vector bundle E on the curve
C, such that the degree of E is r(d + g — 1) and H0(E(—1)) = 0. (See Section 1,
[24])

Recall that the two sections on and av are defined by the action of w on the

OQ[U, v, w]/(wd — f )-module F...(.7-') = 7r...£ (309 On[u, v]. Therefore, the two sections

(an) and i;(av) are determined by the action of w on the k(y)[u,v,w]/(wd —- f)-

'*

zy
module F...(.7:) @09 k(y).

N 0w we have

13:0“) 8‘09 My) 9’ (7&5 8109 Gel”, vl) €09 My)

“2

(may 811(3)) k(y)lui ’0]

38

as graded k(y)[u, v, w] / (wd — f )-modules and (mg )3), which is the restriction of 77.8
to the closed point y E Q, is a vector space of dimension rd. So we have F...(.7-) @09
lr(y) '5’ Sid HIM". ”Ul-

Recall that any vector bundle corresponding to a closed point y E Q is such that
its direct image under q : C —i P1 is trivial of rank rd. So the isomorphism class of 83)
is determined by giving an action of w on the k(y)[u, v]-module $rd k(y)[u, U] such
that wd = f (11., ii). Since the two iii-actions agree, the corresponding vector bundles

are isomorphic. I]

We want to finish this chapter by proving that the U -representation (2]), A) is an

irreducible C f-representation as defined in Section 2.7.

Proposition 3.0.18. The pair (1]), A) is an irreducible U -representation of dimension

rd of the Clifford algebra Cf.

Proof. First we have to prove that A is an Azumaya algebra. Using Corollary 8.3.6
of [15], we can cover U with étale maps pi : V,- -+ U and PCL(N)-equivariant maps
r,- : V,- X PCL(N) —> D such that the diagrams

V, x PGL(N) 3;. e

mi l7

V. LU

are cartesian.
Now it is obvious that the composition p,- o prl is flat. The pullback of A along

this map is isomorphic to the pullback of A along q 0 Ti. But we have:

(.1 0 am e T,*q*snd(7r.e)PGL(N)
9—: Tiltgnd(7r*£)

E En.d(r,*7r*£)

39

It now follows from Proposition 2.7.3 that A is an Azumaya algebra. The fact that
the dimension is rd follows from the fact that the pullback of A along the quotient
map It —i U is a rank rd vector bundle. Finally, the map 10 : Cf —> H0(A) was
constructed in Theorem 3.0.16 and irreducibility follows from Proposition 3.0.17 and

Lemma 2.7.7. CI

40

Chapter 4

The Moduli Problem

As stated in the introduction, Procesi proved (see Theorem 1.8, Chapter 4, [21])
that the functor Reprd(Cf, —) is representable. In this section, we want to prove
the main theorem; which states that Reprd(Cf, —) is represented by U and the U-
representation (7,1), A) as defined in the previous chapter. Recall that U is defined to
be the open subset of the moduli space M(r, r(d + g — 1)) consisting of stable vector
bundles E over C such that H 0(E(—1)) = 0. These correspond to the irreducible
representations of Cf. See [24]. We constructed a sheaf of Azumaya algebras A
on U by considering the direct image 7r*8 of the vector bundle 8 on C X D under
the projection map 7r : C x Q —i Q and then considering the action of PCL(N) on
8nd(7r...8). Taking the quotient gives us A. In Theorem 3.0.16, we constructed an
algebra homomorphism 7/2 : Cf —-i H 0(A) and in Proposition 3.0.18 we proved that
this makes the pair (w,A) into a C f—representation. In this section, we will prove
that this representation is the universal representation for the functor ”Reprd( C f, —).

If S is a kescheme, then by an S -representation of dimension n of C f we mean a

2 over S and

pair (11), CA), where O A is a sheaf of Azumaya algebras of dimension n
7,!) : Cf —» H 0(S, O A) is a k-algebra homomorphism. Two S-representations (ilil, 0A1)

and (1220/12) are called equivalent if there is an isomorphism 6 : 0A1 —i 0A2 of

41

sheaves of Azumaya algebras such that 7,02 = H 0(S , 6) o it’l-

We call an S-representation of C f is called irreducible if the image of C f generates

(9 A locally.

Let Repn (C f, —) be the functor that assigns to a k-scheme S the set of equivalence
classes of irreducible S-representations of degree n of C f. Since Azumaya algebras
pull back to Azumaya algebras and irreducible representations are stable under pull-

back, it follows that Repn(Cf, —) is indeed a functor.

It is known that this functor is representable in Sch] k. See, for example, Theorem
4.1, [25]. Our goal in this section is to identify the scheme which represents it. Recall
that we have the open subset U of M(r, r(d+ g — 1)) before and the sheaf of Azumaya
algebras A = 8nd(7r..£)PGL(Nl on it. We will prove that this pair represents the
functor Reprd(Cf, —); and here we use the morphism 7]) defined and proven to exist

in the previous section. (See Theorem 3.0.16 and Proposition 3.0.18.)

We also consider representations of C f into endomorphism sheaves of vector bun-
dles. Let g,d(Cf, —) be the subfunctor of Reprd(Cf, —) that assigns to a k-scheme
S the set of equivalence classes of irreducible S-representations into endomorphism
sheaves of vector bundles of rank rd. Again, since endomorphism sheaves of vector
bundles pull back to sheaves of the same kind, it follows that this is also a functor.
It is not a sheaf (with respect to the fppf topology); however, it can be proven that
its sheafification Q_,.d is isomorphic to Reprd. (See Lemma 4.2, [25]). This fact will

be useful to us later when we prove the main theorem.

Let r) : C f —-> Mrd(K) be a representation of C f, where K is a field extension of
k. Denote the images of the two generators of C f under 77 by a; and 07,. Consider

the morphism:

42

K[u,v, w]
(wd _ f(u, 1)))

u. v H uIrd, vITd

 

SK = -’ Mrd(Kluivl)

wi—iufifi+va—v.

With the natural grading on N = $rd K [u, v], the above morphism is a graded
homomorphism. This makes N into a graded S K-module. So N is a (coherent) sheaf
on X = Proj SK = CK. Note that (qK),.N E @111 01Pl' (See the Introduction.) But

we can prove more:

Lemma 4.0.19. (Lemma 4.3, [13].) N is a rank r, degree r(d+g — 1) vector bundle
on X. This bundle is stable, and it has H0(N(—1)) = 0.

Proof. For the first statement, by Proposition 2.5.1, Vol. 4 Part 2, and Lemma 12.3.1,
Vol. 4 Part 3, [8]; it is enough to prove the statement for N with the assumption
that K is algebraically closed. We will prove that for any closed point a: E X,
dim K(N 830 X1]: K) = r. By the usual upper semicontinuity argument, this is suf-
ficient. Furthermore, it is obvious that u and v cannot be both in a homogeneous
maximal ideal of S K- So it is enough to prove the dimension condition above for any
closed point a: in Xv = Spec (SK)(

.0), because the argument is the same for X 7,. Now

 

we have:
_ K[fi. 75]
(5K)“) “ (ii—id — f(fi. 1))’
and
N(,,) = 69 K[u].
rd

Here, 5 = u/v and a = w/v. 5 acts in a natural way and a acts as uau + '53.

Since K is algebraically closed, any closed point in Spec (S K)(,,) can be written as

43

m = (E — aw — b) for some a. b E K. So we have:

K[u, 177] )
(ed — f(a, 1)) (HE—b)

IIZ

 

(

OSpcc (SA/)(l’TI’

and:

(N(v))l' g 03 8“(51")(v) (a; Kl-fil)
T

Using these, we get:

$rd K[El g $rd K
(’17. — (1,57 - b)($rd Klfil) (“W + a? — (MGM K)

 

 

(FV )1: 8'10qu K 2

So the required dimension is dim K(ker(aa—u + (TU — b)). We have to prove that this
dimension is equal to 7'. Let us assume that f (a, 1) 7t 0 at first. In this case, there
are exactly d points (a, b) such that bd = f (a, 1). Over each of these points, the rank
of the stalk (N);- ®0X£ K is at least r by upper semicontinuity. Since they must
add up to rd, each of them must be equal to r.

Consider a5; + E} E Mrd(K). We compute the dimension of its eigenspace of
eigenvalue b. The characteristic polynomial of a5: + 575 is trd — f (a, 1). This follows
from the fact that when f (a, 1) sé 0 all the roots are distinct (remember our initial
assumption that char(K) does not divide d); and if f (a, 1) = 0, then the matrix is
nilpotent. Also, if b 75 0, then f (a, 1) ¢ 0 and b is an eigenvalue of multiplicity 1.

Next, let b = 0. Then we have f(a,1) = 0. We can find a matrix B E Cer(K)
such that B(aa—u+'(YJ)B’1 is in Jordan form. If dimK(lcer(aEI+ Ft? — b)) > 1, then

we can write, for w E Allrd(K[u,v]), det w = det 8th"1 = (av — u)ldet w], where:

44

(av—u.. . ..\l

Cl’U—U.

BwB—1= . .. . 1.... w-

 

 

( . . . . . . . . 1 )
There are l diagonal entries (av — u) in the above matrix, and l 2 r—l— 1. But then, we
have det wd = (av — u)lddet(w,)d = f(u, v)d. But we immediately see that (av — u) is
a repeated factor of f (u, v) with multiplicity at least I Z 2, which is a contradiction.

So the required dimension condition is proved.

For the second part, consider the projection qK : CK —> Pk. Then we have

x(CK, N) = x(IP’}{, (qK)...N), and by Riemann-Roch:
r(i — g) + degW) = rd<1— 0) + dammit).

But (qK)...(N) E 69rd 0P1 and so its degree is 0. This gives us deg N = r(d+g— 1).
K

For the statement that h0(CK, N (—1)) = 0, note that we can use the projection
formula to get:

h0(CK. New) = hoes, (mm—1)) = 0.

Lastly, we have to prove that the vector bundle N as constructed above is stable.
For this, we follow the discussion in [24]. As in the introduction, we have a vector

bundle N on C such that (qK)...N E 017;]. We will make use of the following formula:
[C

45

d€9((qri')eN)/Tk((QK)eN) = deg“) 1:]:ng + 9 — 1)

1deg(N) 1 - g
= — ~ +
d rk(N) d

 

(4.0.1)

 

— 1 (4.0.2)

Suppose N is strictly semistable. Then let .7 g N be a subbundle such that
deg(f)/rk(.7-') = deg(N)/rk(N). It follows from a formula similar to 4.0.1 that
deg((qK)...f') = 0. Recall that since (qK)...7-' is a subsheaf of the torsion-free sheaf
(q K)...N E 01;: it is torsion-free itself and hence is a vector bundle. See Lemma 5.2.1,
[15]. By Theorem 2.2.6, it is a sum of line bundles on the projective line. Hence,
((1K)...}' E $§=10P1(n,-), where Zn,- = 0. Now note that H0((qK)...N) = 0 and
hence HO((qK)....7-') = 0 as well. Hence, n,- g 0 for all i, and since Zn, = 0, we
have n,- = 0 for all i. So we have (qK)...7: E 03,1. It follows that the corresponding

representation is reducible, which is contrary to our assumptions. This finishes the

proof. I El

Next, we want to prove a relative version of Lemma 4.0.19. Let S be a k-scheme,
and let (4!, 0) be an element of g,d(Cf, S) so that O = EndOS(ES) for some vector
bundle ES of rank rd on S. We will first construct a rank r vector bundle on C X k S.

So consider the graded sheaf homomorphism

05(71, ii. ’11)]

—_—’ TI. 5 II. I)
w,_f(u,v) 8 de )1 . 1

11.1} H ”(LU

u: i—i iii/r(d?) + iii/)(y)

where x and y are the standard generators of C f. This is a graded homomorphism

of sheaves of 03hr, v]-algebras because the degree of u and v is 1 on the right side. We

46

can view the right hand side of the above morphism as £nd05(53 (805 05[u, v]). So
it allows us to view £58303 03hr, v] as a sheaf of graded (05hr, v, w])/(wd—f(u, v)))—
modules. Since C xk S E Proj(05[u,v,w])/(wd — f(u,v)), we get a sheaf M over
C Xk S. For the rest of the section, for any point 3 in S, qs denotes the morphism
C xkk(s) —> Pi“) induced by the inclusion k(s)[u, v] -—> (k(s)[u, v, w])/(wd—f(u, v)),
p3 denotes the projection of the second factor of P1143) onto Spec k(s), and us denotes
the composition pS o (13. We use a similar notation for an arbitrary field K instead of

k(s).

Lemma 4.0.20. The sheaf M is a rank r vector bundle on C xk S of fiberwise
constant degree of r(d + g — 1). Moreover, for any closed point 3 E S, the rank r
vector bundle M3 = M (805 Spec k(s) on CNS) is stable, has degree r(d+g -— 1) and
satisfies h0(Ck(s),M3(—1))= 0.

Proof. It is sufficient to prove these assertions when S is affine. So let S = Spec R.
Let CR = C Xk S. Assuming further (without loss of generality) that 8 is trivial on
S, we have H0(S,ES) E $rd R, because 8 is a vector bundle of rank rd. In this
case, let us denote the graded (R[u, v, w]/(wd — f(u, v))-module H0(S,€) ®R R[u, v]
by M and the corresponding sheaf M by M.

Note that M is flat over S and that 77 : C X S —i S is a flat morphism. So by
Lemma 12.3.1, Vol. 4 Part 3, [8], it will be enough to prove that for any 3 E S, Ms
is a stable, rank r vector bundle on S. Let M3 = (1 X is)*M, where 1 X is is the
morphism C3 = C X k k(s) ——> C Xk S. Consider the representation associated to the
point 5 via ”(4’):

1,03 : Cf —i Alrd(k(s)).

Using Lemma 4.0.19, we obtain a sheaf N which is isomorphic to Ms. So by Lemma
4.0.19 we know that M3 is a rank r, stable vector bundle. Also, along the fibers of

the projection onto the second factor, the degree is r(d + g — 1).

47

The second assertion follows from Lemma 4.0.19. El

Let S be a k—scheme and V be a vector bundle on S. Recall that V[u, 17]“ defines
a coherent sheaf on PIS. It can be shown that pg~(V) is isomorphic to V[u, v]"’. (See,
for example, Exercise III-47, [6].) We prove the following lemma which will be used

in the main theorem.
Lemma 4.0.21. We have F...(V[u,v]"’) E V[u. v].

Proof. Using the definition of F...(V[u, v]~), we get

I‘e(Vl’uivl~) = $(rs)i((VIuivl~)(n))

7162

a €B(Ps)*(P§(V)(n))

nEZ

g ® V @(ps)*(OP1(nll
nEZ S

g V®P*(Olpl)
S
E V®Os[u,v]

E V[u, v].

We are now in a position to prove the main theorem of this section:

Theorem 4.0.22. (M ain. Theorem) Any given degree rd irreducible S —representation
(111,0,4) can be obtained as the pull-back of the representation (1]), 8nd(7r*8)PGL(N)) by

a unique map f : S —+ U. In particular, (7b,A) represents the functor Rep,d(Cf, —).

Proof. It is already known (see Theorem 4.1, [25]) that the functor Reprd(Cf, —)
is representable. Let T denote the scheme that represents it and let (\II,B) be the

universal representation. Since we have a U -representation (1]), A), we obtain a unique

48

map a : U —i T such that (1]), A) E a*(‘Il, B). We will construct another map
B : T —> U and prove that a and B are inverses of each other.

Let (¢,£nd(£s)) be a degree rd S—representation in g,d(cf, —). Using Lemma
4.0.20, we can construct a vector bundle M on C x S such that for any point 3 E S,
M3 is stable, has rank r and degree r(d + g — 1) and H0(M3(-1)) = 0. Recall that
M(r, r(d+g—1)) is a coarse moduli space of vector bundles and U Q M(r, r(d+g—1)).
So using the coarse moduli property, we obtain a map f : S —i M(r, r(d + g — 1))
whose image lies in U.

Since it will be necessary to use the vector bundle 8 on C X (I later in this proof,
we use the local lifts Of f to I). Using the local universal property of 0, Definition
2.4.3, we see that S can be covered by Zariski Open sets S,- such that the restriction f,-
of f to S,- can be lifted (not uniquely) to Q. In other words, we get maps 9,- : S,- —i Q
with f,- = q o 9,. (Recall that q : (I —-i U is the good quotient map.) These maps

satisfy (id X g,)*£ E M,- on S), where M,- is the restriction of M to C X S,:

 

S,- — Si
...] in
Q i»

We claim that g; (x,£'nd(7r...£)) is equivalent to (see Definition 2.7.4) (¢,End(85i)),
where 852' is the restriction Of 83 to C X 5,. To prove this claim, we will make

extensive use of the following diagram of maps:

49

On top of this diagram, we have the vector bundle M,- constructed over C X S,- as
in Lemma 4.0.20 and the bundle 8 on C X It as in Grothendieck’s theorem 2.4.2. In
Lemma 4.0.20, M,- was defined to be (8375 (80% 052.[u, v])~, which was viewed as a
graded (051. [u, v, w])/(wd — f(u, v))-module. We can see from this that (QS,)*Mi E
ESi[u, 77]”, viewed as a graded 052.]71, v]-module. Next we compute F...(M,-). Recall

that this is a graded (031. [u, v, w])/(wd - f (u, v))-modu1e. Then we have

PM.) = EB(PS,-)e(qs,)e(Mi(n))

nEZ
e EB(PS,)e((qs,)e(Mil 690,... 0e}, W)
nEZ Si i

g P*((QS,-)*Mi)
g P*(£Si [11, YIN)

2 £57 (..., 7)].

Here, the last line follows from Lemma 4.0.21. Similarly, we compute F...((idC x

. g,)*£). Recall also that this is a graded (052. [77, v, 117]) /(’(1}d — f(u, v))-module.

Fe((idc x are) = EB M(idc x dram)
nEZ

g $(p3i)*((IS,-l*(((idC X 111)?) ®0Cx5i OCXSi(n))
nEZ

50

e ®1ps,).(<qs,).<idc x 97)”? 690,1 Gel, ("ll

nEZ Si i
E @(psi)s((idfis1 xgi)*(QQl*£ 8'0 1 Opl ("ll
nEZ Si Psi Si
E @(psi).((idpg X 9i)*(P12)*(7re5) 690?, 0111145,.(nll
nEZ 1 Si 1 z
E $(ps,)e((ps,)*gf(7re5) @014 Opals ("ll
nEZ Si 2
’5 9:05:67 ® @(PS,)*(OP1 (71))
nEZ Si
g 9;(7T*8) 8 F*(Opl )
52'

E 9? (ref) 8 Os, In. vl

9-" g;(7r*8)[u, “I-

Since M,- and (Mg X g,)*£ are isomorphic as sheaves, it follows that I‘*(Mi) and

F...((idC X g,)*8) are isomorphic as (051. [u,v,w])/(wd — f(u, v))-modules.

Recall that the representation (d,£nd(85’i)) is defined by the action of w on
551' [71, v] and the representation g; (x, 8nd(7r...£')) is similarly defined by the action of
w on g;(7r,..8)[u,v]. Now let 7) : I‘...(M,i) E 832'[u, v] —> F..((idC X g,)*£) E g?7r...8[u,v]
be an isomorphism of (051. [u, v,w])/ (wd — f (u,v))-modules. Let w act on 851' by

up}, + vgbj, and on gz‘rrt‘) by 11.33 + veg. SO we have

r}(ui) = 71777

that is,

dune}. + viii) = (net + visa)

Comparing the u and v components, we see that 7195111774 = 95,2, and gain-1 = 3,2,.

Hence the two representations (¢,£nd(85i)) and g," (x,8nd(7r...8)) are equivalent.

51

N ow we have

(a o f,)*(\Ii.B) a f,*(ili,.4) (4.0.3)
E g{q*(i/2,A) (4.0.4)
2 g;(x,£nd(7r.£)) (4.0.5)
2: (d, 8nd(55i)). (4.0.6)

Next, define morphisms Of functors as follows:
grd(Cf, S) —i HomSCh/k(S, U) —> HOmSCh/k(S,T)

by sending (qt, £nd(£3)) to the map f defined as above first, and then by sending f
to the composition a o f. Hence the second map is induced by a : U —i T. Recall
that we are using the flat topology on (Sch / S). Since the sheafification of g,d(Cf, —)
is Reprd(Cf, —-), and H 0mSch /k(—,T) is a sheaf; we get morphisms of sheaves as

follows:
Reprd(Cf, —) ’5 HomSCh/k(-,T) —* HO‘mSCh/k(—,U) —* HomSCh/k(—,T). (4.0.7)

This sequence induces a sequence of morphisms of schemes T —i U ——> T; and any
(ab, £nd(£S)) is mapped to a o f by this composition. Recall that the second map is

07, and we will call the first map B.

Next, let S be a k-scheme and (7, g ) be an rd—dimensional irreducible S-representation
of C f. Since g is Azumaya, by Proposition 2.7.3 we can cover S by flat maps S, —i S
such that the pullback of ('7, Q) is of the form (3, £nd(£3i)) for some vector bundle
852' on S). This gives us a map f,- : S,- —> U as stated earlier in the proof. Then

we can cover S - b Zariski open subsets S j - over which we can lift the restrictions
7. y 7,]

52

fi,j : SM —> U to Q. Following the formula 4.0.3, we see that the pullback of the
universal representation (\IJ, 3) along a o fw- is isomorphic to (oz-J, £nd(£Siaj )). This
means that given any section of the sheaf Reprd(Cf, —) corresponding to an irre-
ducible rel-dimensional representation (7, 9) over a k-scheme S, there is a flat cover
of S by maps Sid —> S such that the morphism of functors defined in the formula
4.0.7 maps the restriction of (7,9) to Sm- to the unique map Sm- —> T that pulls
the universal representation (‘11,8) back to (7,9). Since HomSCk/k(—,T) is a sheaf,
this proves that the composition in 4.0.7 is the identity. In other words, a o [3 is the
identity.

Since a- 0 ,3 is the identity on T, and since (111,11) pulls back to (‘11,3) under fl, it
is clear that (11),.4) must pull back to (\II, 3) under S.

We claim that [3 is the inverse of a. We only need to prove that the composition
3 o a : U —> T —+ U is the identity on U. It is clear that the U -representation (11), A)
pulls back to itself under 5 o a.

Denote 6 = fl 0 a : U —> U for brevity. We claim that 6 maps a closed point
:1: E U to itself. To see this, consider the composition 6 0 2'3; : Spec [C(33) —i U —-> U.
Then 12015.4) is the irreducible rel-dimensional representation of C f corresponding
to the closed point x by Proposition 3.0.17. Let y = 6(1‘), it is clear that the inclusion

morphism of y is 6 0 ix. We have

(6 0 mm. .4) 2 am, A)

i$(w.A)-

IIZ

Again, by Proposition 3.0.17, y must be the closed point corresponding to the rd-
dimensional representation i;(w,.A). But this is the point :5. It is now clear that
the map 6 is the identity on closed points. Since U is a variety, it now follows from

Lemma 2.1.1 that fl 0 a is the identity map on U. This finishes the proof. El

53

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