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COHEN-MACAULAY BLOWING-UP ALGEBRAS
AND CONSTRUCTIONS IN LINKAGE

presented by

Mark Ray Johnson

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MSU Ie An Affirmative Action/Equel Oppommlty Inetltuton
Wan-w

COHEN-MACAULAY BLOWING-UP ALGEBRAS
AND CONSTRUCTIONS IN LINKAGE

By

Mark Ray Johnson

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1995

ABSTRACT

COHEN-MACAULAY BLOWING-UP ALGEBRAS
AND CONSTRUCTIONS IN LINKAGE

By

Mark Ray Johnson

In this work, we study the Rees algebra and the associated graded ring of an ideal
in a local Cohen-Macaulay ring. For ideals with small enough reduction number
and sufficiently good residual intersection properties, we show that these blowing-
up algebras are Cohen-Macaulay and we compute the number and degrees of their
defining equations.

It is shown that a power of an ideal (locally a complete intersection in codimen-
sion one) coincides with its symbolic power whenever they coincide after deforma-
tion of the ideal, if the symbolic power defines a Cohen-Macaulay ring.

We give various constructions, including the tensor product of algebras of finite
type over a field and the sum of two geometrically linked ideals, which produce
Cohen-Macaulay ideals which are either strongly Cohen-Macaulay (or the entire
linkage class has this property), strongly nonobstructed, or both, but are not in the

linkage class of a complete intersection.

ACKNOWLEDGMENTS

I am very grateful to my thesis advisor Bernd Ulrich in ways that are too
numerous to mention. I have enormously benefitted from his guidance, both
mathematically and otherwise. It has been a great pleasure to study with him.

I would also like to thank William Heinzer, Craig Huneke, and Wolmer Vascon-

celos for their support and encouragement over the past few years.

iii

TABLE OF CONTENTS

INTRODUCTION .................................................................................................. 1
CHAPTER 0: PRELIMINARIES ......................................................................... 11
CHAPTER 1: COHEN-MACAULAY BLOWING-UP ALGEBRAS ..................... 17

1.1 Artin-Nagata properties

1.2 Cohen-Macaulayness of the associated graded ring
1.3 Number of defining equations

1.4 Expected reduction number

1.5 Analytically independent elements

CHAPTER 2: SYMBOLIC POWERS AND DEFORMA’I‘IONS .......................... 55
CHAPTER 3: CONSTRUCTIONS IN LINKAGE ................................................ 65

3.1 Licci ideals

3.2 Tensor products

3.3 Invariants of geometric linkage

3.4 Sums of links

3.5 Intersections of complete intersections

BIBLIOGRAPHY ................................................................................................. 96

iv

INTRODUCTION

Let R be a noetherian local ring and let I be an R-ideal. The Rees algebra
’R = R[I t] and the associated graded ring G = gr1(R) are two graded algebras
that encode various algebraic and geometric properties of the ideal I . For example,
Proj(’R) is the blow-up of Spec(R) along V(I) and Proj(G) corresponds to the
exceptional fiber of the blow-up. It is particularly interesting to know when these
“blowing-up algebras” ’R and G are Cohen-Macaulay. Other than being important
in its own right, this property often facilitates the study of various other properties
of these algebras, such as their normality ([9]), the depth of their graded pieces
([17]), or the number and degrees of their defining equations ([46],[5], or Section

1.3).

The relationship between the Cohen-Macaulay property of ’R. and G is fairly well-
understood: Huneke observed some time ago that G is Cohen—Macaulay whenever
’R is (at least when R is Cohen-Macaulay and I is not nilpotent), but recently
various criteria have been found for the converse to hold as well (e.g. [46], [5], [61],
[49]). This shifts the focus of attention, at least in principle, to the study of the

Cohen-Macaulayness of G.

Using their so—called approximation complexes ([27]), Herzog, Simis and Vascon-
celos established the Cohen-Macaulayness of the associated graded ring of any ideal,
in a local Cohen-Macaulay ring, satisfying sliding depth and the condition Goo. (We

refer to Chapter 0 for the definition of the terminology.) At around the same time,

1

2
Huneke ([32]) proved a similar result, using his theory of d-sequences. These works
demonstrated a principle that the Cohen-Macaulay property of blowing-up alge-
bras is related to Cohen-Macaulay properties of the Koszul homology of the ideal.
Other than just satisfying the condition Goo, these ideals have an even more notable
property: their Rees algebras coincides with their symmetric algebras; ideals with
this property were then said to be of linear type. Until fairly recently, relatively
little was known about the Cohen-Macaulayness of blowing-up algebras of ideals
which are not of linear type (except for equimultiple ideals, which were studied by
Sally (e.g. [57], [58]) sometime earlier; see [24] for a comprehensive treatment of

this case).

One approach, which goes back to a classic paper of Northcott and Rees ([54]),
is to pass to a minimal reduction J of I and study the finite extension of Rees
algebras R[Jt] C R[It]. Philosophically, the reduction J is “simpler” in that its
minimal number of generators is no worse than that of I, being at most the analytic
spread [ of I (at least if the residue field of R is infinite). The reduction number
1' is then seen as an important way to measure how far I and J differ. Of course,

this approach is fruitful only if r > O.

The recent work [30] and [31] of Huckaba and Huneke made successful use of
this approach in the first nontrivial cases: they proved the Cohen-Macaulayness
of G (and ’R) for ideals with r = 1 and having analytic deviation one and two.
This work quickly inspired many others: under various additional assumptions
(see Section 1.2 for precise statements) G was shown to be Cohen-Macaulay by
Vasconcelos ([74]) and Ulrich ([68]) when r S 1, by Goto and N akamura ([20],[21]),
Aberbach and Huneke ([4]), and Aberbach, Huckaba and Huneke ([3]) when r S 2

and the analytic deviation is at most two, by Aberbach and Huckaba ([2]) when

3

r S 3 and the analytic deviation is at most two, by Simis, Ulrich and Vasconcelos
([61]) when u(I) = 3 +1 and r S 2 — 9 +1 (where g = grade I), and by Tang
( [63]) when r S E — g + 1 (and sufficiently many powers have high depth). One of
our main results is the following theorem which includes essentially all of the above

results as a special case (subsequently, futher generalizations were also obtained by

Aberbach ([1]) and Goto, Nakamura, and Nishida ([22],[23])):

Theorem 1.2.8 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R-ideal with analytic spread I and reduction
number 1‘, let k 2 1 be an integer, and assume that r S k, that I satisfies sliding
depth and Goo locally in codimension I — 1, that depth R/ I j Z d —- E + k — j for
1 Sj S k, and that I satisfies ANG—max{2,k}'

Then G is Cohen-Macaulay.

The condition “AN ,’ ” in the statement of the theorem denotes the requirement
that certain residual intersections of I are Cohen-Macaulay. This property was
first observed for ideals satisfying sliding depth ([28], [36]) and now better explains
the philosophy involved here: “The Cohen-Macaulayness of a blowing-up algebra
is related to the Cohen-Macaulayness of the residual intersections of the ideal.”
Indeed, our main technique is to exploit the residual intersection properties of the
ideal, which facilitate the computation of various intersections and ideal quotients.
Although most of the previous works made no explicit reference to residual intersec-
tions, they often made at least implicit use of these residual intersection techniques.
We will study these so-called Artin-Nagata properties in Section 1.1. Section 1.2 is

devoted to the proof of Theorem 1.2.8.

Once one knows that the blowing-up rings are Cohen-Macaulay, one can ask

4
about the nature of their defining equations; we study this question in Section 1.3.
Of course, if I is of linear type, then R, being a symmetric algebra, has its equations
completely determined by a presentation matrix of I. In general there will be higher
order relations A, namely elements in the kernel of the natural surjection from R
to the symmetric algebra of I, which are usually difficult to determine explictly.
However, one could at least ask about the number and degrees of these relations.
The latter question was answered under the assumptions of the works [30] and [61].
Similarly, under somewhat stronger assumptions than in Theorem 1.2.8, we are able

to compute the number and degrees of the defining equations:

Theorem 1.3.3 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R-ideal with analytic spread E and reduction number
r and assume that I satisfies G.>0 locally in codimension E — 1, that SJ-(I) E I j and
depth R/Ij Z d — 8 + r —j whenever I S j S r, and that I satisfies ANF—z-

Then A is minimally generated by (22:21) forms of degree r + 1.

In particular, A is cyclic if in addition I has second analytic deviation one. In
some special circumstances, one can find explicitly the generator of A: Vasconcelos
studied this problem for perfect ideals of grade 2 ([73]), in which case the required
equation is essentially the determinant of a J acobian dual of a presentation matrix.
This cannot hold for ideals of larger grade simply by degree reasons, but we are
able to show an analogous result for perfect Gorenstein ideals of grade 3 which
satisfy the so—called row condition ([4],[61]) (we later learned that a similar result

was shown by S. Morey ([52]) in case I has linear presentation):

Theorem 1.3.7 Let R be a local Gorenstein ring with infinite residue field, let

I be a perfect Gorenstein ideal of grade 3, with analytic spread 8, minimal number

5

of generators n = 3 + 1, assume that I satisfies G.>0 locally in codimension E — 1,
let ()5 be an n by n alternating matrix presenting I with last row (—~:rl, ..., -$[,0)
which generates the ideal of entries of qfi, let 1,!) be the I? by 8 alternating submaxtrix
of (I) obtained by deleting the last row and column, let 1,15 denote the jth column
of 1/2, for 1 S j S 8, write 1,12]- : Aj(g:_)‘, where AJ- is an E by 8 matrix whose jth
row consists of zeros and whose ith row, for any 1 S i S 8, is the negative of the
jth row of A5, let T1, ..., T( be variables over R, and let B be the matrix whose jth
column is A§(I)‘.

Then A is generated by F = T; 1X3(Tn), where X denotes the characteristic

polynomial of B in the variable Tn.

For ideals having second analytic deviation one, one can prove converses to the
results of Section 1.2 asserting that G and R are Cohen-Macaulay. This was done
earlier by Aberbach, Huckaba and Huneke ([3]), and by Aberbach and Huckaba
([2]) for ideals of small analytic deviation. The idea is that one can relate (n — 1)-
residual intersections of I to the ideal I1(¢) of entries of a minimal presentation
matrix of I. For example, one can show the following, which builds on one of the

main results of [61]:

Theorem 1.4.10 Let R be a local Gorenstein ring with infinite residue field,
let I be a strongly Cohen-Macaulay R-ideal of grade g 2 2, analytic spread 8, and
minimal number of generators n = E + 1 satisfying Goo locally in codimension Z — 1,
and assume that I C Il(¢)2, where (15 is a matrix with 72 rows presenting I. Then
the following are equivalent.

(a) After elementary row operations, 11(q5) is generated by the last row of ¢;

(b) I has reduction number S I - g + 1;

6
(c) A is generated by forms of degree S 6 — g + 2;
(d) A is generated by a single form of degree 6 — g + 2;
(e) R is Cohen-Macaulay;

(f) G is Cohen-Macaulay;

This result, which can be applied in particular to perfect ideals of grade 2 and to
perfect Gorenstein ideals of grade 3, is proved in Section 1.4 as well as other weaker
results requiring only sufficiently good Artin-Nagata properties rather than strong
Cohen-Macaulayness.

In Section 1.5 we make some remarks about ideals generated by analytically
independent elements, and take the opportunity to show that there exist Cohen-
Macaulay homogeneous prime ideals of grade 3 and deviation 3 in k[:c1 , ..., are] which
are locally generated by analytically independent elements but are not of linear type,

answering a question of Ulrich ([66]) in the negative.

Let I be a prime ideal in a regular local ring R. An important problem is to
determine when the power I ” of I coincides with its symbolic power I (n) (e.g.
[29],[33]). One might expect the symbolic power, being at least unmixed, to have
better depth properties than the ordinary power in general. For example, when
R/ I has dimension one, R/ I (n) is Cohen-Macaulay, a trivial but often useful fact
to know. One could ask in general: when is R/ I (n) Cohen-Macaulay?

In Chapter 2 we study this question in case R/ I has a deformation 5' / J for which
J (n) = J". (For example, any ideal in the linkage class of a complete intersection,
licci for short, in particular any perfect ideal of grade 2, or any perfect Gorenstein
ideal of grade 3, always admits such a deformation.) This problem is related to

asking when the property that the power coincides with the symbolic power is

7
preserved after specialization. In that sense, it is somewhat analogous to the study
of the arithmetically Cohen-Macaulayness of an algebraic variety via its hyperplane
sections. Indeed, our approach was motivated by the recent work of Huneke and
Ulrich ([42]) on that topic. Our result shows that in many cases symbolic powers
do not possess better depth properties (beyond being unmixed) than the ordinary

powers:

Corollary 2.4 Let R be a local Cohen-Macaulay ring, let I be an R-ideal of
height 9, assume that I is a complete intersection locallyin codimension g + 1, that
R/ I has a deformation 5 / J which is equidimensional and satisfies J (n) = J n for
some n, and that N") 75 I".

Then R/I(”) does not satisfy (52).

One application is the following analogue of an older result of Huneke and Ulrich:

If I is a licci ideal satisfying (C11) but not a complete intersection, then for every

n 2 3, R/ I (n) is not Cohen-Macaulay.

In Chapter 3 we study some constructions which produce examples of Cohen-
Macaulay ideals having certain specified properties with respect to linkage. We
are interested in the property of being strongly nonobstructed, which was proved
to be a linkage invariant by Buchweitz in his Paris thesis ( [12]), and the property
of being strongly Cohen-Macaulay, which was shown to be an invariant of even
linkage by Huneke ([34]). We have seen that the latter property is important in the
study of blowing-up algebras. On the other hand, the property of being strongly
nonobstructed plays a role in deformation theory: it implies that there are no
obstructions to lifting infinitesimal deformations ([12],[25]).

The best known examples of strongly Cohen-Macaulay and strongly nonob-

8

structed ideals are the licci ideals. Although a great deal is known about licci
ideals, there are otherwise relatively few known classes of strongly Cohen-Macaulay
and strongly nonobstructed ideals. This work arose in trying to better understand
how these properties are related. It was known for some time that-(perfect) strongly
Cohen-Macaulay ideals are not necessarily licci, while Ulrich ([67]) constructed ex-
amples demonstrating that the properties of being strongly Cohen-Macaulay (or
even that the entire linkage class enjoys this property) and strongly nonobstructed
do not imply each other. There was, however, no known example of a strongly
Cohen-Macaulay, strongly nonobstructed ideal that was not licci.

We will answer this question in a particularly natural way. After recalling in
Section 3.1 some of the basic machinery of linkage, developed mainly by Huneke
and Ulrich, in Section 3.2 we study the tensor product of two algebras over a field.
It is well-known that the tensor product inherits many good homological properties

from each of its factors. Despite this, however, we show the following:

Corollary 3.2.3 Let A and B be complete local licci algebras over a field k

which are not complete intersections and let G = A®kB.

Then G is strongly Cohen-Macaulay and strongly nonobstructed, but not licci.

This result demonstrates that homological conditions alone, like being strongly
Cohen-Macaulay or strongly nonobstructed, will never be sufficient to guarantee
that an ideal lies in the linkage class of a complete intersection. One obtains explicit
counterexamples in any codimension at least 4. We leave it as an open question
whether such an example can exist in codimension 3. The proof of 3.2.3, although
using a stande reduction technique of [38], is fairly technical (partly because we

prove the result in the complete case); a much simpler proof is available in case

9

neither A nor B is Gorenstein: in that case, the tensor product is directly linked

to an algebra which is not strongly Cohen-Macaulay (Theorem 3.2.2(a)).

In Section 3.2 we observe some new invariants of geometric linkage. It is known
that the depth of the twisted conormal module I @wR/I (the first Koszul homology
module H1(I), respectively) is an invariant of the linkage class ([13]) (the even
linkage class ([34]), respectively). It turns out that the depth of these modules
modulo their torsion submodule is an invariant of their geometric linkage classes

(of their even geometric linkage class, respectively).

In Section 3.4 we study the sum of two geometrically linked Cohen-Macaulay
ideals. Peskine and Szpiro observed that such an ideal is Gorenstein, and it turns
out to be an interesting way to construct ideals with the properties we are concerned
with. Kustin and Miller ([47]) used this construction in their study of Gorenstein
ideals of codimension 4 and Ulrich ([67]) showed that if I ~ J are geometrically
linked licci ideals then the sum I + J is also licci, giving a nontrivial way to obtain
new licci ideals from old ones. He also showed that if I is strongly nonobstructed
and H1(I) and H1(J) are Cohen-Macaulay, then K = I + J is strongly nonob-
structed (equivalently, the conormal module K / K 2 is Cohen-Macaulay, as K is
Gorenstein). Using this, and other results from [67], Ulrich constructed a codimen-
sion 5 Gorenstein ideal which is strongly nonobstructed but is not even syzygetic.
He does not obtain such an example as a sum of links however, as his assumptions

force K to be syzygetic.

We are able to construct many such examples directly as sums of links. To do
this, we needed to generalize Ulrich’s result to determine the precise conditions for

a sum of links to be strongly nonobstructed and to be syzygetic:

10

Corollary 3.4.6 Let R be a local Gorenstein ring and let K = I + J be a sum
of two geometrically linked Cohen-Macaulay ideals.

(a) K is strongly nonobstructed if and only if H1 (I)/1', H1(J)/T and (I®wR/1)/T
are Cohen-Macaulay.

(b) If I satisfies (CII), then If is syzygetic if and only if I and J are syzygetic,

and I <8) wR/I is torsion-free.

In fact we prove a more general result (Theorem 3.4.4) which gives the depth of
K / K 2 and the depth of H1(K) directly in terms of invariants of I and J. Using
the tensor product construction of Section 3.2 together with Corollary 3.4.6, one
may construct fairly general strongly nonobstructed Gorenstein ideals which are

not syzygetic.

We conclude in Section 3.5 with a naive construction, which is somewhat dual
to linkage, which produces Cohen-Macaulay ideals of type 2: given any Cohen-
Macaulay ideal I of codimension g whose deviation is at most g, intersect any
two complete intersections inside I of codimension g which together generate I.
Although this may not seem very promising, it actually turns out to be related to
a construction of Section 3.2 describing a link of the tensor product. We make use
of this construction to give a more “generic” example of a perfect homogeneous

ideal of codimension 3 and type 2 having a pure resolution which is not strongly

Cohen-Macaulay ([48] ).

CHAPTER 0

PRELIMINARIES

In this short preliminary chapter, we establish the notation and terminology that we
will use throughout this work. As general references for any undefined terminology,

we refer to [10] and [75].

0.1 Basic Invariants

Let R be a noetherian local ring with maximal ideal m and residue field It and
let I be an R—ideal, by which we always mean a proper ideal. The minimal number
of generators of I is denoted by 11(1); equivalently, V(I) = dimkI <8) R k. We let
g = grade I, and let d(I) = u(I) -— g be the deviation of I. If d(I) = 0 (respectively,
d(I) S 1), I is called a complete intersection (respectively, an almost complete
intersection). We will virtually always be working in a local Cohen-Macaulay ring.
In this case the codimension and height coincide with the grade.

If R is Cohen-Macaulay, the canonical module of R is denoted by w R (if it exists);

if R is Gorenstein, I has grade g, and A = R/ I is Cohen-Macaulay then one may
take em = ExtflA, R). The type of R is defined by (where d = dim R)

r(R) = V(wR) = dimk Ext‘f3(k,R).

11

1 2

0.2 Local Properties

We often say that I has a property ’P if R/ I has property ”P. For example, we say
that I is a Cohen-Macaulay R-ideal if R/ I is Cohen-Macaulay, and I is a Gorenstein
R-ideal if R/ I is Gorenstein. Similarly, we say I has type t if R/ I does. An ideal
is called perfect if it is Cohen-Macaulay and has finite projective dimension. Recall
that an ideal is unmixed if every associated prime has the same height.

Now let 3 be an integer. We say that I has a property ’P locally in codimension
3 if II, has property 7’ for every p E V(I) with dim RP S 3. If I is a complete
intersection locally in codimension s+ht I, we say that I satisfies (C I ,). An ideal
is generically a complete intersection if it is a complete intersection locally at every
associated prime; if it is unmixed, then this is equivalent to saying that I satisfies
(CID). We say that I satisifes the condition G , if its number of generators is at most
the dimension locally in codimension s — 1, i.e. if V(Ip) S dim Rp for all p E V(I)

with dim R, S s — 1. We say that I satisfies G00 if I satisfies G, for every 5.

0.3 Blowing-up Algebras

The Bees algebra of I is

R = R[It] '_—*: $11,

120

and the associated graded ring of I is

G = gr1(R) = R®RR/I a: @Ij/Ij“.

120

The analytic spread 8 of I is defined by
€(I) = dim R®R k = dim G®R k,

and satisfies the inequalities

l3
ht I S 8(1) S min{dim R,V(1)}.

One also defines the analytic deviation to be the difference L’(1)—ht I, and the second
analytic deviation to be V(I) — 6(1).

Burch ’s inequality ([15]) states that
infj depth R/Ij S dim R — 6(1);

however we will also use this to refer to the improved version due to Brodmann (e.g.
[24, 23.11]) which replaces the “inf” by “lim inf”. It is also the case that equality

holds in Burch’s inequality whenever G is Cohen-Macaulay ([17]).

Let 51(1) denote the jth symmetric power of I and let

5(1) = EB SJ-(I)

1'20
be the symmetric algebra of I. There is a natural surjective homomorphism
a : 5(1) —+ R; let A denote the kernel. If A = 0, equivalently if a is an iso-
morphism, we say that I is of linear type. Otherwise, we define the relation type,
rt(I), to be the maximal degree occuring in a homogeneous minimal generating set
of A. If 01 is an isomorphism in degree 2, equivalently if 52(1) 2 12, then I is said

to be syzygetic.

0.4 Reductions

An R-ideal J C I is called a reduction of I if R[I t] is a finite R[Jt]—module, or
equivalently if I"+1 = J I ' for some integer r 2 0. Denote the smallest such r by
r 1(1); it is the reduction number of I with respect to the reduction J. A reduction
J C I is called a minimal reduction if it is minimal with respect to inclusion among

all reductions of I. If the residue field I: is infinite, every minimal reduction is

14

minimally generated by [(1) elements; in this case we define the reduction number

r of I by
r(1) = min{rJ(I) I J is a minimal reduction of I}.

It holds that r(1) = 0 if and only if V(I) = K.

0.5 Ideals of Minors

If M is a matrix with entries in R, I¢(M) denotes the R-ideal generated by all t
by t minors of M. If M is an alternating matrix, and t is even, Pf¢(M) denotes the
R-ideal generated by all t-th order Pfaffians of M (obtained by deleting the same
rows and columns of M).

A matrix (respectively, an alternating matrix) M is said to be generic (over a
ring A) if its entries (respectively, its upper triangular entries) are indeterminants
over A. If M is a generic 12 + 1 by n matrix, then In(M) is a perfect ideal of grade
2, and if M is a generic alternating n by n matrix, and n is odd, then Pf.._1(M)

is a perfect Gorenstein ideal of grade 3.

0.6 Pairs

Let (R, I ) and (S, J) be pairs, where R and S are noetherian local rings, and
I C R and J C S' are ideals (possibly 1 = R and J = S). We say that the
pairs (R, 1) and (S, J) are isomorphic, and write (R, I) E (5, J), if there is an
isomorphism d) : R —> S with d(I) = J. The pairs are said to be generically
equivalent, written (R, I) z (5, J) if there are finite sets of variables X over R and
Y over S such that (R(X),IR(X)) g (S(Y),JS(Y)), where R(X) = R[leR[X]-

We say that (S, J) is a deformation of (R, I) if there is a sequence g = a1, ...,a,

in S, which is regular on S and S/J, such that (S/(g),(J,g)/(g)) E (R, I). Then

15

we also say that S / J is a deformation of R/ I (or even that J is a deformation of
I ), while the canonical surjection S / J —> R/I is called specialization.

We say that a pair (5, J) is essentially a deformation of (R, I ) if there is a
sequence of pairs (S;,J.-), for 0 S i S n, with (So,Jo) = (R,I) and (SmJn), such
that for all 0 S i S n — 1, one of the following conditions holds:

(a) (5.4.1, J§+1) is a deformation of (5;, J;);

(b) (S;+1,J.-+1) = ((S;),,, (J;),,) for some p 6 Spec 5.- ;

(C) (5i+1,Ji+1) z (Sink):

0.7 Koszul Homology

If a1,...,an E R, we let H.- = H;(a1,...,an) denote the ith Koszul homology
of the Koszul complex built on a1,...,an. If I = (a1,...,an), then we also write
H;(I) for H;, although this module depends on the generators a1, ..., on. However,
if R is Cohen-Macaulay, then the property that H i is Cohen-Macaulay in a range
0 S i S k, for some fixed 11:, is independent of the generating set of I . In addition,
H;(I) is an R/I-module, either H.- = O or dim H.- = dim R/I, and H; = 0 for every
i > n—ht I .

We say that I satisfies sliding depth if depth H .-(I ) Z dim R —— n + i for every 2',
and that I is strongly Cohen-Macaulay if H .-(I ) is Cohen-Macaulay for every i. The
class of strong Cohen-Macaulay ideals contains the class of licci ideals ([34]). If I
is strongly Cohen-Macaulay and satisfies G , then by [27, proof of 4.6] one has that
depth R/Ij Z dim R/I —j + 1 and that 51(1) 2:“ Ij whenever I Sj S s —g +1.

There is a natural exact sequence
111(1) —+ (R/I)" —> 1/12 ——> 0

induced by the first syzygies of I. It turns out that I is syzygetic if and only if the

16

first map in this sequence is injective. This implies that 111(1) is torsionfree as an
R/I-module, and the converse holds if I is generically a complete intersection.

Elements a1,..., an generating 1 form a d-sequence if

((a1,...,a,-) : (a;+1)) fl 1 = (01,...,a,-)

for every 0 S i S n — 1. Any ideal satisfying sliding depth and Goo is generated by

a d-sequence, and any ideal generated by a d-sequence is of linear type ([27]).

CHAPTER 1

COHEN-MACAULAY BLOWING-UP ALGEBRAS

In this chapter we study the blowing-up algebras R and G of ideals having good
residual intersection properties. In Section 1 we develop certain technical results
about residual intersections that we will make use of throughout the chapter. In
Section 2, for ideals having sufficiently many Cohen-Macaulay residual intersections,
we prove the Cohen-Macaulayness of R and G when the the reduction number is
sufficiently small. We study the defining equations of R is Section 3. In particular,
under conditions similar to those of Section 2 that guarantee that R is Cohen-
Macaulay, we compute the degrees and the number of defining equations of R. For
certain Gorenstein ideals of grade 3, we find the equation defining R explicitly in
terms of the presentation matrix. In Section 4, for ideals having second analytic
deviation one, we obtain partial converses to the'results of Section 2 by showing
that, under certain assumptions, the Cohen-Macaulayness of G forces the reduc-
tion number to be “small”. We make some remarks about ideals generated by

analytically independent elements in Section 5.

The results of the first two sections will appear in the joint paper [44], while the
material of the later sections will appear in [45].

17

 

18

1.1 Artin-Nagata Properties

We begin by defining the notion of a residual intersection, the properties of which

will play an important role in this chapter.

Definition 1.1.1 Let R be a local Cohen-Macaulay ring, let I be an R-ideal of
grade 9, let K be a proper R-ideal, and let 3 2 g be an integer.

(a) K is an s-residual intersection of I if there exists an R-ideal a C I with
K=a:1andhtKZsZz/(a).

(b) K is a geometric s-residual intersection of I if K is an s-residual intersection

of I and if in addition ht I + K > 3.

If R is Gorenstein, I is unmixed and s = g, then the notion of residual intersec-
tion corresponds to “linkage”, and geometric residual intersections correspond to
“geometric linkage”. Thus the study of residual intersections is a generalization of
the study of linkage.

It is particularly interesting to know when residual intersections are Cohen-
Macaulay. This problem was studied by Artin and Nagata ([7]), and more recently
by Huneke and Ulrich ([36], [40], [68]). The term “Artin-Nagata” , for this property,

was coined by Ulrich in [68].

Definition 1.1.2 Let R be a local Cohen-Macaulay ring, let 1 be an R-ideal of
grade 9 and let s be an integer.
(a) 1 satisfies AN, if for every g S i S s and every i-residual intersection K

of I , R/ K is Cohen-Macaulay.

(b) I satisfies AN 3' if for every g S i S s and every geometric i-residual inter-

section K of I , R/ K is Cohen-Macaulay.

19

Clearly the Artin-Nagata property AN 3‘ is weaker than the property AN ,. It
can be strictly weaker in that it can happen that an ideal admits s-residual inter-
sections but no geometric s-residual intersections (for s = g take any unmixed ideal
which is not generically a complete intersection). Of course, if s is sufficiently large
an ideal will not even admit any s-residual intersections. To avoid this type of triv-
ial obstruction, one usually assumes the condition G, of [7] which guarantees the
existence of s-residual intersections (and geometric (s — 1)-residual intersections).
The following two theorems are the major results known to guarantee that an ideal

has Artin-Nagata properties.

Theorem 1.1.3 (Herzog-Vasconcelos—Villarreal [28]) Let R be a local Cohen-
Macaulay ring and let 1 be an R-ideal satisfying G, and sliding depth.

Then 1 satisfies AN,.

Theorem 1.1.4 (Ulrich [68]) Let R be a local Gorenstein ring of dimension d, let
I be an R-ideal of grade 9, and assume that 1 satisfies G, and that
depthR/IIZd—g—j+l forlSsz—g+1. Then

(a) 1 satisfies AN ,;

(b) for every g S i S s and every i-residual intersection K = a : I of I,

“JR/K r___\_a i-g-t-l/aIi—g.

In particular the assumptions of Theorem 1.1.4 hold for a strongly Cohen-
Macaulay satisfying G,. In that case the result was first proved by Huneke
[36] (for geometric residual intersections).

We will need the following basic, but important, lemma. It states roughly that
if an ideal has a sufficiently good Artin-Nagata property, then one can build up any

residual intersection as a sequence of “links.”

20

Lemma 1.1.5 ([68, 1.7]) Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal of grade g satisfying G,, and let K = a : I be an
s-residual intersection or let K = R and V(I) S s.

(a) There exists a generating set a1, ..., a, of a such that for every g S i S s — l
with a.- = (a1, ..., a;), one has that K.- = a; : I is a geometric i-residual intersection
or K.- = R. Moreover, any permutation of a1, ..., a; enjoys the same property.

(b) If I satisfies AN,.:2 then a1, ..., a, forms a d-sequence.

(c) If 1 satisfies AN,“ for some t S s — 1 and K # R, then the following hold for
0 S i S t + 1:

(i) K.- =a,-:(a.-+1) and a;=K;flIforiSs—1.
(ii) depth R/ai = d — 1'.
(iii) K .- is unmixed of height 1'.

(iv) ht I = 1 where “ - ” denotes images in R/K;, for i S s — 1.

The next two lemmas show that the Artin-Nagata property is preserved after
performing certain standard operations. These are analogous to results observed
by Huneke [37] in the case of strongly Cohen-Macaulay ideals. We will state them

for the case AN 3‘ , but they also hold for AN ,.

Lemma 1.1.6 Let R be a local Cohen-Macaulay ring, let I be an R-ideal,
let a: e I be R-regular, let “ "‘ ” denote images in R/(r), and assume that 1

satisfies AN ,— .

Then 1‘ satisfies AN 3:].

Proof. Let (a;,...,a;) : 1‘ be a geometric i-residual intersection of 1", with

ht 1" S i S s — 1, and let K = (a1,...,a,-,:c) : I. Since K“ coincides with the

21

given residual intersection of 1", it follows that K is a geometric i-residual inter-

section of 1. Hence R’/(af, ...,a?) : 1‘ ’5 R/K is Cohen-Macaulay. Cl

Lemma 1.1.7 Let R be a local Cohen-Macaulay ring, let 1 be an R-ideal, let
“ 7 ” denote images in R/O : I, assume that 1 fl (0 : I) = 0, that I satisfies G1 and
that 1 satisfies AN ,- .

Then I satisfies AN 3’ .

Proof. We may assume that s 2 0. Since I I) (0 : I) = 0 and 1 satisfies G1, 0 : I is a
geometric O-residual intersection of 1, and R is Cohen-Macaulay as I satisfies ANO— .
Let (‘51,...,‘&;) : I be a geometric i-residual intersection of I, with ht I S i S s. We

may assume that a.- E I. Set a: (a1,...,a.-) andK=a:I. Then
Ozchc(a+0:I):I=(a+In(0:I)):I=K,

hence R/K E R/E : I and R/I + K E R/I +6 : I. It follows that K is a geometric

i-residual intersection and that R/E : I ”—_‘—’ R/ K is Cohen-Macaulay. C]

The following lemma is a generalization of a similar result of [68]. It extends

Lemma 1.1.5(c) to the case involving higher powers of 1.

Lemma 1.1.8 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R-ideal, let I: and t be integers, assume that I
satisfies G, and AN 3:3 locally in codimension s — 1, that I satisfies AN,. , and that
depth R/Ij 2 d—s+k—jfor 1 Sj S k, and let K = a : Ibeans-residual
intersection (or K = R and V(I) S s), and let a.- be the ideals as defined in Lemma
1.1.5. Then the following conditions hold:

(a) depth R/aglj Z min{d—i,d—s+k—j} for O S i S s, max{0,i-t—1} S j S k.

(b) (a.- : (a;+1)) fl 11 2 adj-1 for 0 S i S s -— 1, max{1,i — t} Sj S k.

22
Proof. We first show that if (a) holds for i, then so does (b). However, to prove (b)
one may check the equality locally at every p E Ass(R/c1.~IJ"1 ). By (a), p has height
at most d— min {d—i,d—s+k—j+1} = max{i, s—k+j— 1} S s-l. By assumption
Ip satisfies AN:_3. But then by Lemma 1.1.5 (c.iii), any geometric (s — 2)-residual
intersection is unmixed of height .9 - 2, hence is Cohen-Macaulay because it is at
most one-dimensional. We conclude that 1,, even satisfies AN 8:2. Now by Lemma
1.1.5(b), a1,...,a, form a d-sequence in Rp- Moreover, since ht K 2 s, we have
that IF = a, 75 Rp- Replacing R, by R, we have that (a,- : (a;+1)) n a = a.- for

0 S i S s — 1 since a1, ...,a, form a d—sequence. Hence

(Di 1 (a;+1))fl Di 2 (Di 1 ((14.11)) 0 an aj

=agfla’.

Thus to prove (b) it is enough to show that

(1.1.9) a.- 0 a1: agaj-l.

But this follows by standard properties of d-sequences. Indeed, if we let b =

(a;+1, ...,a,) then to show (1.1.9) it is enough to show that

a.- 0 hi c ami-l.

Since a;+1,..., a, form a d—sequence modulo 11,-, this follows from [35, 2.1].
Thus it is enough to prove (a), which we do by induction on i. Since the result
is trivial for i = 0, we may assume that 0 S i S s — 1 and that (a) and (b) hold for

i. We show that (a) holds for i + 1. If j = 0, then i S t and the result follows from

23

Lemma 1.1.5(c.ii). Thus we may assume that j Z 1. Now using (b) for (i),
ai+1ang—1 C ain 0 ai+11j
= 0.41““in 1(ai+1))n Ijl
C ai+l[(ai 3 (ai+1)) fl Ijl
: a;+1a;1j_l,

from which it is clear that all the containments are equalities.

But using part (b) with i = 0 shows that
(0 : (a;+1)) 0 cup-1 c (0 : (a;+1)) n I)" = 0,

and hence that Chump—1 2 (1,1)."1 and c.4111 E II. Thus the required depth
estimate for R/a;+111 follows if i = 0 from the latter isomorphism, while if i > 0 it

follows by induction from the exact sequence

0 —+ adj-1 —> (1,1169 11' —+ amli —+ 0. a

We want to apply these results to minimal reductions.

Remark 1.1.10 Let R be a local Cohen-Macaulay ring with infinite residue field,
let I be an R-ideal with analytic spread I which satisfies G3, and let J be a minimal
reduction of I such that ht J : I Z 8. Then there exists a generating set al, ...,at
of J with the following property: a.- : I is a geometric i-residual intersection of I
for ht I S i S I — 1, where a.- = (a1, ...,a,). Moreover, any permutation of a1, ...,ag
enjoys the same property. If in addition, I satisifes AN (_- 3 locally in codimension

Z — 1, then ht J : I Z 3 holds for every minimal reduction J of I .

Proof. Using Lemma 1.1.5(a), we only have to prove the last statement, which

follows from [68, 1.11] since I is of linear type locally in codimension I — 1. C]

24

Using Lemma 1.1.8, and applying the previous remark, we prove the following

important technical result, which generalizes a result of [63].

Lemma 1.1.11 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R-ideal with grade g, analytic spread f, and
reduction number r, let I: and t be integers with r S k and t Z 8 — k — 1,
assume that I satisfies G; and AN (_— 3 locally in codimension E — 1, that I sat-
isfies AN,-, and that depth R/Ij Z d— 6+ k —j for 1 S j S 1:, let J be a minimal
reduction of I with r 1(1) = r, and let a.- be the ideals in Remark 1.1.10.

Then a.- fi Ij = adj-1 for 0 S i S €- 1 andj Z max{l,i — t}.

Proof. If 3' S k the result follows from Lemma 1.1.8(b) with s = 8. Hence we may
assume that j 2 k + 1. We will prove the result by decreasing induction on i.

For i = 2 note that the result holds automatically since j Z k + 1 Z r + 1 and
hence 11 = JIj'l. For 0 S i S €— 1, note that i—t S 6— 1 —t S 1:. Since the result
clearly holds for j = 1, we may assume that j 2 max{2,i — t + 1}, and that the
equality holds for i + 1 by decreasing induction on i, and that a.- fl 1’"1 = ang‘2
holds by increasing induction on j. Then

aiflIj =a,na.-+1 011'
= a.- 0 (a;+111'1)
= a.~fl[a.‘11'_l + ai+11j‘1]
= (1,114+ a;+1[a; : (a;+1) 0 11"]
= adj-1 + a;+1[(a.- : (a;+1)) fl Imaxli't'll fl Ij’l]
= 0.1171 + ag+1[a.- 0 11-1] by 1.1.8(b)
= ain_1+ a1+1[a.-Ij’2]

= ain-l. E]

25

We point out that one may compute the reduction number by checking it locally

in codimension 3.

Remark 1.1.12 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R-ideal with analytic spread 6, let I: 2 1 be an
integer, assume that 1 satisfies G1 and AN (_- 3 locally in codimension I) — 1, that 1
satisfies AN;k_1, that depth R/Ij _>_ d — €+ k —j for 1 Sj S k, and that J is a
minimal reduction of I with er(Ip) S k for all p E V(I) with dim R,D = 8.

Then r(1) S 11:.

Proof. It is enough to show that I“+1 = J I k , which may be checked locally at every
p 6 Ass (R/JI"). But by Lemma 1.1.8(a) depth R/JI" 2 d—é, and thus any such
prime has height at most 8. Since I satisfies AN (_— 3, I and J coincide locally in
codimension 2 — 1 by Remark 1.1.10. Hence it is enough to check the equality at

primes of height exactly I, in which case the result holds by assumption. CI

1.2 Cohen-Macaulayness of the associated graded ring

For the rest of this chapter we will fix the following notation: R will be a local
Cohen-Macaulay ring of dimension d with infinite residue field, and I will be a
proper R-ideal with grade g, minimal number of generators n, analytic spread Z,
and reduction number r; G and R will denote the associated graded ring and the
Rees algebra of I . Moreover, we will always assume that 1 satisfies the condition
G1.

The passage of the Cohen-Macaulay property from G to R is well-understood

by the following result:

Theorem 1.2.1 (Simis-Ulrich-Vasconcelos [61, 3.6]) Let R be a local Cohen-
Macaulay ring with infinite residue field, and let I be an R-ideal of grade 9 and

analytic spread 8, and assume that 1 satisfies G: and is not nilpotent. Then the

26

following conditions are equivalent.

(a) R is Cohen-Macaulay;

(b) G is Cohen-Macaulay, g > 0 and 7(1) S I — 1.

Slightly different results have also been obtained by Johnston and Katz ([46])

and by Aberbach, Huneke and Trung ([5]), without the assumption of Ge.

These results allow us to focus our study on the Cohen-Macaulayness of the

associated graded ring G; it has been shown to be Cohen-Macaulay under any of

the following additional assumptions:

r g 1, 2 39+ 1 and depth R/I 2 d—e ([30])

r g 1, e g g + 2, R/I is Cohen-Macaulay, I satisfies (011) and R is
Gorenstein ([31])

r S 1 and I satisfies sliding depth ([74])

r S 1, 1 satisfies AN;2 and depth R/I Z d — E ([68])

r S 2, I = g + 1, R/I is Cohen-Macaulay and depth R/I2 2 d — E ([20])
r S 2, E = g + 2, R/I is Cohen-Macaulay, depth R/I2 Z d — E, 1 satisfies
(C11) and R is Gorenstein ([21])

r S 2, E = 3, n S 4 and I is perfect of grade 2 ([4])

r S 2, E = 4, n S 5 and I is perfect Gorenstein of grade 3 ( [3])
rS3,€=g+2anddepthR/Ij2d—g-j+1for1SjS3([2])

r ge—g,ezg+3, depth R/IJ' Zd-g—j+1for 1 Sj SE—g—l,

depth R/Il’g Z d — I, 1 satisfies (CI(_9_1) and R is Gorenstein ([63])

27

e r S E—g-l-l, n S 8+1, 1 is strongly Cohen-Macaulay and R is Gorenstein ([61])
e rSZ—g+1,€Zg+2,depthR/Ij Zd—g—j-i-lfor 1 SjSE—g+l,

I satisfies (C1(_g_1) and R is Gorenstein ( [63])

One of our main results will be a theorem (Theorem 1.2.8), which includes essen—
tially all of the above results as special cases. The key is to systematically exploit
the Artin-Nagata properties of the ideal. Indeed, although most of the above results
make no specific mention of Artin-Nagata properties, they are implied by Theorems

1.1.3 and 1.1.4. (Subsequently, futher generalizations have recently been obtained

by Aberbach ([1]) and by Goto, Nakamura and Nishida ([22]), ([23]).)

We begin with a general result about when a graded ring is Cohen-Macaulay. By

[M ]2,- we denote the truncated submodule 69,2;Mj of a graded module M = EBij.

Proposition 1.2.2 Let S be a homogeneous noetherian ring of dimension d
with 50 local, let I = 5+, let b1, ...,b( be linear forms in 5', set 5; = (b1, ...,bg) for
—1 S i S E (where ((0) = 0), J = be, let g be an integer with 0 S g S e, assume
that I”1 C J (i.e. J is a reduction of I with 1' 1(1 ) S k) and that the following
conditions are satisfied:

(a) [b.- 3 (bi+1)lzi—g+1 = [Mai-9+1 for 0 S i S e — 1,

(b) depth [S/bg];_g+1 _>_ d— i — 1 for g — 1 S i S 13— 1,

(c) depth [S/Jlj Zd-EforE—g+1SjSk.

Then S is Cohen-Macaulay.

Proof. Note that by (a), b1,...,bg form an S-regular sequence and that one has
[S/bg-1]o = [S/b-1]o = 50. Hence we may factor out by and assume that g = 0.
Now conditions (a), (b), (c) are now

(a’) [b. I (bi+1)]2,'+1 = [bi]Zi+1 for 0 _<_ t S g -1,

28

(b’) depth [S/b.-_1].- 2 d —i for 0 Si 3 e,

(c’) depth [S/J]j Z d— E for 6+1 Sj S k.

For 0 g i g 2 consider the graded S-modules M(,-) = [$1142.11 = 1‘“ /b,-I‘
and N(,-) = I'u/b.-_11'."l + b.'1i (where 1"] = 10 = 5). Observe that MU) can be
obtained as a trunction of N(,-), namely that III“) = [N(i)l2i+1' In addition, M(,-_1)
coincides with N“) in degree i, that is to say lN(i)li = [S/b;_1].-. Thus we have a

exact sequence
(1.2.3) 0 —> M“) —> N“) —) [S/b,‘_1].‘ —') 0.

Alternatively, we may view N(i+1) as a quotient of III“), namely N(,-+1) =
M(,)/b.-+1M(,-), for 0 S i S I — 1. Moreover, bi.“ is regular on MU) by (a’). Hence

for 0 S i S E — 1 we have an exact sequence

(1.2.4) 0 -—) M(,')(—l) 31}) M(,') -—-> N(;+1) —> 0.

We will prove by decreasing induction on i, 0 S i S E, that depths N(,-) Z d — i.
Since N0 = 5', it will follows that S is Cohen-Macaulay.

First consider the case where i = 8. Then N“) = [57111-1]: EB $f=c+1lS/Jlj-
By (b') and (c'), this has depth at least d — 8 as an So-module and hence as an
S-module.

Now let 0 S i S 8 — 1 and suppose that we know the induction assumption holds
for i + 1, i.e. that depths N(i+1) Z d— i — 1. Then depths MO) 2 d —i by (1.2.4).
But by (b’) we also have depths [S/b;_1].- = depths0 [S/b;_1],~ Z d — i. Hence by

the sequence (1.2.3) we conclude that depths N“) Z d — i. D

If we let S = G be the associated graded ring, then this proposition gives a

criterion for G to be Cohen-Macaulay. Moreover, in the previous section we have

29

essentially shown that conditions like those of the proposition hold under suitable

conditions. We obtain the following theorem:

Theorem 1.2.5 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R-ideal with grade g, analytic spread E, and reduc-
tion number r, and assume that I satisfies G3 and AN (_- 3 locally in codimension
E—l, that depth R/Ij _>_ d-Q-j-tl for 1 Sj S €—g+1, and that r S6—g+1.

Then G is Cohen-Macaulay.

Proof. Let J be a minimal reduction of I with rJ(I) = 1', let a1, ..., a; and a.- be as
in Lemma 1.1.10, and let a: be the image of a.- in [G]1. We apply Proposition 1.2.2
to the ring 3 = G, linear forms b,- = a; and with k = max{r,€ — g}.

We first check that [b; : (b;+1)]2;_g+1 = [bi]z.~_g+1 for O S i S E — 1. Let
u E [b.- : (b,+1)],-, with j 2 i— g +1, and write u = a: + Ij'H for some element
a: 6 11. Then a;+1:r E a.- +Ij+2. Ifi S 8—2 then by Lemma 1.1.11 (with t = g— 1)

we have

(154.15!) 6 0,44 0 (0i + Ij+2)
= 01+ ai-H O 1H2
= 0i + CHIP-H

= 0" + ai+11j+l.

On the other hand if i = E — 1 then this holds automatically since 1“"2 = J11"H
as j+2 2 €—g+2 2 r+ 1. In any event, we conclude that ai+1(:c—y) E a.- for some
element y E I j“. But since a: — y coincides with :c modulo I “'1 we may as well

assume that ai+1r 6 (1.4.1. Now we may use Lemma 1.1.8(b) and Lemma 1.1.11 to

3O
conclude that
1‘ E (03' I ((1144)) F1 Ij
=(0i 3(ai+1)lfl IFgH fl Ii
= air—9 I) Ij
C 01' n Ij
= ain—l.
It follows that u 6 [b.],-.
Finally, we have to check that the conditions depth [S/b;].-_g+1 Z d — i — 1 for

g — 1 S i S I - 1 and depth [S/J]g_g+1 2 d — 6 hold. The assumption that

depth R/ I j 2 d — g — j + l for 1 S j S k and the exact sequences
0 —+ 11/11"r1 —. R/IJ'+1 ——> R/IJ' —+ 0

give by induction the estimates depth 11 /Ij+1 : depth [5],- 2 d — g — j for
lstE—s

Since [b.- : (bi+1)l2i—g+1 = [11.]?in for 0 S i S E- 1, there are exact sequences
0 —-> [S/b.]j —> [S/b.],-+1 -—> [5/b1+1lj+1 —‘> 0

for 0 S i S I — 1 and j _>_ i — g + 1. These sequences then give by induction that
depth [S/b;]j Zd-g—jforOSiSI—landi—g+1SjS€—g.Inparticular
we obtain that depth [S/b;];-g+1 Z d — i — 1 for O S i S E — 1 as required. Finally
ifi = B, then
lS/Jlt-gH = It-g‘H/Jfl-g “*1 1&9“
= Il-9+1/J]l-g + JIt-g-H

___ If—g-i-l/JIt-g

31

since I has reduction number r S 8 — g + 1. Since by assumption depth R/ I [’9“

Z d — e, the required depth estimate follows from Lemma 1.1.8(a). CI

The above result was proved by Tang ([63, 55]) when R is Gorenstein (in which
case one can delete the local Artin-Nagata condition by Theorem 1.1.4) and I
satisfies (011-94). This latter condition is much stronger than the condtion G1
when 2 is not very small. Theorem 1.2.5 would actually follow from Tang’s proof,
which was a major motivation for our work. But as his proof is rather computa-

tional, we have instead used Proposition 1.2.2.
We immediately obtain the following application.

Corollary 1.2.6 Let R be a local Cohen-Macaulay with infinite residue field
and let I be a strongly Cohen-Macaulay R-ideal satisfying G; and r( I ) S 6 — g + 1.

Then G is Cohen-Macaulay.

Proof. As I is strongly Cohen-Macaulay and satisfies Gt, one knows that
depth R/Ij Z dim R—g—j+1 for 1 S j S €—g+ l and by Theorem 1.1.3,

1 satisfies ANg. The result now follows from Theorem 1.2.5. C]

When the ambient ring R is Gorenstein, it turns out that the assumptions in
Theorem 1.2.5 force the equality r = I — g + 1 (or r = 0) (Corollary 1.2.12).
But it is important to know that the Theorem holds in any Cohen-Macaulay ring,
even if one is solely interested in a regular ambient ring R. This is because we
want to use this result as a base case of an even more general result, in which we
will reduce to Theorem 1.2.5 by factoring out certain residual intersections. (This
method has been used by several authors, and goes back to Huneke ([32]).) Under

suitable Artin-Nagata properties, we can maintain only the Cohen-Macaulayness

32
of the ambient ring, and thus one needs to know results in that generality.
We will need to isolate a special case of [30, 2.9]. For convenience we include a

proof, using an argument of [74].

Proposition 1.2.7 Let R be a local Cohen—Macaulay ring of dimension d,
let 1 be an R-ideal with I 2 = a1 for some a E I, assume that 1,, = 0 for every
p E Ass(R) flV(I), and that depth R/I Z d — 1.

Then G is Cohen-Macaulay.

Proof. The R-homomorphism from the polynomial ring R[T] to the Rees algebra
R[I t], mapping T to at, induces an R[T]-homomorphism I R[T] —) I R[I t] which is
surjective because 12 = a]. To see that it is also injective it is enough to show that
(0 : (0’)) fl 1 = O for all j. But this is clear, as one may check it locally at every
p E Ass(R)flV(I), and I is zero at any such prime. It follows that IR[It] E IR[T]

and hence has depth at least d + 1. But then from the fundamental exact sequences
0—~>IR(—1)——>’R——>R-—>O

0—>I’R——>R—+G—>0

and a depth chase, it follows that G has depth at least d. Cl

We are now ready to prove one of our main results. As we mentioned, this result
generalizes most of the previously known work. By performing certain standard
operations, we will reduce this result to Theorem 1.2.5. The important fact is that

the Artin-Nagata property is preserved in the process, which we previously verified

in Lemmas 1.1.6 and 1.1.7.

Theorem 1.2.8 Let R be a local Cohen-Macaulay ring of dimension d with

infinite residue field, let I be an R-ideal with analytic spread 8 and reduction

33

number r, let k 2 1 be an integer, and assume that r S k, that I satisifes G3
and AN;3 locally in codimension I — 1, that 1 satisfies AN 13—— max {2,1}, and that
depthR/Ij Zd—E-l—k—j for 1 Sj Sk.

Then G is Cohen-Macaulay.

Proof. By settingj = 1 we see that depth R/I 2 d — 6 + k — 1. It follows that
k SE—g-l-l sincedepth R/I S d—g. We set 6: 6(1) :8—g+1—k _>_ 0. We will
proceed by induction on 6. If 6 = 0 then I: = 6 — g + 1 and the result is precisely
Theorem 1.2.5. Hence we may assume that 6 > 0 and that the Theorem holds for
all smaller values of 6. Note that we now have that 6 2 g + k 2 g + 1.

Let J be a minimal reduction of I with TJ(I) = r, and let a; be the ideals as in
Lemma 1.1.10. If we set t = €- max{2,k} then g —t = max{g + 2 - 6, 1 — 6} S1.
Hence by Lemma 1.1.11, we have that a.- n Ii = aJi-l for 1 g 2' _<_ g and all j 21.
It follows that the images a’1,...,a; of a1, ...,a9 in [G] 1 form a G-regular sequence
([71]). We first show that we may factor out (19 and assume that g = 0. Let us
denote by “ ‘ ” images in R/ag.

As G(I‘) 9:“ G/(a’l, ...,a;)G, it follows that G is Cohen-Macaulay if and only if
G(I‘) is Cohen-Macaulay. Now dim R“ = d — g and [(1‘) = 8(1) — g, whereas
11: may be taken as unchanged. By Remark 1.1.10, a.- : I is a geometric i-residual
intersection for every g S i S E — 1, hence II, = (a1, ...,a,)p for all p E V(I) with
dim R S i. It follows that 1* satisfies G“ 1.). We next show that the condition on
the depth of the powers is preserved. Since R” /I’j E R/ag + I j, by Lemma 1.1.11

We have an exact sequence
0 ——> R/ang‘l —> R/a, e R/Ii —> R*/I:i ——> 0.

Now by Lemma 1.1.8(a) and since 6 > 0, we have that depth R/agI-l—1 Z

 

34

d — f + k —- j + 1 for l S j S k. Hence our assumption that depth R/Ij Z
d — E + k — j together with this sequence implies that depth R’/I‘j Z d — Z + k — j
= dim R‘ — 6(1‘) + k —j for l S j S k. Finally the Artin-Nagata conditions are
preserved by Lemma 1.1.6. Hence we have reduced to the case where g = 0. Now
6(1) 2 8 + 1 — 1:.

Now we have that 8 Z k 2 1. Suppose that I = 1. Then r S k = l and hence
I2 = a1 for some a E 1. Further, since 1 satisfies G1, it holds that 1 is locally
zero at every associated prime of R containing I . In this case, we are done by
Proposition 1.2.7.

Thus we may assume that E 2 max{2, k}. In particular, we have that 1 satisfies
AN; . Now let K = 0 : I and let “ ‘ ” denote images in R/K. We will show
that our assumptions are again preserved, and that 6 decreases when passing from
R to R. First, note that by Lemma 1.1.5(c.i), we have that 1 H K = 0. Since 1
satisfies G1, again it holds that 1,, = 0 for every prime p containing 1 with dim
R, = 0. It follows that ht I + K > 0, and hence that K is a geometric O-residual
intersection of 1. Since 1 satisfies ANO- , it holds that R is Cohen-Macaulay. We
have dim R = d, [(1) = I, and again k may be taken as unchanged. To see these
last claims, it is enough to see that any minimal reduction of I lifts to a minimal
reduction of 1. But that is clear since given an equation (I)’+1 = :1:(I)" in R for

some integer s 2 0 and where we may assume J1 C I, then it follows that
11+1 c (J11: + K) n 1 = J11: + 1 n K = .1118.

Now by Lemma 1.1.5(c.iv), I has grade 1, and clearly still satisfies G; since the

dimension is unchanged. Again since 1 n K = 0 we have an exact sequence

(1.2.9) 0 —> K —-) gr1(R) —-> g77(—R) —> 0.

35
Since R is Cohen-Macaulay, it follows that depth K = d. Hence the degree 0 piece
of this sequence gives us the estimate depth R/I 2 min{depth K — 1, depth R/I}
2 min{d—1,d— 6+ 13— 1} = d—6+ k — 1 since I: S 6. On the other hand, for all

j _>_ 2 we have the isomorphisms
71-1/71' e 11-1 m + 11-1 n 1; e 11-1/11-1.

It follows that 1 satisfies the same depth estimate on its powers, namely that
depth R/Ij 2 d — 6 + k — j for 1 S j S k. Finally, the Artin-Nagata proper—
ties are preserved by Lemma 1.1.7. Thus we have shown that I satisfies all of the

assumptions of the theorem.

We have that 6(1) = 6(1)— grade 1+1— k = 6 — k < 6 — k + 1 = 6(1). Hence
we may conclude by our induction on 6 that ng-(R) is Cohen-Macaulay. But as

depth K = d, by the exact sequence (1.2.9) it then follows that G = gr1(R) is

Cohen-Macaulay. C]

We now give several corollaries.

Corollary 1.2.10 Under the assumptions of Theorem 1.2.8, R is Cohen-

Macaulay if and only if g 2 2, g = 1 and r # 6, or I is nilpotent.

Proof. We know that G is Cohen-Macaulay and that r S 6 — g + 1. Hence by
Theorem 1.2.1 the result if clear if I is not nilpotent. However, if I is nilpotent, one
has 6 = 0, hence r S 1, and thus 12 = 0. In this case R = REE I is Cohen-Macaulay

since dim R = d and depth 1 Z depth R/I Z d — 6 = d. C]

Corollary 1.2.11 Let R be a local Cohen—Macaulay ring with infinite residue
field and let 1 be a perfect R-ideal of grade 2 satisfying G;.

Then R is Cohen-Macaulay if and only if r(I) S 6 — 1.

36

Proof. Since I is strongly Cohen-Macaulay, this follows immediately from Theorem

1.2.1 and Corollaries 1.2.6 and 1.2.10. Cl

Corollary 1.2.12 Let R be a local Gorenstein ring of dimension d with infinite
residue field, let I be an R-ideal of grade g and analytic spread 6 and assume that
I satisfies G: and that depth R/Ij 2 d— g — j + l for l S j S 6 — g and that
r(I) S 6 — g.

Then 1 is strongly Cohen-Macaulay and satisfies G00 (and r = 0).

Proof. It will be enough to show that r = O, for then 1 satisfies G00 and the result
follows from [68, 2.13]. Thus we may assume that 6 > g.

By Theorem 1.1.4 we know that 1 satisfies AN[_1, and hence G is Cohen-
Macaulay by Theorem 1.2.8. Since then equality holds in Burch’s inequality, one
knows that in particular that depth R/ 1"9+1 2 d — 6. Together with our assump-
tion, we nowhave that depth R/Ij 2d—g—j+1for 1 Sj S6—g+1.

Let J be a minimal reduction of I with r 1(1 ) = r. We must show that J = 1.
Suppose that this is not the case. Then by Remark 1.1.10, K = J : I is an
6-residual intersection of I. We may now apply Theorem 1.1.4 to compute the
canonical module of R/ K ; we find that wR/K E 1“9+1/JI"9. Since this module
cannot vanish, it follows that r 2 6 — g + 1, which is a contradiction. Hence I = J

holds and thus r = 0. [:1

Example 1.2.13 Let k be an infinite field, let X be a generic alternating 5 by 5
matrix, let Y be a generic 5 by 1 matrix, put R = k[X, Y] (possibly localized at the
irrelevant maximal ideal), and let I = P f4(X ) + 11(X Y) be the R-ideal generated
by the 4 by 4 Pfaffians of X and the entries of the product matrix X Y.

It is well-known that R/I is the associated graded ring of the ideal P f4(X ) in

37
k[X] ([34, 2.2]) and it follows that I has grade 5, that 1 satisfies (C14)
([33, proof of 2.1]), and that R/I is Gorenstein ([34, 2.2 ). Futhermore, 6(1) = 9,
and a computation using the computer algebra system MACAULAY shows that
R/ I 2 and R/ I 3 are Cohen-Macaulay. By Theorem 1.1.4, it follows that I satisifes
AN7, and that r(I) = 1 by Remark 1.1.12. We may thus apply Theorem 1.2.8 to

conclude that R is Cohen-Macaulay.

1.3 Number of Defining Equations

In this section we study the defining equations of the Rees algebra via the canonical

homomorphism

a : 5(1) —+ 72(1)

from the symmetric algebra of I onto the Rees algebra. Recall that A denotes the
kernel of this homomorphism.

Now let 1 be an ideal with analytic spread 6 which satisfies G1 and we fix a
choice of a minimal reduction J (say with r 1(1 ) = r) satisfying ht J : I Z 6, and
ideals oi, as in Remark 1.1.10. We may extend a1, ...,a: to a minimal generating
set a1, ...,a,. of 1. Let S = R[T1,...,Tn] be a polynomial ring over R, and present

the Rees algebra R = R[It] E S/Q, by mapping T.- to ait.

Lemma 1.3.1 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I an R-ideal with analytic spread 6, let I: Z 1 be an integer
and assume that 1 satisfies G; and AN;3 locally in codimension 6 — 1, that 1
satisfies .4N;_k_l and that depth R/Ij 2 d - 6 + k —j for 1 Sj S k.

Then [(T1,...,T() 0Q]k+1 C [Q]k5.

38
Proof. It will be enough to prove that [(T1,...,T,-) fl Q]k+1 C [Q]kS, for every
0 S i S 6, which we do by induction on i. Since the case i = O is trivial, we may
assume that i Z 1 and that the result holds for smaller i. Let F 6 (T1, ...,T,) O Q
be a form of degree 11: + 1 and write F = 23:1 GjTj where Gj 6 [S]k. Evaluating

F at (01,...,an) gives O : Z;=1Gj(a1,...,a,,)aj, hence
Gi(g.) 6 (Oi—1 I (01)) F] [k : Gi_11k_l

by Lemma 1.1.8(b). It follows that there are forms H1, ..., 11,--1 E [Sh--1 for which
P = G.- — 33,11,73- e [Q].s. But then
i—l

F — T.P = [(0, + T.H,-)T,- 6 [(TI, T;_1) n Q]...1 c [(21.5

i=1

by induction. It follows that F E [Q] 1.5. [3

Proposition 1.3.2 In addition to the assuptions of Lemma 1.3.1, set n = V(I)

and assume that 51(1) E II for 1 Sj S k and that r(I) S 1:.

Then [A]k+1 is minimally generated by (gift) forms.
Proof. Consider the exact sequence
O—>A——>S(I)——>R(I)—>0,
which in degree k + 1 is

0 —->[.A]k+1—-> Sk+1(1) -—) Ik+1 —-> 0.

Since we may present R E S / Q and 5(1) E S / L, where L is the ideal generated by

the linear forms in Q, we have A ’5 Q / L. This will induce an exact sequence

0 —> [.4]...H —> Sk+1(I/J) —+ [HI/u“ —> 0

39

once we have shown that [(L,T1,...,Tg) fl Q]k+1 C L or equivalently that
[(T1,...,T() fl Q]k+1 C L. But this is clear from Lemma 1.3.1, since by the as-
sumption on the symmetric powers we have [Q]kS C [Q]13 = L. The result is now

clear since if r(I) S k then the module Ik‘H/JIk vanishes. E]

This proposition allows us to compute the number of defining equations of the
Rees algebras of ideals having the minimal reduction number. We take this to mean
that 5,-(1) E Ij for 1 S j S r, or equivalently that A...“ is the first nonvanishing

component of A, where r = r(1) is the reduction number.

Theorem 1.3.3 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R—ideal with analytic spread 6, minimal number of
generators n, and reduction number r and assume that 1 satisfies G; and AN (_— 2,

and that 31(1) E 11 and depth R/Ij 2 d—6+r —j whenever 1 Sj S r.

Then A is minimally generated by (2:52) forms of degree r + 1.

Proof. If r = 0 then n = 6 and I is generated by a d-sequence by Proposition 1.1.5;
since then A = 0, the result holds. Hence we may assume r 2 1. By Proposition
1.3.2, it is enough to show that the relation type rt(I) is at most 7‘ + 1 since then A

will be generated by its degree r + 1 component. By [56, 2.3] it is enough to show
((a1,...,a,-) : (a;+1)) fl I"H = (a1, ...,a,-)Ir

for all 0 S i S n — 1. Since 1 satisfies AN (_- 2, this follows from Lemma 1.1.11 and

Lemma 1.1.5(c.i) ifO S i S 6 — 1. However, if 6 Si S n -— 1 then

((a1,...,a,-) : (a;+1)) fl I"+1 C I"+1 = JIr C (a1,...,a;)1r. Cl

Corollary 1.3.4 Let R be a local Cohen-Macaulay ring with infinite residue

field, let I be a strongly Cohen-Macaulay R-ideal of grade 9, analytic spread 6 and

4o
minimal number of generators n, and assume that 1 satisfies G: and has reduction
numberr =6—g+1.

Then A is minimally generated by 0:99:21) forms of degree 6 - g + 2.

Proof. Since 1 is strongly Cohen-Macaulay and satisfies G5, one has depth R/ I j Z
d—g—j+ l and Sj(1)E 11 whenever I S j S 6—g+ 1. As 1 satisfies AN: by

Theorem 1.1.3, the result follows from Theorem 1.3.3. [:1

Corollary 1.3.5 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let I be an R-ideal with analytic spread 6, minimal number of
generators n, assume that 1 satisfies Gt.“ and AN [_2 and that depth R/I 2 d — 6.
n-§+l)

Then A is minimally generated by ( quadrics.

Proof. By Theorem 1.3.3, it will be enough to show that r S 1. However, since 1
satisfies AN (_— 2 and G5“, it follows that I is of linear type locally in codimension
6 by [68, 1.11]. Since in particular 1 has reduction number at most one locally in

codimension 6, the result follows from Remark 1.1.12. Cl

The previous corollary had been observed for the class of monomial curves in
P3 lying on a quadric, i.e. the homogeneous codimension two prime ideals in
k[a:,y,z,w] defining :1: = ta+b,y = satb,z = sbt“,w = ta'H’, with (a,b) = 1. It is
well-known that they are minimally generated by b — a + 2 equations. Huckaba
and Huneke ([30]) showed that 6 S 3 and that the Rees algebra is defined by at
most quadrics. Schenzel ([59]) and Morales and Simis ([51]) actually compute the
defining equations explicitly. Since the curves are smooth we may immediately
apply Corollary 1.3.5 (note that the condition AN (.— 2 is vacuous since 6 -— 2 S 1 <
2 = g) to conclude that A is minimally generated by (“g“) quadrics.

More generally, now by Corollary 1.3.5 and [25, 4.1] (and its proof), for any

41

saturated homogeneous ideal I defining a smooth curve in P3 lying on the surface
my = zw, generated by n equations, its Rees algebra is defined by linear and exactly

('72) quadratic equations.

Corollary 1.3.6 In addition to the assumptions of Theorem 1.3.3, assume that
n = 6 + 1.

Then A is cyclic.

It is natural to ask, at least in the case n = 6+ 1, where does this single equation
come from. In other words, can one compute the generator of A explicitly? This
has been studied when 1 is a perfect ideal of grade 2 by Vasconcelos ([73]), and the
answer is completely natural, just given by a determinant of a Jacobian dual of a
presentation matrix of I . When the grade is larger than two, this will not work, as
one can easily see by degree reasons. However, this still seems to be a good place
to look for the equations, so we describe the general idea. We will then show how
it leads to the explicit defining equation of the Rees algebra for Gorenstein ideals
of grade 3.

Let R be a local Gorenstein ring with infinite residue field, and let I be a strongly
Cohen-Macaulay R-ideal satisfying Gg, having second analytic deviation one, i.e.
n = 6 + 1, and having the expected reduction number r = 6 — g + 1 (we will
show in the next section that often this is equivalent to assuming that R is Cohen-
Macaulay). By Corollary 1.3.4 we know that A is cyclic, generated by a form of
degree 6 — g + 2.

Let

Rm-LR"——.I—>0

be a presentation of I, where (6 is an m by n matrix with entries in the maximal

42

ideal of R. Recall that the symmetric algebra 3(1) admits a presentation
5(1) E R[T1,...,Tn]/(61,...,6m)
defined by the equations

(61,...,6m) = (T1,...,T,,)¢.

Our assumptions imply that 1 satisfies the row condition: after elementary row
operations, the entries of 11(4)) can be generated by the last row of d) ([61], or
Proposition 1.4.2). Moreover, it is known that 11(¢) is Gorenstein ([61, 4.13]) and
has height 6 ([27, 9.1]), but let us assume further that it is a complete intersection
and choose a generating set 11(q5) = (1:1,...,a:(). Now we may assume after row
and column operations that 11(¢) is generated by the entries of the last row of
the submatrix consisting of the first 6 columns. Let 211 be the 6 by 6 submatrix

of 4) obtained by deleting the last row and the last m — 6 columns, and over the

polynomial ring R[T1,..., Tl] consider a J acobian dual ([60]) of 1,6:

(T1, ..., T()1,b = ($1, ..., $()B(¢).

Here B = B(1,b) is an 6 by 6 matrix whose entries are linear forms in the variables
T1, ..., Tg. The characteristic polynomial of B seems to be a good place to look for
equations of the Rees algebra. .

To give evidence to this claim, assume now that I is a perfect Gorenstein ideal
of grade 3 satisfying G; and the row condition. After elementary row operations,
11(4)) is generated by the last row of the presentation matrix (15, and by the structure
theorem of Buchsbaum and Eisenbud ([11]), 45 can be chosen as an alternating n by

n matrix. Since 45 is alternating, it follows that 11(q’9) is automatically a complete

intersection. Hence we may apply the arguments above; we obtain the following:

43

Theorem 1.3.7 Let R be a local Gorenstein ring with infinite residue field,
let 1 be a perfect Gorenstein ideal of grade 3, with analytic spread 6, minimal
number of generators n = 6 + 1, and assume that 1 satisfies Ge, let cf) be an n by n
alternating matrix presenting I with last row (—:1:1,..., —.rg, 0) which generates the
ideal of entries of d) and let ti’ be the 6 by 6 alternating submaxtrix of (6 obtained

by deleting the last row and column. Then there exists a Jacobian dual B = B (11)):

(T1,...,T.)¢ = ($1,...,rg)B(¢).

such that A is generated by F = Tn—1X3(Tn), where X denotes the characteristic
polynomial of B in the variable T n. Moreover, if we let 1b,- denote the jth column
of 1,1), for 1 Sj S 6, and write 16,- = Aj(_:g)‘, where Aj is an 6 by 6 matrix whose jth
row consists of zeros and whose ith row, for any 1 S i S 6, is the negative of the

jth row of .45, then B may be taken to be the matrix whose jth column is A;- (If.

Proof. It is enough to prove the second statement. Let a;,- and bgj be the i j th entry
of the matrices 111 and B respectively, and let aijk be the ikth entry of Aj. Then
write a;,- = 21:1 agjkxk and bij 2 25¢, akngk, where a”)c = —aj,-k for i 753' and
am, = O for i = j. Let {zijk} be a new set of variables with zgjk = —zj.-k for i #j
and 2.3), = 0 fori = j, and define 61,-,- = Sig, zijkrk and b;,- = 25ml zkjiTk. Denote
by d; and B the matrices whose i j th entry is (1,, and bi], respectively. It follows

that B is a J acobian dual of 11’:

~

(1.3.8) (T1,...,T()1/3 = ($1,...,rg)B.

~

Now it will be enough to show that det(B) = O. For then by specializing, it
holds that det(B) = 0, and hence that Tn divides the characteristic polynomial

X 3(Tn) of B. Since the characteristic polynomial may be obtained as a minor of a

44

Jacobian dual of d), it is clear that X 3(Tn) is a relation on the Rees algebra. But
by Corollary 1.3.4, A is cyclic, hence since the quotient F = Tn‘IXB(T,,) is monic,
it is the required form of degree 6 — l = 6 — g + 2.

Now to show the claim, multiply equation (1.3.8) on the right by the column

(T1, ...,Tg)‘. Since 11) is an alternating matrix,
(e)B(TI,....T.)‘ = 0.

Since the 33’s form a regular sequence, the entries of the matrix B (T1 , ..., Te)t belong
to the ideal generated by (2:1, ...,IL‘() and the subring over 1: generated by the T’s

and the 2’s. It follows that B(T1,...,Tg)t = 0 and hence that det(B) = 0. Cl

Example 1.3.9 Let I C k[[r,y,z,w]] be the defining ideal of the Gorenstein

monomial curve k[[t5,t6, t7, t8]]. Then I has a presentation matrix 45:

0 z w y y
—z 0 r2 — w w — y w
—w w — x2 O 0 z
—y y — w 0 0 :c
—y —w —z —:r 0

Since 11(¢) = (r,y,z,w), 4) satisfies the row condition. Deleting the last row

and column, we obtain the Jacobian dual B :

—T4 T4 0 T1 — T2
—T3 T3 — T4 T1 — T2 T2
—T2 T1 0 0

0 — 3T3 3T2 0

Dividing the characteristic polynomial x3(T5) by T5 gives us the nontrivial cubic
relation on the Rees algebra:
T53 + 2T4T52 + TfTs + T1T2T5 — T12T5 —— 7"ng — T225111 + 2T1T2T4
—T12T4 + :rT2T3T5 + xT2T3T4 + xT2T32 — rT1T32 — :rT23.

(This answers the query raised in [70, 2.11].)

45

1.4 Expected Reduction Number

In this section we give a converse of Theorem 1.2.5 for ideals having second analytic
deviation one. It turns out that if the ideal in question is contained in a sufficiently
high power of its content ideal 11(6)), then one obtains the expected bound for the

reduction number. We closely follow ideas from [3].

Lemma 1.4.1 With the assumptions as in Lemma 1.3.1, suppose in addition

that n = 6 + 1 and let (I) be a minimal presentation matrix of I.

Then 11(¢)[Qlk+1 C [62115.

Proof. Let F E [Q]k+1 and write F = QT:+1 + G where G E [(T1,...,T()]k+1 and
a E R. Since Lemma 1.3.1 holds for any permutation of T1,...,Tn (by Remark
1.1.10) and :c 6 11(cb), we may assume .1: E (a1,...,an_1) : (an). Hence there is a

linear form 11 = xTn + 25:] rgT; 6 [Q];, with r; E R. But then

:cF = arTf'H + :cG
= ..er — aT,’f(H — e11.) + xG
6 Q1+[(T1,~-,Ttln Qlk+1

C [QhS

by Lemma 1.3.1. C]

The following shows that the row condition is satisfied, under mild conditions,

when I has second analytic deviation one with minimal reduction number.

Proposition 1.4.2 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let 1 be an R-ideal with analytic spread 6, minimal number

of generators n = 6 + 1, and reduction number r, assume that 1 satisfies G; and

46

AN,.;3 locally in codimension 6 — 1, that 1 satisfies AN (_— r_, and that S j(1 ) E 11
and depth R/Ij 2 d — 6 + r —j for 1 S j S r and let ¢ be a matrix presenting 1
with n rows.

Then, after elementary row operations, 11(qb) is generated by the last row of ob.

Proof. Let J be a minimal reduction as in Lemma 1.1.10 and let K = J : 1
= J : (on) which is, after elementary row operations, the ideal generated by the
last row of (6. Hence it is enough to show that K = 11(96). By Lemma 1.4.1,
11(¢)[Q]r+1 C [Q],.S = [Q] 15 by assumption on the symmetric powers (note that
r 2 1). Let F E [Q]r+1 with F = T];+1 + G where G E [(T1,...,Tg)],.+1, and let
a: E 11(gb). Then :rF E [Q]15 and hence there is an equation

rF = erg“ + :rG = Zam-

i=1
where Li 6 [Qh and H ,- E [5],. But by comparing the coefficients of the term T1?“

it is clear that :1: E K. C]

We will need another version of Lemma 1.4.1 for large values of k. This holds

whenever the associated graded ring G is Cohen-Macaulay.

Lemma 1.4.3 Let R be a local CoheneMacaulay ring with infinite residue field,
let I be an R-ideal of grade g, and analytic spread 6, which satisfies Gt, let J be
any minimal reduction of I with ht J : 1 2 6, let a; be the ideals defined in Remark
1.1.10 and assume that G is Cohen-Macaulay.

Then(a.-:(a;+1))flIj=a;Ij’lfor0SiS6—1andei—g+1.

Proof. In the terminology of [4], J is a “special reduction” of 1. Hence by [4, 5.10]

and Remark 1.1.10 one has the equation

((ai + Ij+2) 2 ((1.41)) f] [I = Gin—1 + Ij+1

47

wheneverOSiS6—1andj Zi—g+1. But then
(a.- : (a;+1)) 0 11C £1.11"1 + 11+].
Since 0,21 11+,“ = 0, the result follows. Cl

Lemma 1.4.4 Let R be a local Cohen-Macaulay ring with infinite residue field,
let 1 be an R-ideal of grade g, and analytic spread 6, which satisfies Ge, let J be a
minimal reduction of I with ht J : I Z 6 and assume that G is Cohen-Macaulay.
Then I

(a) [(T1,...,T() fl Q]k+1 C [Q]kS for k 2 max{6 — g, 1}.

(b) If V(I) = 6 +1 then 11(¢)[Q]k+1 C [Qh-S for k 2 max{6 — g, 1}.

Proof. This follows immediately from Lemma 1.4.3 as in the proof of Lemma 1.3.1

and Lemma 1.4.1. [:1

We are now ready to show one of the main results of this section.

Theorem 1.4.5 Let R be a local Cohen-Macaulay ring of dimension d with
infinite residue field, let 1 be an R-ideal, of grade g, analytic spread 6, and minimal
number of generators n = 6 + 1, satisfying Gg, let I: Z 1 be an integer such that
5,-(1) E II for 1 S j S k, let c5 be a minimal presentation matrix of I, assume that
1 C 11(45)e for some e Z 2, that G is Cohen-Macaulay, and that one of the following
conditions holds:

hlkZE-g;

(b) depth R/Ij 2 d—g —j — 1 for 1 S j S 6— g — 2, 1 satisfies ANF—k-v and
1 satisfies AN (.- 3 locally in codimension 6 — 1.

Thenr(1)S6—g+1+£:%-‘:—ll-—k.

48
Proof. If necessary, one may pass to the faithfully fiat extension R( X ) of R to ensure
that there is a minimal reduction J ofI with ht J : I 2 6 ([4, 24]). Let r = r(1) 2 k
and choose a form F 6 [62],.“ with F = Tl,“ + G where G E [(T1,...,T()],~+1.
Since F 6 [62],“, repeated application of Lemma 1.4.4 (in case (a)) or Lemma
1.4.1 (in case (b)) gives 11(¢)r"k+lF E [Q]kS. (Note that in case (b), we may
assume that k S 6 — g — 1 and note the condition on the depth of R/1['-"‘l is
automatically satisfied since G is Cohen-Macaulay.) By the assumption on the
symmetric powers we have [Q]1.S C [Q]1S, hence Il(¢)r‘k+1F 6 [Q]IS. Now as in
the proof of Proposition 1.4.2, it follows that 11(cf))"_k‘H C J : I. Let C = [1153-].

e

It will be enough to show that r S 6 — g + 5. For then

 

 

 

 

 

 

rS6—g+£
r—k+l
=e—g+1——1
—k'
=6—g+1+r
=6—g+l+r/e—-k/e
from which it follows that
r<6—g+1—kk
- l—l/e
_ (6-g+1)e—k
_ e—l
__M—g+1Xa—U+6-g+l—k
_ e-l
6— 1—k
=6—g+1+ 9+ .
e—l

Now by the assumption that I C 11(45)‘, we have

1£+1 = 1‘1 c (11(45):)51 c 1,(¢)*-e+11 c (J : 1)1 c J.

49

In particular, 1514""+1 C J1["9. The result now follows from the next lemma. Cl

Lemma 1.4.6 Let R be a local Cohen-Macaulay ring with infinite residue
field, let I an R-ideal with analytic spread 6 which satisfies G1, let J be a minimal
reduction of I with ht J : 1 2 6, and assume that G is Cohen—Macaulay and that
I"+1 C JIl'g, for some integer s 2 1.

Then I"+1 = JI".

Proof. Using Remark 1.1.10 this follows from [4, proof of 5.2]. Cl

Theorem 1.4.5 was inspired by the works of Aberbach, Huckaba and Huneke ([3])
and Aberbach and Huckaba ([2]), where it was shown when the analytic deviation

is one or two, respectively.

Theorem 1.4.7 Let R be a local Gorenstein ring of dimension d with infinite
residue field, let 1 be an R-ideal of grade g 2 2, analytic spread 6 and minimal
number of generators n = 6 +1 and assume that I satisfying G(, that depth R/Ij Z
d-g—j +1 for 1 Sj S 6—g+1 and that 1 C hwy—9+2, where qb is a matrix

with n rows presenting I. Then the following conditions are equivalent.

(a) After elementary row operations, 11(6) is generated by the last row of 43;
(b) r(1) S6—g+1;

(c) R is Cohen-Macaulay.

Proof. Since 1 has second analytic deviation one, (a) and (b) are equivalent by
[69, 5.1]. Now by Theorem 1.1.4, 1 satisfies AN; and thus (b) implies (c) by
Corollary 1.2.10. Now assume (c) holds. Then in particular G is Cohen-Macaulay

([33, 1.1]) and hence (b) follows from Theorem 1.4.5. Cl

50

Naturally we can obtain stronger results by assuming the vanishing of the torsion

of sufficiently many symmetric powers.

Corollary 1.4.8 Let R be a local Cohen-Macaulay ring with infinite residue
field, let I be an R-ideal of grade g, analytic spread 6, minimal number of
generators n = 6 + 1, assume that 1 satisfies Ge, that SJ-(I) E Ij for
l S J S 6 — g + 1, that G is Cohen-Macaulay and let (6 be a minimal presen-
tation matrix of I. Then

(a) fit—9+2 9* 0;

(b) ifI C 11(qb)2 then r(1) = 6— 9 +1.

Proof. To prove (a) put I: = max{j | 5,-(1) E 1’}. Since k 2 6 — g +1, 1 satisfies
the assumptions of Theorem 1.4.5(a). But by the proof, putting e = 1 shows that
k S 6 - g + 1. As for (b), by the assumption on the symmetric powers, r(1 ) = 0 or

r(1) 2 6 — g + 1. Since n > 6, the result follows from Theorem 1.4.5. Cl

Remark 1.4.9 Under the assumptions of Theorem 1.4.5 (a) or (b), assume that
e2[fi+l]andthatg22.

Then R is Cohen-Macaulay.

Proof. Since 1 satisfies Ge, it is enough to show that r < 6 by Theorem 1.2.1. Since

3%? 2 1 by Corollary 1.4.8(a), this follows immediately from Theorem 1.4.5. Cl

We should point out that J. Lipman has shown that G Cohen-Macaulay implies
that R is Cohen-Macaulay whenever R is a regular local ring (or more generally is

pseudo-rational ([49] )).

The following theorem complements one of the main results of [61]. It applies
immediately to grade 2 perfect ideals and grade 3 Gorenstein ideals satifying G3

and having second analytic deviation one.

51

Theorem 1.4.10 Let R be a local Gorenstein ring with infinite residue field,
let I be a strongly Cohen-Macaulay R-ideal of grade g 2 2, analytic spread 6, and
minimal number of generators n = 6 + 1, and assume that 1 satisfies Ge and that
I C 11(¢)2, where d) is a matrix with 11 rows presenting I. Then the following
conditions are equivalent.

(a) After elementary row operations, 11(6)) is generated by the last row of gt;

(b) r(1) =6—g-l-l;

(c) rt(1)= 6—g+2;

(d) A is generated by a single form of degree 6 — g + 2;

(e) R is Cohen-Macaulay;

(f) G is Cohen-Macaulay.

Proof. Since 1 is strongly Cohen-Macaulay and satisfies G:, we have that 31(1) E 11
and depth R/Ij Z d—g—j+1 for 1 S j S 6—g-l-1, and 1 satisfies AN; by Theorem
1.1.4. In particular, r(I) _>_ 6—g+ 1. Now (a) and (b) are equivalent as in Theorem
1.4.7, (b) implies (c) and (d) from Corollary 1.3.4, (c) implies (b) since 1 has
second analytic deviation one, and trivially (d) implies (c). By Corollary 1.2.10,
(b) implies (e), while (e) implies (f) by [33, 1.1]. Finally (f) implies (b) by Corollary

1.4.8(b). El

Remark 1.4.11 ([69, 211]) Under the assumptions of Theorem 1.4.10,

rm) = V(11(¢)) + g - 2 and 1‘(G) = V(11(¢)) + 1-

Remark 1.4.12 The condition 1 C 11(qu)2 in Theorem 1.4.10 is really essen-
tial. A. Simis and B. Ulrich have discovered examples of strongly Cohen-Macaulay
generically complete intersection prime ideals with second analytic deviation one in

a Gorenstein ring (the diagonal ideal of a certain codimension 3 Gorenstein algebra)

52
for which G is Cohen-Macaulay but R is not, and (after adjoining variables to the

ideal) R is Cohen-Macaulay but the reduction number is not the expected value.

Example 1.4.13 Let 1 be the prime ideal defining the Gorenstein monomial
curve k[[t1°,t“,t“,t19]]. A computation on MACAULAY shows that R is nor-
mal. However, using Theorem 1.4.10, it is not hard to show that it is not Cohen-

Macaulay.

1.5 Analytically Independent Elements

Suprisingly, one can use the results of the previous sections to say something about
ideals having second analytic deviation zero, that is to say ideals generated by an-
alytically independent elements. We also show that a question of Ulrich, prompted
by a prior one of Valla about when such ideals have linear type, has a negative

answer.

Proposition 1.5.1 Let R be a local Gorenstein ring of dimension d, let 1
be an R-ideal of grade g and assume that 1 satisfies Goo, that I is generated by
analytically independent elements and that G is Cohen-Macaulay. Then

(a) rt(I) S d(I).

(b) Assume further that depth R/Ij Z d — g —J + 1 for 1 S j S d(I) - 2.

Then 1 is of linear type if and only if I is syzygetic.

Proof. We may assume the residue field is infinite. Part (a) follows from Lemma
1.4.3 and [64, 3.3] (or Lemma 1.4.3 and the proof of Theorem 1.3.3) since n :-
V(I) = 6(1). Now for (b), since G is Cohen-Macaulay, depth R/I"‘-"’1 2 d— 6, and

1 satisfies AN;3 by Theorem 1.1.4. Since I is syzygetic, by part (a) it is enough

53

to show 5,-(1) E 11 whenever 3 S j S d(1). But since 1 = J for any reduction J of

I, this follows from Lemma 1.3.1 by induction on J'. D

Corollary 1.5.2 Let R be a local Gorenstein ring and let 1 be a Cohen-Macaulay
ideal of deviation three satisfying Goo with G Cohen-Macaulay.

Then I is of linear type if and only if 1 is syzygetic.

Proof. It is enough to show by Proposition 1.5.l(b) that V(I) = 6(1). But if
V(I) = g + 3 > 6 then I would automatically satisfy AN;2 by Theorem 1.1.4.
But since 1 satisfies Goo, it would follow that 1 has reduction number at most

one by Remark 1.1.12. Since I is syzygetic, this would contradict the fact that

r(1) 9e 0. Cl

C. Valla asked if a prime ideal in a regular local ring which is generated by
analytically independent elements is necessarily of linear type. A counterexample
was produced in [60, 4.5]. It is a normal homogeneous Cohen-Macaulay prime ideal
of codimension three and deviation three in a polynomial ring in nine variables
which satisfies G,,o and whose Rees algebra is Cohen-Macaulay. By Corollary 1.5.2
the ideal is not even syzygetic.

Ulrich asked in [66] whether such an ideal is of linear type if it is locally generated

by analytically independent elements.

Proposition 1.5.3 There exist homogeneous perfect prime ideals in k[:z:1, ..., 1:6]
of codimension three and deviation three which are locally generated by analytically

independent elements but are not of linear type.

Proof. Take the above counterexample of [60] and specialize it by three general

linear forms. This produces by Bertini’s theorem ( [18]) the required prime ideal,

54

in a six dimensional polynomial ring, which still satisfies Goo, is generated by ana-
lytically independent elements and is not of linear type since the associated graded
ring specializes ([17]). To check that the ideal is locally generated by analytically
independent elements, note that the ideal has deviation at most two on the punc-
tured spectrum, hence is strongly Cohen-Macaulay ([8]). It follows that the ideal

is even of linear type on the punctured spectrum. C]

We include one other criterion for linear type, for ideals having second analytic

deviation at most one.

Proposition 1.5.4 Let R be a local Cohen—Macaulay ring and let I be an R-
ideal of grade g and analytic spread 6 with G Cohen-Macaulay. Then the following
are equivalent:

(a) 1 is of linear type;

(b) V(I) S 6(1) + 1, 1 satisfies G: and 51(1) E Ij whenever 1 SJ S 6— g + 2.

Proof. We may assume the residue field is infinite. It is enough to show that (b)
implies (a). By Corollary 1.4.8(a) it follows immediately that n = V(I) E 6 + 1.
Hence n = 6, and by Proposition 1.5.l(a) we conclude that I has relation type at
most 6 — g. But by the assumption on the symmetric powers it holds that rt(I) = 1

or rt(I) 2 6 — g + 3. Hence I is of linear type. D

CHAPTER 2

SYMBOLIC POWERS AND DEFORMATIONS

In this chapter we prove a criterion for the power of an ideal to coincide with
its symbolic power: they should coincide locally in codimension one, they should
coincide after deformation, and (the quotient ring by) the symbolic power should
satisfy Serre’s condition (52). Requiring the power to coincide with its symbolic
power after deformation is in general much weaker than requiring them to coincide

on the nose.

Definition 2.1 Let R be a noetherian ring, let 1 be an R-ideal, and let n be a

positive integer. The nth symbolic power of 1 is

1‘") = I“Rw n R

where W is the complement in R of the union of the minimal primes of 1.

The following result is well-known.

Remark 2.2 Let R be a local Cohen-Macaulay ring and let I be a complete
intersection R-ideal.

Then 1(") = 1" for all n.

55

56
Proof. Since I is a complete intersection, the associated graded ring gr1(R) is a
polynomial ring over R/ 1, and hence the modules 1"/1"+1 are R/I-free for all n.

It follows that R/I" is Cohen-Macaulay, hence unmixed, for all n. E]

Proposition 2.3 Let R be a local Cohen-Macaulay ring, let I be an R-ideal
and let (S, J) be a deformation of (R, 1) with S / J equidimensional. Then

(a) R/I is equidimensional;

(b) if R/I satisfies (5),) then so does S/J;

(c) if 1 satisfies (C1).) then so does J.

Proof. Let g = a1, ...,a, C S be a sequence regular on S and on S/J with
(R, 1) = (S/(g),(J,_a_)/(_q)). Since R = S/(g) is Cohen-Macaulay, so is .S', and in
particular is equidimensional and catenary. Hence to show (a), it will be enough to
show that every minimal prime p of (J, g) has height at most ht J + r. But since
both 5' and S / J are equidimensional and catenary, it holds that
htp = dim S- dim S/p = dim S-(dim S/J— ht p/J) = ht J + ht p/J S ht J+r.

To prove (b), let p E Spec S/J with dim (S/J)p = s and let q be a mini-
mal prime of (p, g). Since 5/ J is equidimensional, as above ht q S s + r.Then .
11' = q/(g) E Spec R/I and ht q S 5. Now (Sq,Jq) is a deformation of (R7,17), and
(S/J)p is a localization of (S/ J )q. If s S k then (R/I), is Cohen-Macaulay and
hence so is (S/J)p. Otherwise, depth (R/I); Z k and hence depth (S/J), Z k + r.
The inequality dim (S/J)p- depth (S/J)p S dim (S/J)q— depth (S/J), now im-
plies that depth (5/ J )p Z k. This proves that S/ J satisfies (5).).

Since the property of being a complete intersection is preserved under deforma-

tion or localization, (c) follows as in the proof of (b). Cl

Our main result depends on an observation of Huneke and Ulrich; we include the

57
short proof for convenience. Recall that H ,‘n is the ith local cohomology functor.
Proposition 2.4 ([42, 2.1]) Let (B, m) be a local ring such that B is unmixed, let
a: 6 B be a regular element, set A = B/rB, and assume that depth A/H2,(A) 2 2.

Then H3,(A) = 0.

Proof. We may assume that B is complete. Since H 9,,(A) is supported at {m}, the

exact sequence

0 ——> H3,(A) ——> A —> A/H2,(A) —+ 0
shows that H,‘,,(A/Hg,(A)) = Hjn(A) for all i 2 1. Since depth A/Hg,(A) Z 2, we
have that an(A) = 0.

The exact sequence

0——>B—x>B—+A—>O

induces an exact sequence in local cohomology
0 ——> 11,2,(A) —> H},,(B) is 113,,(13) —> H},,(A).

Since 11,1n(A) = 0, it follows that 113,,(B) = rHEn(B). Thus to show Hg,(A) = 0, it
is enough by Nakayama’s lemma to show that H 3,,(B) is finitely generated. Since

B is complete and unmixed, this follows from local duality (e.g. [10, 358]). E]

The following is the main result of this chapter.

Theorem 2.5 Let R be a local Cohen—Macaulay ring, let 1 be an R-ideal
satisy'fying (CID), let n and t be positive integers, assume that (R, 1) has a
deformation (5', J) with S' / J equidimensional and J (n) = J ”, that 1;") = 1; holds
for all p 6 Spec R/I with dim (R/I)p S t and that depth (R/I(")lp Z 2 for all
p E Spec R/I with dim (R/I),,' > t.

Then 1(") = 1".

58

Proof. Note that the assumptions are preserved after localization at a prime ideal
(the condition (C10) is preserved since by Proposition 2.3(a) R/I is equidimen-
sional). We may proceed via induction on the dimension of R/I and assume that
1(") = 1" holds locally on the punctured spectrum. We may also assume that
dim R/I > t.

Let g = a1, ...,a,. C S be a sequence regular on S and on S/J with
(R,1) = (S/(g),(J,g)/(_q)). For 0 S i S r consider the sequence of deforma-
tions (Si,J.') = (S/(a1,...,a.-),(J,a1,...,a;)/(a1,...,a.-)). Then (Sr,J,-) = (R,I)
and (So,Jo) = (S, J). We will show that the pair ($1-1,J,-(:'[) is a deformation
of (5.,J§")) and that the assumptions are preserved from J.- to J.-_1. Each 5.- is
Cohen-Macaulay, and (S, J) is a deformation of every pair (5.1, J.) By Proposition
2.3, Si/J; is equidimensional and J.- satisfies (GIG) for all i.

We first consider the case say i = r, i.e. the pairs (R, 1) and (T, K) = (SQ--1, Jr_1).
We will denote by “ ’ ” reduction modulo a = or.

There are inclusions

I" C KW C 1‘").

To see the rightmost containment, it is enough to show it locally at every p 6 Ass
R/I(") = Min R/I. Let q be the preimage in T of p. Since 1 satisfies (CID), K, is a
complete intersection. It follows that, locally at p, both ideals in question coincide

with the power 1".

Moreover, it holds that
(*) KW“) = 1W.

The same argument above gives the containment “C”, from which equality follows.

59
Now we are in the position to apply Proposition 2.4. Let B = T/ K (n) . Since
K ('0 is unmixed (as T / K is equidimensional) it follows that B is unmixed (e.g.
[53, 34.10]), and a is regular on B since it is regular on T/K. Because 1(") = 1"
holds on the punctured spectrum, the inclusions I" C W C 1”) imply that KIT)-
is unmixed on the punctured spectrum. Hence if we set A = B /.1:B E R/K’GII it

holds that

A/Hg,(A) = R/KWU)

which by (*) implies that

A/H,‘3,(A) = 1.2/1‘").

By hypothesis, depth R/ 1(") Z 2. Thus we may apply Proposition 2.4 to conclude
that 113, (A) = O, or in other words that Km = I”). It follows that (T, K(”)) is a
deformation of (R, 1(")) via a.

We now observe that K satisfies the same hypothesis as I. Let p E Spec T / K
and let q be a minimal prime of (p, a). If dim (T/K)p > t then since T/K(")
is a deformation of R/1(“) it follows as in the proof of Proposition 2.3(b) that
depth (T/K("))p 2 2. Now suppose that dim (T/K), S t. We must show that
K’s") = K:. By localizing at q, we may assume that q = m. Thus we have a
deformation (T, K (")) of (R, 1(")). But then as dim R/I S t we have by hypothesis
that 1(") = I". Since K”) and K n then have the same image in R, it follows that

Kl") C (K",a), and as a is regular on T/K("),
K‘") = (K",a) n 11‘") = K" + (o) rt KW = K" + em").

Hence by Nakayama’s lemma, we conclude that K ("l = K" and hence this equality

holds locally at p.

60

By descending induction on i, we conclude that (S, J (")) is a deformation of
(R, 1(")) via the sequence g. However, by assumption J ("l = J ”, and hence by

specializing we must have that I”) = I". C]

We separate the two important extreme cases of the theorem.

Corollary 2.6 Let R be a local Cohen—Macaulay ring, let 1 be an R-ideal
satisfying (011), assume that (R, 1) has a deformation (5, J) such that S/J is
equidimensional and that J (”l = J n for some n, and further assume one of the
following conditions holds:

(a) R/I(") satisfies (52) ;

(b) depth R/I(") Z 2 and I(”) = 1" holds on the punctured spectrum.

Then 1(") = I".

An unmixed ideal I is called normally torsionfree if 1(") = I" for every n.

Corollary 2.7 Let R be a local Cohen-Macaulay ring, let 1 be an R-ideal
satisfying (011) and having a normally torsionfree deformation, and assume that
R/1(") satisfies ($2) for infinitely many n.

Then 6(1p) S max{ht I, dim RP — 2} for all p E V(I).

Proof. By Corollary 2.6, we can conclude that R/I" satisfies ($2) for infinitely

many n. The result now follows from Burch’s inequality. C]

For an ideal I, which satisfies (Clo), in a local Cohen-Macaulay ring R, one can
characterize the property that I is a complete intersection by the condition that
R/I" is Cohen-Macaulay for infinitely many n ([16]). Using Theorem 2.5, we obtain

an analogous result for symbolic powers:

61

Corollary 2.8 Let R be a local Cohen-Macaulay ring and let I be an R-ideal
satisfying (C11) and having a normally torsionfree deformation. Then the following

are equivalent:
(a) 1 is a complete intersection ;

(b) R/I(”) is Cohen-Macaulay for infinitely many n.

Proof. By Remark 2.2 (and its proof), it is enough to show that (b) implies (a).
But if (b) holds then by Corollary 2.6 it follows that R/I" is Cohen-Macaulay for
infinitely many n. By Burch’s inequality one has that 6(1) 3: ht 1. Since 1 satisfies
(010), and has a reduction which is a complete intersection, by localizing it follows

that 1 is a complete intersection. Cl

Under stronger assumptions, one can prove sharper versions of Corollary 2.8.

The following partly generalizes a result of Huneke and Ulrich on the powers of

licci ideals ([41]).

Theorem 2.9 Let R be a regular local ring and let I be a licci R-ideal satisfying
(011). Then the following are equivalent:
(a) 1 is a complete intersection;

(b) R/1(") is Cohen-Macaulay for some n 2 3.

Proof. Since 1 is licci, (R, 1) has a normally torsionfree deformation by
[40, 2.6 and proof of 5.3]. The result now follows from [41, 2.8] using Corollary

2.6. CI

We now illustrate how one may actually compute the depth of symbolic powers

in certain cases.

62

Theorem 2.10 Let (R, m) be a local Gorenstein ring and let I be a licci ideal
with dim R/I > O satisfying V(Ip) S max{ ht I, dim Rp— 1} for all p E V(I)—{m}.
Then depth R/I(") = max{dim R - V(I), l} for all sufficiently large n, and for

n > d(I)if1/(1)S dim R.

Proof. Since 1 is licci it is strongly Cohen-Macaulay and has a normally torsionfree
deformation ([40, 2.6 and proof of 5.3]). By the numerical condition on the local
number of generators, it follows that 1 is normally torsionfree, and satisfies Goo, on
the punctured spectrum ([70, 4.2]).

First, if V(I) S dim R — 1, then again by [70, 4.2] I is normally torsionfree and
1 satisfies Goo. Now the result follows from [41, 2.7]. Thus we may assume that
V(I) Z dim R.

Now suppose to the contrary that depth R/1(”) Z 2 for infinitely many n. Then
by Corollary 2.6, I”) = I" and Burch’s inequality implies that 6(1) S dim R — 2.
But then [70, 4.2] implies that V(I) = 6(1) S dim R — 2, which is a contradiction.
Finally, if V(I) S dim R, then again by [41, 2.7] the equality must hold for any

n > d(I). C]

We now give an application to the module of differentials.
Let R = k[[:rl, ...,rn” be a formal power series ring over a perfect field 11:, and

let 1 be a reduced R-ideal. There is a natural exact sequence

0 —. 1/1”) —> 11,.(12) a3 11/1 —> 121(12/1) —. o

where Qk(—) denotes the universally finite module of differentials. From this se-
quence it is clear that the depth of R/ I (2) is intimitely related to the depth of the

module of differentials Q(R/I).

63

Proposition 2.11 Let k be a perfect field, let A = k[[.r1, ...,:cn]]/1 be normal,
and assume that A has a deformation B = k[[y1, ..., ym]] / J such that the conormal

module J / J 2 is a torsionfree B-module. Then the following hold:
(a) 1/12 is a reflexive A-module if and only if Qk(A) is torsionfree.

(b) If A is Cohen-Macaulay, then 1/12 is a Cohen-Macaulay A-module if and

only if depth Qk(A) Z dim A — 1.

Proof. Let R = k[[r1,...,rn]]. Now if 1/12 is reflexive or Cohen-Macaulay, it is
torsionfree and hence 1(2) = 12. It then follows from the fundamental exact sequence
above and the fact that A is normal that Qk(A) is torsionfree (respectively satisfies
depth Qk(A) Z dim A — 1 if A is Cohen-Macaulay).

Conversely, assume that Qk( A) is torsionfree (respectively satisfies depth Qk(A) Z
dim A — 1 and that A is Cohen-Macaulay). Since A is normal, Qk(A) satisfies (5'1)
and hence 1/ 1(2) satisfies (52). It follows that R/ 1(2) satisfies (52), and hence by
Corollary 2.6 that 1(2) = 12. It then follows that 1/12 is reflexive (respectively

Cohen-Macaulay). [I]

We point out an example where the above equivalences fail.

Example 2.12 Let k be a perfect field, let X be a symmetric 3 by 3 matrix of
variables, put R = k[[X]], and let I = 12(X) be the R-ideal generated by the 2 by
2 minors of X. Then A = R/I has an isolated singularity and Qk(A) is torsionfree
(e.g. [68, 3.6]). However, the conormal module 1/12 is not even torsionfree; in fact

det(X) 6 1(2) — 12.

Finally, we give an application to the study of the Artin-Nagata properties of

Section 1.1.

64

Proposition 2.13 Let R be a local Gorenstein ring of dimension d, let 1 be an
unmixed R-ideal of grade g S s satisfying (CIO) and 6(1,) S max{g,dim Rp — 1}
for all p E V(I) with dim Rp S s + 1, and assume that (R,1) has a deformation
(5, J) with J”) = J1 for 1 S j S s — g + 1. Then the following are equivalent:

(a) 1 satisfies AN, ;

(b)depthR/1j_>_d—g—j+1for1Ssz-g+l.

Proof. By [68, 2.9], it is enough to show that (a) implies (b). Now if I satis-
fies AN,, the local assumption on the analytic spread implies by [68, 3.3] that
depth R/IU) Z d — g — j + l for 1 S j S s — g +1. Note that this already
implies that R/I is Cohen-Macaulay since 1 is unmixed. Thus to prove (b), it
will be enough to show that I”) = Ij for 1 S j S s — g + 1. It follows that
depth (R/Im), 2 2 for 1 S J S s —g+ 1 whenever dim (R/I)p > s —g+ 1. But by
[68, 1.11] and [70, 4.2], I is normally torsionfree locally in codimension s + 1. Hence

Theorem 2.5 with t = s —g+ 1 implies that I”) = Ij for 1 Sj S s —g+ 1. Cl

CHAPTER 3

CONSTRUCTION S IN LINKAGE

In this chapter we present ways to construct Cohen-Macaulay ideals having certain
linkage properties. In Section 1 we review some of the basic facts about linkage, and
in particular about the class of ideals which lie in the linkage class of a complete
intersection. In Section 2, taking the tensor product of two algebras over a field
which lie in the linkage class of a complete intersection, we obtain algebras which
are strongly Cohen-Macaulay and strongly nonobstructed, but which do not lie in
the linkage class of a complete intersection. The depth of the twisted conormal
module and the first Koszul homology module are well-known to be invariants of
the linkage class (respectively even linkage class), and we observe in Section 3 that
moreover the depth of these modules modulo their torsion submodule is an invariant
of the geometric linkage class. In Section 4, we study the sum of two geometrically
linked ideals. We prove results which relate the depths of the first Koszul homology
modules and the twisted conormal modules of the two linked ideals to that of
their sum. In particular, we are able to obtain classes of Gorenstein ideals which
are strongly nonobstructed, but are not syzygetic. In Section 5 we construct, by

intersecting two complete intersections, some Cohen-Macaulay ideals of type 2.

65

66

3.1 Licci Ideals

In this section, we recall the most important results about linkage, mainly due to
Huneke and Ulrich, that we will need in this chapter. We first recall the basic

notion:

Definition 3.1.1 Let R be a local Gorenstein ring, and let I and J be two
R—ideals.

(a) I and J are linked, denoted by 1 ~ J, if there is an R-regular sequence
9;: 01,...,ag in IDJ such that J = (g) :1 and I = (g) : J.

(b) 1 and J are geometrically linked if 1 and J are linked and have no common

associated primes.

It follows that if 1 and J are linked, then 1 and J are unmixed ideals of

grade g. In addition, if they are geometrically linked then 1 n J = (94).

Proposition 3.1.2 (Peskine-Szpiro [55]) Let R be a local Gorenstein ring, let I
be an unmixed R-ideal, let g be a maximal regular sequence properly contained in
1 and set J = (g) :1. Then

(a) 1 and J are linked (via the sequence g);

(b) R/I is Cohen-Macaulay if and only if R/ J is Cohen-Macaulay;

(c) if R/I is Cohen-Macaulay then (UR/1 E J / (_a_); in particular,
r(R/I) = V(J/(gll-

By iterating this process, one is led to the following:

Definition 3.1.3 Let R be a local Gorenstein ring and let 1 and J be two

R—ideals.

(a) I and J are in the same linkage class if there is a sequence of links

67

I = 10 ~ 11 ~ ~ 1n = J. If n is even, 1 and J are in the same even link-
age class (or are evenly linked, for short). If all the links are geometric, 1 and J
are in the same geometric linkage class.

(b) 1 is licci if 1 is in the linkage class of a complete intersection ideal.

Proposition 3.1.2(b) can now be rephrased as saying that “Cohen-Macaulayness
is an invariant of the linkage class.” We will study various other linkage invariants
in the sequel. The class of licci ideals turns out to be a particularly nice one,
inheriting many good properties trivially observed for complete intersections. We

recall the classic examples:

Theorem 3.1.4 Let R be a local Gorenstein ring and let I be an R-ideal.
(a) (Apéry [6], Gaeta [9]) If I is perfect of grade 2, then I is licci.

(b) (J. Watanabe [77]) If I is a perfect Gorenstein ideal of grade 3, then 1 is licci.

Indeed, one also has structure theorems for these ideals. By the Hilbert-Burch
theorem ([14]) any ideal in (a) is the ideal of n-sized minors of an n + 1 by n
matrix, while by Buchsbaum-Eisenbud ([11]) any ideal in (b) is the ideal of 2nth
order Pfaffians of an alternating 2n + l by 2n + 1 matrix.

The following two results give some of the most important properties which are

invariant under linkage:

Theorem 3.1.5 (Huneke [34]) Let I and J be two ideals which are evenly linked.

Then 1 is strongly Cohen-Macaulay if and only if J is strongly Cohen-Macaulay.

However, strongly Cohen-Macaulayness is usually not an invariant of the entire
linkage class. For example, any non-strongly Cohen-Macaulay Gorenstein ideal is
linked to an almost complete intersection (by 3.1.2(c)) which is of course strongly

Cohen-Macaulay.

68

Theorem 3.1.6 (Buchweitz-Ulrich [13]) Let 1 and J be two perfect R—ideals in
the same linkage class.

Then depth 118mm” = depth J @wR/J.

In particular, the property that the so-called twisted conormal module 1 ® wR/I
is Cohen-Macaulay is a linkage invariant. This was first shown (for generic complete
intersections in a regular k-algebra) by Buchweitz ([12]). This property has been
shown to be important in deformation theory (e.g. [12], [25]) and that motivated

the following definition:

Definition 3.1.7 Let R be a local Gorenstein ring. A Cohen-Macaulay R-ideal

I is strongly nonobstructed if 1 (8) w R / I is Cohen-Macaulay.

If I is Gorenstein, then I @103” = 1/12; in this case strongly nonobstructedness
is simply the Cohen-Macaulayness of the conormal module.

Since any complete intersection is trivially both strongly Cohen-Macaulay and
strongly nonobstructed, we obtain the following (using the fact that any complete

intersection is linked to itself):

Corollary 3.1.8 Every licci ideal is strongly nonobstructed and its entire linkage

class is strongly Cohen-Macaulay.

Part of our motivation was to understand whether the converse of this corollary
holds. Before our investigation began, there was apparantly no known examples of
nonlicci ideals with these properties. We will construct such examples in the next
section.

The following main result of [38] allows one to reduce many questions about licci

ideals to the simplest examples described in Theorem 3.1.4.

69

Theorem 3.1.9 (Huneke-Ulrich [38, 4.3]) Let R be a regular local ring and let I
be a licci R-ideal which is not a complete intersection. Then (R, 1) has a essentially
a deformation (5, J) with S/J E (P[X]/K)(mp,x), where P is a regular local ring,
K is a P-ideal and either

(a) X is a generic 2 by 3 matrix and K = 12(X), or

(b) X is a generic alternating 5 by 5 matrix and K = P f4(X )

We will need the following results as well.

Proposition 3.1.10 ([39]) Let R be a local Gorenstein ring with infinite residue
field, let S be a faithfully fiat local Gorenstein extension, and let I be an R-ideal.

Then I is licci if and only if 15 is licci.

Proposition 3.1.11 ([38, 216]) Let R be a local Gorenstein ring, let 1 be an
R-ideal, let (S, J) be a deformation of (R, I) and let I = 10 ~ 11 ~ ~ In be a
sequence of links in R. Then there exists a sequence J = Jo ~ J1 ~ ~ Jn in S

such that (S, J.) is a deformation of (R, 1;) for all 0 S i S n.

It follows from the above result that if 1 is licci and (S, J) is a deformation of

(R, I), then J is also licci. Moreover, one has the following:

Theorem 3.1.12 ([40, 5.1]) Let k be a field, R = k[[:r1,..,:rn]], and let I be a
licci R-ideal. Then (R, I ) has a deformation (5, J), where .S' is a power series ring

over k, such that J satisfies Goo.

Finally we recall the “shift condition” satisfied by any homogeneous licci ideal.

Theorem 3.1.13 ([38, 5.13]) Let S = k[:r1, ...,:rn] be a positively graded poly-

nomial ring over a field k, let I be a homogeneous S-ideal, such that I(r1....,:cn) is

7O

licci, having homogeneous resolution
0 —> ef;,S(—n,,) —+ ——. @f;,5(—n1.)—> s —> 5/1 ——> 0.

Then max {ngg} > (g — 1) min {nu}.

3:2 Tensor Products

In this section we work in the category CM(k) of complete local Cohen-Macaulay
lc-algebras with residue field 11:. Given any A 6 CM (1:), we say that A is licci,
strongly Cohen-Macaulay or strongly nonobstructed if there exists a presentation
A E k[[rl,...,rn]] /1 where I has the corresponding property. For the property
of being strongly Cohen-Macaulay and strongly nonobstructed, this definition is
independent of the given presentation. The property of being licci is independent
of the presentation if k is infinite, as can be seen from [39, 4.7].

We say that two algebras A and B in CM(k) are linked if there exists a regular
R E CM(k) and presentations A E R/I, B E R/J, where 1 and J are linked
R-ideals.

Given A, B E CM(k), one may form the (complete) tensor product A®kB. The

following statements are by and large well-known.

Proposition 3.2.1 Consider the following properties of an algebra A E CM(k):
(a) strong Cohen-Macaulayness;

(b) strong nonobstructedness;

(c) Cohen-Macaulayness of Tor,R(A,wA) for every i.

If two algebras in CM(k) enjoy any of the above properties, then so does their

complete tensor product.

71

Proof. Let A,B 6 CM (k) and let R,S E CM(lc) be regular local rings mapping
onto A and B respectively. Now A®kB has a resolution over R®kS given by the
complete tensor product of the resolutions of A and B. In particular it follows that
A®kB is Cohen-Macaulay so A®kB 6 CM(lc).

Now (a) is proved in [36, 1.9] and (c) follows from the Kunneth formula ([50])
Tor?®*S(A®kB,wA®kB) E 633:0 Tor?(A,wA)®k Tor?_J-(B,w3).

Since 1 (8111),; E Torfz(A,wA), (b) follows from (c) with i = 1. C]

One might expect that virtually any reasonable homological property would be

preserved from A and B to the tensor product. However one has the following:

Theorem 3.2.2 Let A and B be complete local licci algebras over the residue
field It.

(a) If A and B are generically complete intersections, then the entire linkage
class of A®kB is strongly Cohen—Macaulay if and only if A or B is Gorenstein.

(b) A®kB is licci if and only if A or B is a complete intersection.

Corollary 3.2.3 Let A and B be complete local licci algebras over the residue
field I: which are not complete intersections and let C = A®kB. Then
(a) C is strongly Cohen-Macaulay and strongly nonobstructed, but not licci;

(b) Tor?®*S(C,wC) is Cohen-Macaulay for every i.

Proof. This follows from Proposition 3.2.1 using Theorems 3.1.5, 3.1.6 (and [13] for

part (b)), and Theorem 3.2.2(b). D

This gives a natural construction of algebras which are not licci but enjoy many

of the properties usually only observed for that class. This lends support to the

72
belief that it is very difiicult to find necessary and sufficient conditions for an ideal
to lie in the linkage class of a complete intersection. Before giving the proof of

Theorem 3.2.2, which is somewhat technical, we give two explicit examples.

Example 3.2.4 Let A = k[[X]]/12(X) where X is a generic 2 by 3 matrix,
and let B = k[[Y]] / P f4(Y) where Y is a generic 5 by 5 alternating matrix. Then
A®kB is a Cohen-Macaulay k-algebra of embedding codimension 5, deviation 3 and
type 2 which is strongly nonobstructed and whose entire linkage class is strongly

Cohen-Macaulay, but which is not licci.

Example 3.2.5 Let A = k[[X]]/Pf4(X) and R = k[[X]], where X is a generic 5
by 5 alternating matrix. Then G = A®kA is a Gorenstein k-algebra of embedding
codimension 6 and deviation 4 which is strongly nonobstructed, strongly Cohen-

Macaulay, satisfies Exti (C, C) is Cohen-Macaulay for every i, but is not licci.

R®hR

We introduce an explicit construction, which we call the transversal link, which
we will use in the proof of Theorem 3.2.2(a).

Let I; C R1 = k[[X]] and 12 C R2 = k[[Y]] be unmixed ideals which are
generically complete intersections and let R = k[[X,Y]]. Let 11 ~ J1 and 12 ~ J2
be any two geometric links, linked by the regular sequences g1 and g2 respectively.
Set L = ((gl)R,(g2)R) : (11R + 12R). By Proposition 3.1.2(a), L is linked to
11R + 12R. Moreover, this link is again geometric (by transversality). We call L a

transversal link. We summarize its properties in the following proposition.

Proposition 3.2.6 Let R1 = k[[X]],R2 = k[[Y]],R = k[[X,Y]], let 11 C R1
and 12 C R2 be Cohen-Macaulay ideals, let J.- = (g,) : I,- be geometric links
and set A.- = Ri/I; and B; = Ri/J; for i = 1,2. Let I = 11R + 12R and let

L = ((g1)R, (g2))R : 1 be its transversal link. Then

73
(a) d(Ll = 71441)"le-
(b) r(R/L) = r(B1)+ r(B2).
(c) If depth Rl/Jl2 2 dim B1 - 1 and depth R2/J22 Z dim B2 — 1 then
depth R/L2 2 dim R/L — l.
(d) If A1 and A2 are not Gorenstein, then L is not syzygetic.
(e) If A1 is Gorenstein, r((A2)p) S dim(A2)p for every p E V(J2), and J2 is

strongly Cohen-Macaulay, then L is strongly Cohen-Macaulay.

Proof. If “ ' ” denotes reduction modulo g1 , {12 then we have L' = J1' J2' by transver-
sality. Since ghgz are part of a minimal generating set of L (as can be seen by

factoring out the variables X or Y), one has

aKL) = V(L') = V(J{)V(Ji) = r(411)?"(1‘12)

by Proposition 3.1.2(c). This proves (a). Similarly, for (b) we have that

1‘(R/L) = ”(I/(21.22)) = UNI/(21)) + NIB/(9.2)) = 1‘(Bll + r(132)-

Now we show (c). Let g.- = grade 1;, g = gl + 92 and Q = $11,952. Since

(a) + Lz/L2 E (_l/(s) 0 L2 = (4)“le g (R/ng,

as g generically generates L, it follows that there is an exact sequence

0 —> (R/L)9 -—> L/L2 ——+ L’/(L’)2 —-+ 0.

If we let “ ‘ ” denote reduction modulo _q,, then similarly there is an exact sequence
for i = 1, 2

0 —) (Ri/Jg)!“ ——> Ji/J,2 -—-> JI/(Jf)2 —> 0.

74

But since (L’)2 = (Jl’ )2(J§)2, there is also an exact sequence
0 —+ 12’/(L')2 —. R’/(J,’)2 ea R'/(J;)’~’ —> R'/(.1{)2 + (12')2 —-> 0.

By the transversality of J1 and J2, the result follows by chasing depths in these
sequences.

To show ((1), it will be enough to show that L’ is not syzygetic ([34, 1.4]). Chang—
ing notation, we write R for R’. Then L = J1 J2 is a product of two transversal
ideals. Moreover, since neither 11 nor I2 is Gorenstein and V(Ji) = r(Ri/ 1;) by
Proposition 3.1.2(c), it follows that neither J,- is principal. Suppose that a1,b1
and a2,b2 are part of a minimal generating set of J1 and J2 respectively. Then
a1a2,a1b2,a2b1, b1 b2 is part of a minimal generating set of L. We conclude that L
is not syzygetic since it admits the quadratic relation T1T4 = T2 T3 on these minimal
generators.

It remains to show (e). It will be enough to show that L’ is strongly Cohen-
Macaulay ([34, 16]). Changing notation again, we write R for R’. It follows
from Proposition 3.1.2(c) that J2 satisfies Goo. Now since 11 is Gorenstein, by
Proposition 3.1.2(c) we have that J1 is cyclic. Write J1 = (b) for some element
b E R1. Then L = bJ2 also satisfies Goo. By [68, 2.13] it will be enough to show
that depth R/Lj Z dim R — g — j + 1 for 1 S j S d(L) — 1. But since J1 is
an almost complete intersection, it holds that depth R1/ Jlj Z dim B1 — 1 for all
j 2 1, and as J2 is strongly Cohen-Macaulay and satisfies Goo it follows that

depth R2 / J2” 2 dim B2 — j + 1 for all j Z 1. Hence from the exact sequence
0 —. R/LJ' —> R/JljR ea R/JgR —+ R/JljR + 12112 —> 0,

we conclude that depth R/Lj Z dim R/ L — j + 1 for all j 2 1. This completes the

proof. Cl

75

Corollary 3.2.7 With the notation of Proposition 3.2.6, assume that 11
and 12 are licci ideals which are not Gorenstein. Then L is strongly nonobstructed,

depth R/L2 Z dim R/L — l, but L is not syzygetic.

Proof. Since 11 and 12 are licci, they are strongly nonobstructed (Corollary 3.1.8).
By Proposition 3.2.1 it follows that I = (11 + I2)R is strongly nonobstructed since
it defines the complete tensor product A1 ®k A2. Now Theorem 3.1.6 implies that its
transversal link L is also strongly nonobstructed, and Proposition 3.2.6(d) implies
that L is not syzygetic.

By Theorem 3.1.5, it holds that J,- are strongly Cohen—Macaulay and since they

are generically complete intersections, we have the exact sequence

0 ——> H1(J) —> B" ——> J/J2 —+0
with J = J5, B = B.- and n = V(J.) for i = 1,2. Since H1(J.-) is Cohen-
Macaulay we have that depth R.-/J,-2 2 dim B.- — 1. Hence by Proposition 3.2.6(c),

depth R/L2 Z dim R/L — 1. Cl

Example 3.2.8 Let R = k[[:r1, ...,r4,y1,...,y4]] and let 11 = (rlr2,x1r3,r3r4)
and 12 = (y1y2,y1y3,y3y4), which are grade 2 perfect ideals, and in particular are
licci. Then I = 11 + 12 is a perfect R-ideal of grade 4 and deviation 2, which is
strongly Cohen-Macaulay, strongly nonobstructed, but not licci by Corollary 3.2.3.

Linking via the regular sequences 2311;243:134 and mm, mm produces a transver-
sal link L = (r1x2,r2y2,$2y4,x32:4,r4y2,:r4y4,y3y4,y1y2). By Proposition 3.2.6
and Corollary 3.2.7, L is a perfect R-ideal of grade 4, deviation 4 and type 2 which

is strongly nonobstructed, depth R/L2 Z dim R/ L — 1, but L is not syzygetic.

Proof of Theorem 3.2.2. We first prove part (a). Since A and B are licci, by

Proposition 3.1.12, they have deformations A and B in CM(k) which satisfy Goo.

76

Then A®kB is a deformation of A®kB. Now A or B is Gorenstein if and only
if A or B is Gorenstein. Since the property of being strongly Cohen-Macaulay is
preserved under deformation (by [41, proof of 2.1], since A and B are generically
complete intersections), by Proposition 3.1.11 it follows that the entire linkage class
of A®kB is strongly Cohen-Macaulay if and only if the entire linkage class of A®kB
is strongly Cohen-Macaulay. Hence we may assume that A E R1 /11 and B E R2 /12
where R; = k[[X]], R2 = k[[Y]] and 11 and I2 satisfy Goo. Let R = k[[X,Y]] and
let L be a transversal link of [IR + 12R.

Now assume that A or B is Gorenstein. By symmetry, let us assume that A
is Gorenstein. To show that the entire linkage class of A®kB is strongly Cohen-
Macaulay, it is enough to show by Theorem 3.1.5 that a single link is strongly
Cohen-Macaulay. In particular, it is enough to show that L is strongly Cohen-
Macaulay. Since 11 is Gorenstein and J2 is strongly Cohen-Macaulay, this will follow
from Proposition 3.2.6(e) if 12 satisfies the local condition on the type. However,
as 12 is licci, by [40, 2.5] and [38, 2.17] it always has a deformation satisfying this
condition. By the previous deformation argument, it follows that L is strongly
Cohen-Macaulay.

For the converse, if the entire linkage class of A®kB is strongly Cohen-Macaulay
then L is strongly Cohen—Macaulay, and in particular L is syzygetic. But then by
Proposition 3.2.6(d), we immediately obtain that either A or B is Gorenstein.

It remains to prove part (b). First observe that by transversality that if A
or B is a complete intersection, then A®kB is licci. Indeed, if say B is a com-
plete intersection and A ~ A1 ~ An where An is a complete intersection, then
A®kB ~ A1®kB - .. ~ An®kB and of course An®kB is a complete intersection.

Conversely, we must show that if A and B are not complete intersections then

77
A®kB is not licci.

If k’ denotes the algebraic closure of k, then let A’ = A®kk’ and B’ = B®kk’. If
A®kB is licci then so is A’ ®ktB’ . If one of A’ or B’ is a. complete intersection then
the same holds for A and B. Hence we may assume that k is algebraically closed.
In particular, since I: is infinite, the property of being licci is independent of the
presentation of the algebra.

Choose presentations A E R/I and B E S/J, where R = k[[r1,...,a:d]] and

S = k[[y1, ..., y,]]. Since I is licci, there is a sequence of links
(3:1,...,:rg) ~ 11 ~ ~ 1,._1~ 1,. = 1.
By [38, 2.17], there is a sequence of links in a polynomial extension T = R[Z]
(2:1,...,:rg)T ~ L1 ~ ~ Ln_1~ Ln = L

with the property that (Tq, L,,) is a deformation of (R, 1) for some prime
q 6 Spec(T). Similarly, there is an ideal L’ C U = 5 [Z ’] and a prime q’ ESpec(U)
such that (W , L9.) is a deformation of (S, J). It follows that there is a deformation

of tensor products

AA

(T/L).®.(U/L').I ——+ 4&3.

Since the property of being licci is preserved under deformation (3.1.11), it will be
enough to show that the former ring is not licci.

Now by construction of the T—ideal L and U -ideal L’ (cf. [38]), we may view
them as an extended ideals from the polynomial subrings k[X , Z] and k[Y, Z ’]
respectively. If we let p = q n k[X, Z], p’ = q’ 0 k[Y,Z’], A0 = (k[X,Z]/L)p
and B0 = (k[Y, Z ’ ] / L’ ),r then there is a faithfully fiat morphism of local rings

AA

(Ao ®k Bo)M —'> (T/L)q®k(U/L')q'a

78

where M is the maximal ideal m A0 (8) B + A0 63377130. Since the property of being licci
descends from faithfully fiat extensions by Proposition 3.1.11, it is then enough to
show that (after localizing at .M) that A0 (81k B0 is not licci, i.e. that there is some
defining ideal which is not licci. Hence we have reduced the problem to the case
where A and B are essentially of finite type over I: and have replaced the complete
tensor product by the ordinary tensor product.

Now we would like to apply Theorem 3.1.9. Since A E R/I and B E S/ J are
licci and not complete intersections, they have essentially a deformation to a pair
(P[W](m,w), K) where P is a regular local ring and K is either the ideal of 2 by
2 minors of a generic 2 by 3 matrix or the ideal of 4 by 4 Pfaffians of a generic
alternating 5 by 5 matrix. Further, if A’ is essentially a deformation of A and
B’ is essentially a deformation of B, where all the rings are essentially of finite
type over k, then (A’ (8)). B’) Mr is essentially a deformation of (A (8);, B) M, where
M’ = m A: (8 B’ + A’ (8) m 8'; here we use the fact that k is algebraically closed to
ensure that if p €Spec(A) and q €Spec(B) then p 18> B + A (8) q is a prime ideal
in A (81. B (e.g. [78, p.198]). Since by Proposition 3.1.11 the property of being
licci is preserved under essentially a deformation, we have reduced to the case
where A and B are either of the algebras described in Theorem 3.1.9. However,
if A E (P[W]/K)(m,w) and B E (P’[W’]/K’)(mt,wt) then there is a faithfully fiat
morphism

(k[W]/1{ ®k k[W’]/K’)MI —-> (A ®k B)M.

Now by faithfully fiat descent, we are reduced to showing that A (8)), B is not licci
(after localizing at the maximal ideal generated by the variables) where A and B
are one of the algebras G = k[X]/I2(X), where X is a generic 2 by 3 matrix, or

D = k[Y]/Pf4(Y), where Y is a generic 5 by 5 alternating matrix. By part (a) of

79
the theorem we know that one of A or B is Gorenstein. Thus it only remains to
show that both (C (8);, D)M and (D (231k DlM are not licci. We show this by checking
that they do not satisfy the “shift condition.”
Now C and D have homogeneous resolutions over R = k[X] and S 2 MY] of the
form

0 ——> R2(—3) —> R3(—2) —-> R ——> C ——> 0
0 —-> S(—5) —> S5(—3) —> $5(-2) ——> S —> D —-> 0.

Thus C (8)], D and D (8)), D have respectively graded resolutions over T = R 18>); S

andU=S®kSoftheform
0——+T2(—8)~-———>T8(—2)—+T——>C®tD——>O

0—>U(—10)—>---——iU1°(—2)-—-+U——>D®kD-—>0.

Now C (8);, D has projective dimension 5 and max{ng;} = 8 = (g — 1)min{n1.-}.
Similarly D®k D has projective dimension 6 and max{ng;} = 10 = (g—1)min{n1,-}.
It now follows from Theorem 3.1.13 that (C (81k D) M and (D (8);. D)M are not licci.

This completes the proof of Theorem 3.2.2. C]

One could ask if the tensor product is essentially the only way to construct the
examples of this section. More specifically, for low codimensional ideals one could

ask the following:

Question 3.2.9 Let R be a regular local ring and let I be a perfect R-ideal of
grade g. Then is I licci if 1 satisfies any of the following conditions?

(a) g = 3, I is strongly nonobstructed and strongly Cohen-Macaulay.

(b) g = 4, 1 is strongly nonobstructed and the entire linkage class is strongly

Cohen-Macaulay.

80

(c) g = 4, I is Gorenstein and strongly Cohen-Macaulay.

(d) g = 5, 1 is Gorenstein, strongly nonobstructed and strongly Cohen-Macaulay.

3.3 Invariants of Geometric Linkage

In this section we point out that the depth of certain modules modulo their torsion
is an invariant of the geometric linkage class. We will often make use of the following
well-known fact, which can be shown by tensoring with the total ring of quotients.

For an R/I—module built from an ideal I, 1' always denotes the R/I-torsion.

Lemma 3.3.1 Let A be a noetherian local ring, let
Ml 1’. M2 —+ M3 —. 0

be an exact sequence of finitely generated A-modules having a rank, assume that
M2 is torsionfree and that rank M2 = rank M1 + rank M3.

Then r(Ml) = ker 45.

The following is an analog of Theorem 3.1.6.

Proposition 3.3.2 Let R be a local Gorenstein ring and let 1 and J be two
Cohen-Macaulay R-ideals which are geometrically linked.

Then depth (I®wR/1)/r = depth (J®wR/J)/r.

Proof. If the ideals have grade 0 then 1123/] E J by Proposition 3.1.2(c) and the
result is trivial. Also, if 1 is a complete intersection then the result would just
follow from Theorem 3.1.6. Hence we may assume that I and J have positive grade
9 and are both not complete intersections. Under these assumptions we will show

that depth (1 (8)1123”) /r = depth 1 J . By symmetry, this will prove the proposition.

81

Consider the natural exact sequence
0 —+ wR/I ——-> R/(g) ——> R/J ——> 0.
Tensoring this sequence with R/I gives an exact sequence
Torfz(wR/1,R/I)——> Term/(41mm —+ Term/mu)
which after natural identifications corresponds to the sequence
1 (8003/1 '—> (gl/(Q)I —‘> (El/[J —* 0-
Since it holds that
(El/(Q)I '5 (Q) 812 R/1 g (El/(Q2 ®R/(_q) R/I g (R/(Qllg cfits/(9,) 13/1 E (R/Il"

is free over R/I and 1 is generically generated by g (since the given link is geomet-

ric), by Lemma 3.3.1 there is a short exact sequence
0 ——> (I ®wR/1)/1- —-> (R/I)9 ——> (g)/IJ —-> 0.

Now from this sequence it will follow that
depth (I®wR/I)/r = depth (_q)/IJ+1 = depth R/IJ +1 1: depth IJ,

as long as R/ 1 J is not Cohen-Macaulay. However, if R/ I J would be Cohen-
Macaulay then it would be unmixed and it would follow that 1 J = 1 H J = (91); but
cutting down to the grade one case, we obtain that 1 J is principal, which implies
I or J is principal since R is local. Since I and J are not complete intersections,

this gives our contradiction and completes the proof. D

Corollary 3.3.3 Let R be a local Gorenstein ring and let 1 and J be two

Gorenstein R-ideals in the same geometric linkage class.

82
Then depth R/1(2) = depth R/J(2).

Proof. This follows immediately from Proposition 3.3.2 since if 1 is Gorenstein then

1®wR/I = 1/12 and (I®wR/1)/r g 1/1(2). Cl

The following application shows that the depth of Q(A) (291.11,; is invariant for two
geometrically linked rigid algebras. Recall that a noetherian local k-algebra is rigid

if every infinitesimal deformation is trivial.

Corollary 3.3.4 Let A and B be complete reduced local Cohen-Macaulay
algebras over a perfect residue field which are geometrically linked and assume that

A and B are both either rigid or not rigid.

Then depth 9(A) ® w(A) = depth 9(B) ® w(B).

Proof. Choose presentations A E S/I, B E S/J where S = k[[:r1, ..., rn]] and I and

J are linked S-ideals. The natural exact sequence of A—modules
1/12 —> A" —-—> 9(A) ——+ 0
induces by Lemma 3.3.1 the short exact sequence
0 ——> (I/I2 (gang/1- —> w}; ——> 9(A) (3101,, ——> 0,
and analogously there is an exact sequence of B-modules
0 ——+ (J/J2 ®wB)/r —) tug —+ 9(B) ®w3 -——+ 0.

The result now follows from Proposition 3.3.2 by chasing depths in these sequences,

using the fact ([25, 1.3]) that A is rigid if and only if Ext],(Q(A) ®wA,wA) = 0. C]

Now we give the analog of Theorem 3.1.5.

83

Proposition 3.3.5 Let R be a local Gorenstein ring and let 1 and J be two

Cohen-Macaulay R-ideals which are geometrically evenly linked.

Then depth H1(1)/r = depth H1(J)/T.

Proof. We follow the proof [34, 1.11] of the corresponding result for H1(1) We may
assume that I and J are doubly linked, say I = (91) : K, K = (g) : J, where both

links are geometric. Since I is generically a complete intersection, Lemma 3.3.1

induces the exact sequence
0 ——> H1(I)/-r -—+ (R/I)" ——+ 1/12 —> 0.

Put t = depth H1(I)/r. Then t = min{depth R/I2 + 1, dim R/I}. We induct on
the grade g. If g = 0 then 1 = J so the result is trivial. If g = 1 then I and
J are isomorphic as ideals and the result follows as well. So we may assume that
g 2 2 and that the result is known for smaller 9. By prime avoidance, one may
choose an element '79, which is a minimal generator of (B), such that 011, ..., 019-1 , 79
is an R-regular sequence which generically generates K. Extend 79 to '71, ...,79, a
minimal generating set of (g) and let L = (01, ..., 019-1 , 79) : K. Note that this link
is geometric, and 1 and J are both doubly linked to L. Hence we may now assume
that 1 and J are doubly linked and that both linking sequences contain a common
element, say :1. Since the links are geometric we have that (:1) fl 12 = 2:1, which can
be checked by localizing at the associated primes of I.

Now let “ " ” denote reduction modulo 2:. Then there is an exact sequence
0 ——> R/I ——+ R/I2 ——> R"/(I")2 —-> 0.

It follows from this sequence that t(I‘) = t and by induction we know that t( J ‘) =

t(I"). Hence t(J) = t(J“) = t. E]

84
One may restate Theorem 3.1.6 superficially as follows:
Remark 3.3.6 Let R be a local Gorenstein ring and let I and J be two perfect

R-ideals in the same linkage class which are both generically complete intersections.

Then depth (H1(I) @wR/1)/T = depth (H1(J) @wR/J)/T.

Proof. The exact sequence
H1(I) —+ (R/I)" —> I/I2 —> 0
induces by Lemma 3.3.1 an exact sequence
0 —-> (H1(I) ®wR/1)/T ——+ 1.1;,” ——> U]2 ®wR/1 ———> 0.

But since depth 1/12 (8) CUR/1 = depth J/J2 (8) 1113/1 by Theorem 3.1.6, the result

follows. C]

We point out that, despite the previous results, depth 111(1) 8) wR/I is not a
linkage invariant. By [65, 2.19] if R is a regular local ring containing the rational
numbers and if I is a licci R-ideal which is generically a complete intersection, then
111(1) @wR/I is torsionfree if and only if I is Gorenstein. However, by Remark 3.3.6
it holds that any for any licci ideal I which is generically a complete intersection,

(H1(1) ® wR/1)/r is Cohen-Macaulay.

3.4 Sums of Links

In this section we use sums of links to construct more nonlicci ideals with certain
specified properties. We begin with the following observation from [55], whose proof

is short enough to be repeated.

85
Proposition 3.4.1 (Peskine-Szpiro) Let R be a local Gorenstein ring and let 1
and J be two Cohen-Macaulay R—ideals of grade g which are geometrically linked.

Then I + J is a Gorenstein ideal of grade g + 1.

Proof. By factoring out the regular sequence defining the link, we may assume that

g = 0. Since then 1 I) J = 0 there is the exact sequence
O——>R—->R/I€BR/J——>R/I+J——>0.

It follows that dim R/I + J = d — 1 where d = dim R. Applying the functor

Hom(k, —), where k is the residue field, we obtain an exact sequence
0 —) Extfiz'l(k,R/1 + J) ——) Ext‘}:(k,R) ——-> Ext‘k(k,R/1)G§Ext‘}’i(k,R/J).

But then r(R/I + J) = dim): Extd‘l(k,R/1 + J) S dim); Ext“(k,R) = r(R) = 1,

hence R/I + J is Gorenstein. C]

This turns out to be an interesting way to construct new examples of Gorenstein

ideals. For example, Ulrich has shown the following:

Theorem 3.4.2 ([67, 2.1]) Let R be a local Gorenstein ring and let I and J be
two licci R-ideals which are geometrically linked.

Then I + J is licci.

In [67], Ulrich also gives conditions for 1 + J to be strongly nonobstructed, but
his assumptions force 1+ J to be syzygetic as well. We generalize his result, making

use of the following result whose proof may be found in [67, proof of 3.1].

Lemma 3.4.3 ([67]) Let R be a local Gorenstein ring and let I be a Cohen-
Macaulay R-ideal. Then exactly one of the following conditions holds:

(a) depth H1(I) = depth 52(1);

86
(b) H1(I) is Cohen-Macaulay and depth 52(1) 2 dim R/I +1.

We are now ready to prove the main result of this section.

Theorem 3.4.4 Let R be a local Gorenstein ring, let 1 and J be two Cohen-
Macaulay ideals which are geometrically linked and let K = I + J. Then
(a) depth K/K2 = min{depth 1/12, depth J/J“, depth (1 ®wR/I)/r — l};

(b) depth H1(K) = min{depth H1(1), depth H1(J), depth I®wR/1, dim R/I-l}.

Proof. We first prove (a). If 1 has grade 0 then K = 169 J and hence K2 = I2 69J2,
since 1 OJ = 0. As (I (8) wR/1)/T = O, the claim follows. Hence we may assume
that 1 has positive grade g, and we may also assume that I and J are both not
complete intersections (for else K is a hypersurface section of either 1 or J by
Proposition 3.1.2(c)). Let 1 and J be geometrically linked by Q. Since g form
a regular sequence, (g)/(g)1 E (R/I)’ and thus depth R/(g)1 = dim R/I (and
similarly for J, depth R/(g)J = dim R/I). It also follows that 12 fl IJ = (g)1
and [K n J 2 = (Q)J, since these equalities are easily seen to hold locally at every

associated prime of (g). Hence there are exact sequences
O—+(g_)I——>12€BIJ——>1K—>O

o —> (2)1 ——. 1K4; .12 —. K2 —> 0.

But since depth 1 J = depth (1 (831.03, 1) / 7' < depth (g)1 by the proof of Proposition
3.3.2, (a) now follows by chasing depths in these sequences.

To prove (b), let “ " ” denote reduction modulo the linking sequence g and note
that depth H1(I) = depth H1(I), depth H1(J) = depth H1(J), and depth 111 (K)

= depth 111(K). Since 1 is generically a complete intersection, there is an exact

87

sequence

O—ng/I

——) [@023/1 —> I®w§fi —-) 0.
It follows that (b) remains unchanged by factoring out (g), and thus we may assume

that g = 0.

Now since K = 1 65 J and wR/I E J by Proposition 3.1.2(c), we have that
(3.4.5) 52(K) E 52(1) 63 52(J) 69(1 (8)1113”).

We now use Lemma 3.4.3 to replace the depth of the symmetric square by the depth
of H1. Indeed, depth 111(1) = depth 52(1) (and similarly for J) since the condition
of Lemma 3.4.3(b) would imply depth 52(1) 2 dim R/I + 1 > dim R. Similarly
we claim that depth H1(K) = depth 52(K). For otherwise, Lemma 3.4.3 implies
that depth 52(K) Z dim R/ K + 1 = dim R, and thus 52(K) would be torsionfree.
But then (3.4.5) implies that I 8) wR/I = 0, hence that I = 0 or J = 0, which is

impossible. Now (3.4.5) implies that

depth H1(K) = min{depth H1(1), depth H1(J), depth 1®wR/1}. Cl

Corollary 3.4.6 Let R be a local Gorenstein ring of dimension d and let
K = I + J be a sum of two geometrically linked Cohen-Macaulay ideals of grade g.

(a) K is strongly nonobstructed if and only if depth R/I2 Z d - g - 1,
depth R/J2 2 d — g -1 and (I ®wR/1)/r is Cohen-Macaulay.

(b) If 1 satisfies (C11), then K is syzygetic if and only if I and J are syzygetic,

and 1 <8) 1.) R / 1 is torsion-free.

Proof. Part (a) follows immediately from Theorem 3.4.4 since by Proposition 3.4.1
K is Gorenstein, hence is strongly nonobstructed if and only if K / K2 is Cohen—

Macaulay.

88

For (b), note that K is generically a complete intersection as 1 satisfies (C11).
Now if 1 and J are syzygetic, then H1(1) and H1 (J) are torsion-free, and if 1 (8)1.) R / I
is torsionfree, so is H1 (K) by Theorem 3.4.4(b). It follows that K is syzygetic. Con-
versely, assume that K is syzygetic, and hence that H1(K) is torsionfree. Since 1
satisfies (011), it follows from Theorem 3.4.4(b) that H1(I) and 1 63> wR/I are tor-
sionfree. To check that H1(J) is torsionfree, it remains to show that J is syzygetic
locally in codimension one. But in codimension one J is linked to 1, which is a com-
plete intersection, hence J is an almost complete intersection (Proposition 3.1.2).

Since J is also generically a complete intersection it follows that J is syzygetic. Cl

Theorem 3.4.4 gives a concrete method to construct Gorenstein ideals whose
conormal module and first Koszul homology module can have prescribed depths.
We point out that Corollary 3.4.6(b) was obtained in [67, 3.10, 3.11], where the
converse was shown under the stronger assumptions that H1(I) and I (8)1123” satisfy
(52).

In [67] Ulrich constructed an example of a grade 5 Gorenstein ideal which is
strongly nonobstructed but is not syzygetic. We can now give a general method to

construct such examples:

Construction 3.4.7 (Sums of transversal links) Let I; C k[[X]] and 12 C k[[Y]]
be two licci ideals that satisfy (01]) but are not Gorenstein and let L be any
transversal link of I = (11,12). Then K = I + L is a strongly nonobstructed
Gorenstein ideal which is not syzygetic. Equivalently, the conormal module K / K 2

is Cohen-Macaulay, but the first Koszul homology module H1 (K) has torsion.

Proof. Since I is licci, by Corollary 3.1.8, it is strongly Cohen-Macaulay and strongly

nonobstructed, and by Corollary 3.2.7 it follows that L is not syzygetic and

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depth R/L2 2 dim R/ L — 1. Hence the result follows by applying Corollary 3.4.6

to the geometric link I ~ L. C]

Example 3.4.8 Applying Construction 3.4.7 to Example 3.2.8 produces a sum
of links K = ($1172, 1:2y2, r2y4, r3y4 , 2:4y2, 1:4y4, mm, mm, $1163, ylyg), which is a
grade 5 Gorenstein ideal generated by 10 quadrics, which is strongly nonobstructed
but is not syzygetic. This is quite similar to the example obtained in [67] by a

somewhat different method.

Examples 3.4.9 Let k be a field, let X be a generic alternating 5 by 5 matrix,
let Y be a generic 5 by 1 matrix, put R = k[[X,Y]] and consider the R-ideal
I = Pf4(X) + 11(XY). We have stated in Example 1.2.13 that R/I2 is Cohen-
Macaulay (i.e. that I is strongly nonobstructed). We will use Example 3.4.8 to give
a proof of this fact.

Write I = (fl , .., f5, 61 , ..., 65), where f.- denotes the Pfaffian obatined by deleting
the ith row and column, and 6.- is the product of the ith row of X with Y. The
elements f5, 62, 63, 64, 65 form a regular sequence and linking via this sequence gives
a link J = (f5,62,63,64,65,y1y5). Now f5,63,64,65,y1y5 is a regular sequence and

linking via this sequence produces the ideal K generated by the polynomials

f5,€5,ylysty1y4,$34314,$34315,$23y4—$13y5,$24y4—$14y5,2345.711—$34y3,$35yi—$34y2-

It is clear that ($11,312,315a324tx35ay3) + I? = (x11,m12,$1s.424,x35,y3) + K.
where K is the ideal of Example 3.4.6 (properly relabed). It follows that K special-
izes to K. Since K / K 2 is Cohen-Macaulay, it follows that K / K 2 is Cohen-Macaulay
([41, 2.2]), and hence K is strongly nonobstructed since it is Gorenstein. But as 1
is doubly linked to K, I is also strongly nonobstructed by Theorem 3.1.6. As 1 is

Gorenstein, this shows that R/ 12 is Cohen-Macaulay.

90

Contrary to the examples above, Gorenstein ideals of grade 4 are much more

well-behaved:

Theorem 3.4.10 (Vasconcelos-Villarreal [76]) Let R be a local Gorenstein ring
and let I be a perfect Gorenstein R-ideal of grade 4.

Then H1(1) is Cohen—Macaulay if and only if 1 is strongly nonobstructed.

By considering a sum of links, this result imposes some genuine restrictions on

two linked perfect ideals of grade 3. One example is as follows.

Corollary 3.4.11 Let R be a local Gorenstein ring, let 1 and J be two geometri-
cally linked perfect R-ideals of grade 3, and assume that I is strongly nonobstructed
and satisfies (C 11). Then the following conditions are equivalent.

(a) H1(I) and H1(J) are Cohen-Macaulay;

(b) depth 111(1) 2 dim R/I — l and depth 111(J) Z dim R/J — 1;

(c) H1(1)/T and H1(J)/r are Cohen-Macaulay.

Proof. Consider K = I + J, which is a perfect Gorenstein ideal of grade 4. Clearly
(a) implies (b). Now by Theorem 3.4.4(b), (b) holds if and only if H1 (K) is Cohen-
Macaulay which holds if and only if K is strongly nonobstructed by Theorem 3.4.10,
which holds if and only if (c) holds by Corollary 3.4.6(a). Hence (b) and (c) are
equivalent. Now assume that (b) and (c) hold. By (b) it follows that H1(I) and

111(J) are torsionfree, hence are Cohen-Macaulay by (c), and this proves (a). C]

Using different methods, the direction (b) => (a) in the above result can be shown
to hold in greater generality: if I is a syzygetic grade 3 perfect ideal then one has
depth H1(J) 7t dim R/ J - 1 for any geometric link J of 1 ([72, 2.1(c)]). This result

somewhat suggests that, for perfect ideals of grade 3, some properties of the Koszul

91

homology H1 may be preserved throughout the entire linkage class. This is in fact
the case: Vasconcelos has shown the property that H1 is Cohen-Macaulay in an

invariant of the entire linkage class:

Theorem 3.4.12 (Vasconcelos [72, 24]) Let R be a local Gorenstein ring and
let 1 and J be two perfect R-ideals of grade 3 which are linked.

Then H1(1) is Cohen—Macaulay if and only if H1 (J) is Cohen-Macaulay.

Using this, one can show a certain rigidity of the twisted conormal module.

Corollary 3.4.13 Let R be a local Gorenstein ring and let I be a perfect R-ideal
of grade 3 satisfying (G11) and with H1(1) Cohen-Macaulay.

Then depth 1 (8)1123” 515 dim R/I — 1.

Proof. Let J be any geometric link of I. Then by Theorem 3.4.12, H; (J) is Cohen-
Macaulay. Now if depth 1 ® “JR/1 = dim R/I — 1 then by Theorem 3.4.4, letting
K = I + J, one has H1(K) Cohen-Macaulay and that K is not strongly nonob-

structed. This contradicts Theorem 3.4.10. Cl

In his recent book [75, p.68], Vasconcelos asked the following question.

Question 3.4.14 (Vasconcelos) Let R be a regular local ring and let I be a

Gorenstein ideal of deviation 3. Is 1 strongly Cohen-Macaulay?

This is equivalent to just asking whether H1(1) is Cohen-Macaulay. Let us
analyze what this would mean for a sum of links I + J. However, we will replace
this sum of links by the ideal K = I + rJ, where a: is an element of R that is
regular on R and on R/I ([47]). Virtually everything we have said for sums of links
holds for K and one has the equation d(K) = d(I) + r(R/I) — 1. Thus if K has

deviation 3, then either I is Gorenstein of deviation 3, or is linked to such an ideal,

92

or else I has type 2 and deviation 2. Of course the latter ideal is automatically
strongly Cohen-Macaulay ([8]). An affirmative answer to Vasconcelos’ question, by
Theorem 3.4.4, would thus imply that any perfect R-ideal I of type 2 and deviation
2 satisfies depth 1 (811.13,, 2 dim R/I -— 1. We have seen in Corollary 3.4.12 that in
grade 3 this would already imply that I is strongly nonobstructed. This suggests

the following question:

Question 3.4.15 Let R be a regular local ring and let I be a perfect ideal of

type 2 and deviation 2. Is I is licci?

This actually includes an older question which is still open in general: Is every
Gorenstein ideal of deviation 2 licci? (For some information about this problem,

see [26] and more recently [43]).

We conclude with an example of a grade 4 perfect ideal with deviation 3 and
type 3, whose entire linkage class is strongly Cohen-Macaulay but whose twisted

conormal module has torsion.

Example 3.4.16 Let R = k[[r1, ...,r3]] and let I be the Cohen—Macaulay R-
ideal I = ($1$2,$2$4,$3$4,$436,$5$6,$6$8,$7$3) which may be viewed as the
edge-ideal associated to a tree. By [62, 3.11], I is strongly Cohen-Macaulay and of
linear type. Linking via the regular sequence 312:2, $334, $5236, $738 gives an ideal
J with the same graph, hence also strongly Cohen-Macaulay. It follows that the
entire linkage class is strongly Cohen-Macaulay. However, K = 1+J is a Gorenstein
ideal of grade 5 generated by 10 quadrics. In fact, it is the ideal of Example 3.4.8.
Since K is not syzygetic, and H1(I) and H1(J) are Cohen-Macaulay, it follows that
1 8) w R / 1 has torsion by Corollary 3.4.6(b). Since K is strongly nonobstructed, it

follows however that (1 <8) wR/1)/r is Cohen-Macaulay by Corollary 3.4.6(a).

93

3.5 Intersections of Complete Intersections

We give a naive method, which is somewhat dual to linkage, to produce Cohen-

Macaulay ideals of type 2.

Construction 3.5.1 (Intersections of complete intersections) Let R be a local
Gorenstein ring, let I be a Cohen-Macaulay R-ideal of grade g satisfying V(I) S 2g
(which always holds after adjoining variables to I), let g and g be regular sequences
of length 9 properly contained in 1 such that 1 =2 (g) + (B), and set I = (g) F) (6).

Then I is a Cohen-Macaulay ideal of type 2.

Proof. The exact sequence
0 ——> R/I ——> R/(g) {BR/(g) ——> R/I ——+ 0
shows that I is Cohen-Macaulay and induces an exact sequence
0 —+ wR/I -—> R/(Ql EBB/(é) —* wR/I —> 0

from which the result is immediate. Cl

Another construction, which can be shown similarly, produces ideals of arbitrary

type, but of smaller grade.

Construction 3.5.2 Let R be a local Gorenstein ring, let 1 be a Cohen-
Macaulay R—ideal of grade g satisfying V(I) _ 29 — 2 (which always holds after
adjoining variables to I), let Q and g be regular sequences of length 9 — 1 in I such

that 1 = (g) + ()3), and set J = (g) n (g).

Then J is a Cohen-Macaulay ideal of grade g - 1 with r(R/j) S r(R/I) + 2.

The proof of Construction 3.5.1 shows moreover that one can construct a reso—

lution of R/I in terms of a resolution of R/ 1. This is especially interesting in the

94

graded case since, as the resolution is essentially built from the Koszul complexes

on g and g, the shifts at the tail end will tend to be fairly large.

Example 3.5.3 Let R = k[X], where X is a generic 2 by 4 matrix, and let
I = 12(X). Write I = (f12,f13,f14,f23,f24,f34), where f;,- denotes the minor
involving columns i and j. Then f12, f13, f24 and f14, f23, f34 are regular sequences
and let I be the intersection of these two complete intersections. Since 1 has a

linear resolution

0 ——> R3(—4) —> R8(—3) —+ R6(—2) —> R ——> R/1 ——> 0,
I has a resolution of the form

0 ——> R2(—6) ——> R9(—4) ——> R8(—3) ——> R ——> R/I —> O.

In particular, I is a perfect ideal of grade 3 and type 2 with a pure resolution. It fol-
lows that H1(I) is Cohen-Macaulay ([72, 2.9]), but 112(1) is not Cohen-Macaulay
([72, 4.2.4]). This ideal has essentially the same properties as the example con-

structed in [48, 2.6].

It turns out, however, that Construction 3.5.1 is somewhat related to the transver-

sal link of Section 3.1.

Example 3.5.4 Let R = k[[:r1, ...,:rn]] be a power series ring over a field 11:,
let I = (1:1,...,:rg), consider regular sequences g = $1,...,rk,q1,...,qg_k and
g = q[,...,q[,xk+1,...,rg where 1 S k S g — 1 and q; (respectively qfi) are gen-
eral quadrics in 2:1,“, ...,rg (respectively 1:1,...,:rk) and let I = (g) 0 (g).

Then I is licci if and only ifk =1 or k = g — 1.

95

Proof. Let q = (q1,...,qg_k) and q’ = (qi, ...,qz). Then

I = ($1, '°'1$k1q) n (q,,1?k+], ...,:L'g)

: q + (I, + ($1, °--9$k)(‘1"k+1’ "°’xg).

This may be viewed as a transversal link of the ideals 11 = q : (rk+1,...,:1:g) and
I2 = q’ : (3:1, ...,:c,). It follows from Theorem 3.2.2 that I is licci if and only if 11
or 12 is a complete intersection, which holds if and only if k = 1 or k = g — 1 (as q

and q’ are general). C]

For example, if we consider the regular sequences g = r1,r2y2,x3y3, ...,:rgyg
and g = r1y1,r2, ...,:r_, in k[[:r1, ...,rg,y1, ...,yg]] (corresponding to k =1) then we
obtain the ideals

I = (r1y1,..., rgyg, 311:2, ..., $1159).

The fact that these ideals are licci would also follow from [62, 2.3].

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