Cohen-Macaulay Rees algebras and degrees of polynomial relations

Math. Ann. 301, 421 444 (1995)

Cohen-Macaulay Rees algebras and degrees of polynomial relations*

Mathematische Annalen �9 Sprmger-Verlag 1995

Aron Simis t,a, Bernd Ulrich 2,b, Wolmer V. Vasconceios 3x

I lnstituto de Matemfitica, Universidade Federal da Bahia, 40170-210 Salvador, Bahia, Brazil (E-mail: [email protected]) 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (E-mail: [email protected]) 3 Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA (E-mail: [email protected])

Received: 5 October 1993

Mathematics Subject ClasM]ication (1991). 13D40, 13D45, 13H15

1 Introduction

Let R be a Noetherian ring and let I be an R-ideal. The Rees algebra of ! is the subalgebra o f the ring o f polynomials R[t],

R[lt] = ~ l ' t ' C R[t]. t>=O

It realizes algebraically the notion of blowing up a variety along a subvariety. In a manner of speaking, it is made up of two parts, R itself and the associated graded ring

grl(R) = }-~1'/I '+' . t>0

One basic set of issues is how the arithmetical properties o f R[lt] depend on those of R and gr1(R). Here we will be concerned with how the property of being Cohen-Macaulay passes around these algebras.

In one case the situation is straightforward: I f I is an ideal o f positive height in a Cohen-Macaulay local ring R and if the Rees algebra R[lt] of I is Cohen- Macaulay, then the associated graded ring grz(R) is also Cohen-Macaulay [21, Proposition 1.1]. The converse however comes in many different flavors, but it is inexorably linked to the degrees of the generators of modules o f syzygies and the polynomial relations amongst the elements of a generating set of the ideal. We want to explore how this relationship is expressed in terms of these sets of degrees and other integers that arise from a specification of the ideal through a presentation matrix.

* Dedicated to Professor Robert Berger on his sixtieth birthday a Partially supported by CNPq, Brazil b Partially supported by the NSF c Partially supported by the NSF

422 A. Simis et al.

A framework for examining the connection between the Cohen-Macaulay- ness of R[It] and gri(R) was developed by Ikeda and Trung [25], in terms of the degrees of the minimal generators of the canonical module of grl(R). To describe it, let us recall the notion of the a-invariant of a graded ring introduced by Goto and Watanabe [11].

Definition 1.1 Let R = (~ i>oR~ be a Noetherian graded ring of dimension d with Ro local, and let M denote the irrelevant maximal ideal of R. The integer

(1) a(R) = max{iI[H~t(R)], =t = 0},

where H~t(R ) denotes the (graded) d-dimensional local cohomology module of R with respect to M, is the a-invariant of R.

That the module H~(R) is Artinian ensures that a(R) is well defined. I f ~oR is the canonical module of R, by local duality, this definition can be rephrased as

(2) a(R) = - min{i I [(DR] , :# 0 } .

When the computation of the sign of the a-invariant of G = grt(R ) is possible one can use the criterion of [25, Theorems 1.1 and 7.1]:

Theorem 1.2 Let R be a Noetherian local ring of dimension d and let I be an ideal that is not nilpotent. The Rees algebra R[It] is Cohen-Macaulay if and only if [H~t(gr1(R))], = 0 ,for i=# - 1 and r < d, and [H~t(gr1(R)) ], = 0 f o r i >=O.

Corollary 1.3 Let R be a Cohen-Macaulay local ring and let I be an ideal of positive height. Then R[It] is Cohen-Macaulay if and only i f grl(R ) is Cohen-Macaulay and a(gr1(R)) < 0.

Among the features of the ideal I that often permit the determination of a(G) are the properties of its reductions. Recall that an ideal J C I is a reduction of I if JI r ~- I r+! for some integer r [29]; the least such integer, rj(I), is the reduction number of I with respect to J . Phrased otherwise, J being a reduction means that

R[Jt] = ~J't'~--* ~ l ' t ' = R[It] i>=O ~>=0

is a finite morphism of the associated Rees algebras. This works as a Noether normalization, and is particularly useful if the ideal J has an amenable character and the module structure of R[It] over R[Jt] is rich.

One type of reductions which have been studied are the minimal ones (for set theoretic containment). They arise as follows. For an ideal I in a Noetherian local ring (R, m), the special fiber of R[It] is the ring F(1) = R[It]@nR/m, an affine algebra over the residue field R/m. The Krull dimension o f F ( / ) is called the analytic spread of I and is denoted by f(1); it satisfies {(I) =< dimR. If R/m is infinite, all minimal reductions of I arise from applying the graded version of the usual Noether normalization to the algebra F(I). In particular

Cohen-Macaulay Rees algebras 423

their minimal number of generators always equals ((I) . The reduction numbers r j ( 1 ) however may change from one minimal reduction to another. The least r(1) among the r j ( I ) is referred to as the reduetion number of I.

In case R is a Cohen-Macaulay local ring of dimension d > 0, and I is an m-primary ideal with a reduction J generated by a regular sequence (always achievable if the residue field of R is infinite), the finite morphisln

grj(R) ~ grr(R )

can be used to determine a(gr1(R)) if grI(R) is Cohen-Macaulay. This was first shown by Goto and Shimoda [10] who proved that

(3) a(gri(R)) = r j ( 1 ) - d ,

which establishes the invariance of r j ( I ) and gives the condition for R[It] to be Cohen-Macaulay, namely r j ( 1 ) < d.

When I is not m-primary, the notion of a minimal reduction loses some of its effectiveness. To make up for it, one substitutes conditions on the behavior of certain localizations of the ideal I . Ours can be described in terms of a matrix presentation

(4) R m ~ i R n ----+ I ~ 0 ,

involving the sizes of the ideals /t(q~) generated by the t by t minors of the matrix qo:

Definition 1.4 The ideal I is sa id to sa t i s f y condition ~.~o i f f o r any p r i m e ideal p containing L v(l~, ) < dim R~, + 1 ; /f ins tead v(I,, ) < dim R v holds, then

1 is said to sa t i s fy condit ion Y l .

We shall mostly use these conditions in a range of integers, when we require them to hold for dimRp < s. With this restriction, ~ l is the condition Gs of [5].

It often occurs that such conditions allow for deriving higher degree relations among generators of 1 through elimination theory applied to their syzygies.

Another source of polynomial relations amongst the elements of a generating set of I lie in the Koszul homology modules of I. We recall two conditions on such modules that are propitious to that end. An ideal I -- ( f l . . . . . f , , ) in a Cohen-Macaulay local ring R of dimension d is said to satisfy sliding depth if the Koszul homology modules Hi -- H i ( f l . . . . . f , , ) satisfy depth H, > d - n + i ; and I is called strongly Cohen-Macaulay if H, are Cohen-Macaulay for all i. (These conditions are independent of the generating set.) Standard examples of strongly Cohen-Macaulay ideals include deviation two Cohen-Macaulay ideals in Gorenstein local rings [6, p. 259], height two perfect ideals [6, Theorem 2.1], height three perfect Gorenstein ideals, or more generally, ideals in the linkage class of a complete intersection [22, Theorem 1.14].

We are now ready to describe our first main results.

424 A. Simis et al.

Theorem 3.5 Let R be a Noetherian local rin9 with infinite residue field, let I be an ideal o f height 9 with analytic spread ef and reduction number r, and assume that I satisfies "~l locally in codimension { - 1. Further assume that G = gr1(R ) is Cohen-Macaulay with a = a(G). Then

a = m a x { - g , r - {} .

In particular, r < a + {, where equality holds i f a ~ - g.

While our work was in progress, results similar to the above theorem have been independently obtained by Johnston and Katz [26], and by Aberbach, Huneke, and Trung [3]. When combined with Theorem 1.2, the theorem yields:

Corollary 3.6 If, with the assumptions o f Theorem 3.5, I is not nilpotent, then R[It] is Cohen-Macaulay i f and only i f 9 > 0 and r < ~.

We are able to derive a sharper bound for the reduction number in case grl(R ) happens to be Gorenstein:

Theorem 3.7 In addition to the assumptions o f Theorem 3.5, suppose that I is" yenerically (i.e., locally' at each o f its associated primes) a complete intersection, R/I is Cohen-Macaulay, and gri(R ) is Gorenstein. Then

r < m a x { 0 , • - 9 - 1}.

Based on this theorem, one can show a converse to the well known fact [14, Theorem 9.1] that a strongly Cohen-Macaulay ideal 1 in a Gorenstein local ring has a Gorenstein associated graded ring if I satisfies o~1:

Theorem 3.8 Let R be a Noetherian local ring, and let I be a strongly Cohen- Macaulay R-ideal that is generically a complete intersection. I f grl(R) is Gorenstein, then I satisfies ~ 1 .

The relationship between the Cohen-Macaulayness o f a Rees algebra R[It] and the reduction number of I may occur without the mediation of the associated graded ring gr/(R). This will be our second main issue for a class of ideals which we now introduce. Let R be a commutative Noetherian ring of Krull dimension d, and let f l . . . . . f n be elements of R. In [35], when n = d + 1, forms

h(Ti . . . . . T,,) C R[Tt . . . . . Tn] ,

were considered such that

h ( f , . . . . . f , ) = O,

with the view to establish the Cohen-Macaulayness o f the Rees algebra R[It] of the ideal generated by the f ' s . When R is a local ring (or R is a polynomial ring and the f ' s are homogeneous forms of the same degree), the focus was on those cases in which the ideal I had analytic spread equal to dim R, a condition which in the case o f forms of k[xl . . . . . xa] translate as: the ring k [ f l . . . . . f , ] has Krull dimension dimR. In particular the ideal o f all the h 's


lying in k[Tl . . . . . T,,] is generated by a single polynomial h0, the so-called resultant of the system of polynomials.

In the case of forms, the determination of the h's goes back a long time (e.g., [38]). Rather recently, Sturmfels and Yu [31] described a general formula for h0 (extending earlier work of Abhyankar [4, Sect. 30], and McKay and Wang [27]):

(5) h q = a discrete resultant

where q = deg(k (x l , . . . , xd ) : k ( f l . . . . . fd+l)) . It requires a non-degeneracy condition to hold.

In contrast, under special conditions, [35] contains a formula for the degree of h0:

(6) deg h 0 - - d - g + 2 ,

with g being the height of the ideal I. The condition is that I satisfies Y l locally on the punctured spectrum of R and that I is strongly Cohen-Macaulay.

Working in a slightly more general setting, we will make heavy use of the natural exact sequence

(7) 0 --~ ~/ ~ S -+ R[It] --~ O,

where S = S( I ) denotes the symmetric algebra of I. Recall that for .z/:#0, the maximal degree occurring in a homogeneous minimal generating set of .~/ is called the relation type of I. If R is a Cohen-Macaulay local ring and I is a strongly Cohen-Macaulay R-ideal of positive height satisfying Y0, then S is Cohen-Macaulay of the same dimension as R[It], and a great deal of other information is known about S [15]; this will provide a platform from which to study R[It].

In the main result of the second part, we show that under assumptions on the reduction number similar to (6), the Rees algebra is Cohen-Macaulay, and that the degrees and the number of its defining equations can be determined:

Theorem 4.10 Let R be a Gorenstein local ring o f dimension d with infinite residue field, let I be a strongly Cohen-Macaulay R-ideal o f height g > 2 with minimal number o f generators n, analytic spread {, and reduction number r, and let ~p be a minimal presentation matrix g f I. Further assume that ( < n and that I satisfies ~1 locally in codimension n - 2. Consider the following conditions:

(a) r <= n - g (in which case r = n - g). (b) r(Iv) < n - g for every p E V(1) with dimR~, = n - 1 (in which case

r(I~,) = n - g). (c) The relation type o f l is at most n - g + 1 (in which case the relation

type equals n - g + 1 ). (d) o~r is generated by one element (in which case ~/ is generated by one

fo rm o f degree n - g + 1 ). (e) Af ter suitable elementary row operations, the entries o f one row o f

r generate l l (q) .

426 A. Simis et al.

(f) R[It] is Cohen-Macaulay. Then (a), (b), (c), (d), (e) are equivalent, and either condition implies (f). All conditions are equivalent i f g = 2.

The above requirement (e) can be readily checked for a given matrix ~0. This type of condition was first introduced by Aberbach and Huneke, who dubbed it row condition [2, Definition 8.1] and used it in their study of four generated perfect ideals [2, Theorem 8.2]. Typical examples of ideals satisfying the assumptions of Theorem 4.10 are provided by strongly Cohen-Macaulay ideals with g > 2 and n = d + 1, satisfying Y l locally on the punctured spectrum of R, by perfect ideals with g = 2 and { < n, satisfying O~l locally in codimension n - 2, and by perfect Gorenstein ideals with g = 3 and { < n, satisfying "~1 locally in codimension n - 2. In that sense, Theorem 4.10 gives a new perspective on some results in [2] and [1] about four generated perfect ideals and five generated perfect Gorenstein ideals. Returning to the theme of [35], we also come up with the following application:

Corollary 4.11 Let R be a polynomial ring over a field, let I be an R-ideal of height g > 2 with a presentation matrix consisting o f linear forms, write n = v(I), and assume that I is strongly Cohen-Macaulay and satisfies o~l locally in codimension n -2 . Then R[It] is Cohen-Macaulay and I is generated

by forms of degree n - I

2 Gorenstein algebras

The aim of this section is to examine some instances of associated graded rings which are Gorenstein - when the determination of its a-invariant is much facilitated.

Cohen-Macaulay Rees algebras

We first record an elementary case of Cohen-Macaulay Rees algebras that has been known to the authors for some time.

Theorem 2.1 Let R be a Noetherian local ring, let I be an ideal, and assume that gri(R ) is Gorenstein. I f height / > 0 and I is a complete intersection locally at one o f its minimal primes, then R[It] is Cohen-Macaulay. I f I is a complete intersection locally at each of its minimal primes, then I is height unmixed.

Proof We may assume that R is complete. Let coc stand for the canonical module of the algebra G = grI(R). By assumption, o9~ = G(a), where a = a(G) [13, (36.8), (34.6)(4) and (33.2)]. To determine a, let p be a minimal prime of I , and suppose 1~, is generated by a regular sequence of g -- height p elements. Note that ~oc | R~, = coa| because G is equidimensional. Since o)c is cyclic, generated in degree - a , the canonical module of G | R v is also


generated in degree - a . But G | R~, is a polynomial ring in 9 variables of degree one, whose a-invariant is - 9 . This proves that a = -c3. Now the Cohen- Macaulayness o f R[lt] follows from Theorem 1.2, and our assertion about the height unmixedness of I is trivial. []

Remark 2.2 The condition on I being a complete intersection locally at every minimal prime is automatically satisfied whenever R is Cohen-Macaulay, 1 is a perfect Gorenstein ideal, and grl(R ) is Cohen-Macaulay [28, Theorem 2.6].

Another case in which the condition that I x, is a complete intersection for some minimal prime t~ of I can be bypassed has been pointed out by Craig Huneke (see [36, Remark 2.3.2]). It suffices that R~, be a regular local ring, as the Brianqon-Skoda Theorem implies that R[lt]~, is Cohen-Macaulay along with G~,, and therefore its a-invariant is negative. Since G is Gorenstein we have a( G ) = a( G~, ).

Symbolic powers

The following result has seen several versions. It can also be derived from Theorem 2.1, but we prefer to give here an independent proof.

Theorem 2.3 Let R be a reyular local rin 9 and let p be a primc ideal such that the ordinary and symbolic powers of p coincide. I f gr~,(R) is Cohen- Macaulay, then R[pt] is also Cohen-Macaulay.

Proof We may assume that height p = 9 > 0. Let ~oa be the canonical module of G = gr~,(R):

coc . . . . | ~oy G co~+l | . . . ,

where (~)y denotes the component of degree 9. We claim that there are no components in lower degree. For that, it suffices to observe two facts. First, since G is an integral domain, cog is a torsion free R/p-module. Then, localizing at p, ~oo localizes to the canonical module o f grp(R~,), with P = pR~,, and the initial degree of the canonical module remains unchanged. But the associated graded ring of P is a polynomial ring and as in the proof of Theorem 2.1 we conclude that a(G) = - 9 < 0. We again use Theorem 1.2 to complete the proof. []

Theorem 2.4 Let R be a Noetherian local rin9 and let I be an ideal of height 9 satisfyin9 ~ . I f gr1(R) is Cohen-Macaulay, then a(gr1(R)) = -9. In particular, R[It] is Cohen-Macaulay in case 9 > O.

Proof We know that R is Cohen-Macaulay [13, ( l l .16)(b)] and we may assume that R has an infinite residue field. Define d = dim R, n = v(1), B -- R[TI . . . . . Tn], and let M be the irrelevant maximal ideal of B. Further write G = grt(R ) = B/K and S(I/I 2) = B/J. Since I satisfies Y l , dim S(I/I 2) = dim G = d [23, Theorem 2.6], and hence grade J = grade K = n. Thus there

428 A. Simis et al.

exists a regular sequence cr = cq,...,~z~ in K consisting of 9 forms of degree 0 and n - g forms of degree one. Now

a(B/(~l . . . . . ~z,,)) -- a(B) + n - g = - g .

Furthermore the natural projection of d-dimensional rings

B/(~I . . . . . ~ , ) ~ G -+ O,

induces a homomorphism of local cohomology modules,

H ~ t ( B / ( ~ t , . . . , ~ ) ) --+ H d ( G ) -+ O,

which shows that

a ( G ) <- a(B/ (oc l , . . . ,CZn))= - g .

On the other hand, a ( G ) > - g , since there exists a prime ideal p of R with I~, being a complete intersection of height g. [Z

R e m a r k 2.5 Let R be a Cohen-Macaulay local ring and let I be an ideal of height 9. If I satisfies ~ 0 and has relation type at most 9 > 0, then the above argument still shows that R[It] is Cohen-Macaulay if and only if gr:(R) is Cohen-Macaulay.

R e m a r k 2.6 Let R be a Cohen-Macaulay local ring and let I be an ideal with height I > 0. The following are equivalent:

(a) gr1(R ) is Cohen-Macaulay.

(b) The Rees algebra of the ideal (I, X I , . . . , X , . ) in the ring R[XI . . . . . Xr](,,,.x,...,x~) is Cohen-Macaulay for some r.

Proof . Adjoining variables X l , . . . , X r to the ideal I has the effect of adjoining linear variables Ti , . . . , T,. to gr:(R), and hence of lowering the a-invariant by r. [-j

3 Ideals whose associated graded rings are Cohen-Macaulay

For an ideal I as in the title, we are going to establish a relationship between a(gr1(R)), { ( I ) and r(1). This requires I to satisfy .~-j in codimension below the analytic spread.


Koszut homology

The aim here is to locate the degrees of the Koszul homology modules of certain associated graded rings gri(R ).

Lenlma 3.1 Let G be a Noetherian positively graded ring over a local ring, let M be the irrelevant maximal ideal of G, assume that G is Cohen-Macaulay of dimension d > O, and let

F . " O--, Fd---+...-+ Fo

be a complex of finitely generated graded free G-modules. Write B. and H. for the boundaries and homology of F., and let j be an integer.

(a) 1f for 1 <= i < d - 1, [H~I(Hi)]j = O, and if [H~(Bd-I)]j = O, then [H~ = O.

(b) 1f for 1 < i <_ d, [H~c(H,)]j = 0 = [H/v~l(Hi)]s, and if [H~t(Fd_, )]j = O, then [H~t(Fa)]j ~_ [H~

Proof The usual sequences relating boundaries Bo, cycles Zo, and homology Ho of a complex, yield the following sequences of local cohomology for 1 < i < = d - l :

[HSM(Ht)]j ._.+ [H~l(Bi)]j ____+ [H~l(z t )] j ___+ [H2~,+l ( H , ) ] j ,

and

. ___+ r H ~ + l r z - ~ ] ; t + l [H~t(F,)], [H~t(B,-1)]i [ M t ,J,, ~ [H~ (F,)] s .

Using these sequences, one sees by decreasing induction on i, that under the assumption of (a), the vanishing of [H~(Ba-t)lj implies that [H~(B0)L = O, and thus [H~ = O. On the other hand, in the setting of part (b),

where

[H~(Ba_~ )]j -~ [H~(F,~)]j

and

[ H ' ( B o ) ] j - [ H ~ �9 []

L e m m a 3.2 Let R be a Noetherian local ring with infinite residue field, let I be an R-ideal satisfying .'~-1, f ix integers g <= height I and n > v(1), assume that G -- gr/(R) is Cohen-Macaulay, and let . . . . . denote images in I/I z = [G]b Then there exists a generating sequence {bl . . . . . b,} of l such that for every i >= 1 and every 0 < k < n, H~(b'l,...,b~;G ) is" concentrated in degrees < k - g.

430 A. Simis et al.

Proo f . Let m be the maximal ideal of R, let M = m G + G+, and write ! !

H , ( b ~ . . . . . b k ) = H,(b'~, " = . . . . b~. ,G). Since R / m is infinite, since grade G+ height G+ = height I, and since I satisfies Y l , we may find a generating sequence {bl . . . . ,b,,} of I such that

( 8 ) ' ' = = b I . . . . . by form a G-regular sequence, and for every g + 1 < k < n, b~ can only lie in associated primes o f (bl, . . . , ! bk _t I ) that contain G+,

(9) for every k with height I < k __< n and every p C V ( I ) with di reR v < k,

12, = (bl . . . . . bk)v [5, Lemma 2.3].

By (8) one has that for every 0 =< k =< n, G+ C v / a n n H l : b ' t I , " ' , b ' k ) " Fur- thermore all our assumptions are retained as we localize at any p E V(1) . We are going to prove by induction on d = dim R that for a sequence {bl . . . . . b ,} satisfying (8) and (9), H ~ ( b ' l , . . . , b ~ ) is concentrated in degrees __< k - g, i f i _ > l .

I f d = 0, then I = 0, and nothing is to be shown. So let d > 0; now our induction hypothesis implies that for every 0 < k < n and every j > k - g, [Hl(b'l . . . . . b2)]: is an R-module of finite length. Since furthermore, G+ C v/ann H,(b~l , . . . ,b~) , we conclude that

( 1 0 ) ' ' [ H l ( b I . . . . . bk)]: _~ [ H ~ . . . . . b~))]j, for j > k - g.

To prove our assertion for d, we use induction on k, the case k = 0 being trivial. For the induction step from k - 1 to k, notice that the exact sequence

H , ( r ' ' ' . . . , l(bl . . . . . b k _ l ) ( - bk_l ) - - -* H i ( b l . . . . . b k ) - - ~ H ,_ ' ' 1)

already implies that H,(b~l . . . . . b' k ) is concentrated in degrees < k - g , i f / > 2. Thus it suffices to consider i = 1.

Case 1: k > d. By (9), (hi . . . . ,bk) = I = (bl . . . . . b k - l ) , hence (b l' . . . . . bk)' = (b'~ ' . . . . . bk 1 ), and

/-/ ,(b'l . . . . . b~) ~_ / - / , (b ' l , . ' hr0(b'~, ' ) ( - 1 ) . . , b k _ l ) O . . . , b k _ l t ~_ H l ( b ' 1 . . . . . b k _ l ) O ( R / I ) ( - 1 ) .

By (8) we may assume that k => g + 1. Thus H l (b l' . . . . , b k' ) is concentrated in degrees __< m a x { k - l - g , 1} < k - g .

/ I Case 2: k _-< d. In this case, the boundaries B d _ l ( b I . . . . . bk_ l ) of the Koszul complex vanish. Furthermore for every i > 1, H, (b 'L , . . . , b ' k_ l ) is concentrated

in degrees < k l - g , and hence ,+l , bk_t = - H~t ( H , ( b I . . . . . )) has the same property (see, e.g., [8, Lemma 2.2]). But then, applying Lemma 3.1(a) to the Koszul complex F ~ K ~ l . . . . . b~_ 1) we conclude that 0 , , = H ~ t ( H o ( b x . . . . . )) is concentrated in degrees < k - 1 - g .

On the other hand, the exact sequence

I I I I I Hi (b i , . �9 �9 bk_ I ) -~ HI (b ' 1 . . . . . bk ) -~ H o ( b l , ' " , b k - I ) ( - 1 )

yields an exact sequence


0 ~ E --+ Hl(b:l , . btk) ~ Ho(btl,. t . . . . . . b k _ l ) ( - 1 ) ,

where E is concentrated in degrees < k - 1 - g . Therefore H ~ . . . , b k ) ) ' is concentrated in degrees < k - g, which by (10) implies that H l ( b ' 1 . . . . ,b~.)

has the same property. []

Theorem 3.3 L e t R be a Noe ther ian local ring, let I be an R-Meal o f he igh t g

sa t i s fy ing ~ 1 , ~ and assume that G = gr: (R) is Cohen-Macaulay . Then f o r any

generat ing set {bl . . . . . b , } o f I and f o r every i, H,(b ' 1 . . . . . b,',) is concen t ra ted in degrees < min{n - g, v ( I ) - g + i}.

P r o o f Since v(1) > g, it suffices to consider the case i > 1. We may further assume that the residue field of R is infinite and that n = v(1). Now apply Lemma 3.2 and notice that our assertion is independent of the chosen generating set.

Reduc t ion number

Lemma 3.4 L e t R be a Noe ther ian local ring with infinite residue f ield, let I = ( f l . . . . . f , ) be an R- ideal with analy t ic spread { and reduction number r, let X be a generic n by : ma t r i x , in R[X] define

{bi , . . . , b t ] = {fi . . . . , fn} �9 X ,

and set S = R ( X ) , J = (bl . . . . . bl)S . Then J is a m in ima l reduct ion o f l S with r j ( I S ) = r.

P r o o f It suffices to show that Is+IS = JISS if and only if there exists an R-ideal K = ( a i , . . . , a t ) C I satisfying I s+l = KU.

Let m denote the maximal ideal of R. Since the residue field o f R is infinite, the equality Is+iS = j l s S holds if and only if there exists an n by / matrix A with entries in R, such that in the ring T = R[X]( , , , ,X_A) , IS+iT = (bl . . . . . b : ) U T . Now given that I s + I s = J I S S , simply define the elements a l , .] . a/ via the equality

[ a l , . . . , a / ] = [ f l . . . . , f n ] �9 A .

Conversely, if we have elements a l , . . . , a : with I s+l = K I s, then take A to be any n by { matrix with entries in R satisfying the same matrix equation. In either case, the entries of X - A form a regular sequence on T and on T/IS+IT, and modulo this sequence, (bl . . . . . b : ) T is mapped onto K. Thus I s+l T = ( b l , . . . b : ) I s T holds if and only if I s+ 1 = K I s. [-1

We are now ready to prove our first main theorem, which generalizes [10, Theorem 3.1], [12, Theorem 4.8] and [8, Proposition 2.4]. Related results were independently obtained by Johnston and Katz [26], and by Aberbach, Huneke and Trung [3], at the same time when our paper was being written (we are grateful to Trung for showing us his earlier paper [33]). Somewhat later, we also became aware of a manuscript by Tang [32] that contains a special case of Theorem 3.5.

432 A. Simis et al.

Theorem 3.5 Let R be a Noetherian local ring with infinite residue field, let I be an R-ideal o f height g with analytic spread f and reduction number r, and assume that I satisfies ~ 1 locally in codimension f - 1. Further assume that G = gr:(R) is Cohen-Macaulay with a -- a(G). Then

a = m a x { - g , r - f} .

In particular, r < a + {, where equality holds i f a # - g.

Proo f First notice that R is Cohen-Macaulay [13, (11.16)(b)]. Now write d = dimR, let m = (xl . . . . . Xs) be the maximal ideal of R and choose a generating set { f l . . . . . . f , } of I. Let X and Z be generic matrices o f sizes n x f and s x (d - f ) respectively, set

[bl . . . . ,b:] = [ f l . . . . . fn] " X, [ Y l , . . . , y a - : ] = [xl, . . . ,Xs] �9 Z ,

and define J = (bl . . . . . b : )R(X,Z) . By Lemma 3.4, J is a minimal reduction of IR(X ,Z) with r j ( IR (X ,Z ) ) = r and by [24, the proof of 3.2], height J : I R ( X , Z ) > {. We may replace I by IR (X ,Z ) and r by r j ( I ) .

Notice that Yl . . . . , Ya - / form a G-regular sequence because grade n~G = d - f [18, Theorem]. Thus, when replacing I and J by their images in R/(y l . . . . . Yd-: ) , we simply have to tensor G with (~RR/(y l . . . . ,Ya - : ) , and g,: , a, r j ( I ) remain unchanged. Furthermore, I still satisfies ~1 locally in codimension f - 1 and height J : l > f . Thus we may from now on assume that : - - d .

Next, let b~ denote the image of b, in [G]I. Since grade G+ = g, we have that b' I . . . . . b,~ form a G-regular sequence of linear forms [18, Theorem]. Thus the associated graded ring of I/(b~ . . . . , b,j) is G/(btl , . . . , bty). Hence, when replacing I by I/(bl . . . . . by), a increases by g, whereas g and f decrease by g. Furthermore, r j ( I ) is unchanged, height J : l >= d, and 1 still satisfies ~ l locally in codimension d - 1 [24, the proof of 3.2]. Thus we may from now on assume that g = 0. Furthermore, our assertion being trivially true for d = 0, we suppose d > 0.

Set M = m G + G+, consider the Koszul complex ~ ~" K.(b 1 . . . . , b d , G), and set H~ = H.(b~l , . . . ,b~;G). Since v/ (b ' l , . . . ,b~) = v/G~+, one has that G+ C a x / ~ Hi and hence H~(H~) --- OiH~I([H,]j) for every i and k [8, Lemma

2.2]. Furthermore recall that for p E S p e c ( R ) \ {m}, J~, = 1~, satisfies ;~1. Thus by Theorem 3.3, H~@RR ~, is concentrated in degrees < d, which yields that [H~(Hi)]; = H~,([H,]:) = 0 and [H~I(H~)]: = H~+I([Hi]j) = 0, for every i > 1 and j _>- d + 1. Since furthermore

max{j[[H~t(Fa_l )]j 4=0} < max{j I[H~(Fd)] / + 0 } ,

we may then use Lemma 3.1(b) to conclude that

(11) max{d} O {j[[H~t(Fd)]j+O } = max{d} tO {j l[H~

The left hand side of (11 ) is max{d, d + a} = d + a, where the latter equality follows from the fact that a _>_ - g = 0 (cf. the proof of Theorem 2.1).


As to the right hand side, recall that I~, = J~, whenever p + 111, which yields [Ho]: = H,~ for j > 1. Therefore

r = max{jl[Ho] / + 0 } = max{0} U {jlH,~,([Ho]/)+O}

= max{O} u {jl[/-/~ + 0 } ,

and the right hand side of (11) becomes max{d,r}. Now (11) reads as d + a = max{d,r} , or a = max{0, r - d} -- m a x { - 9 ,

r - : } . D

Corollary 3.6 If, with the assumptions of Theorem 3.5, I is not nilpotent, then R[It] is Cohen-Macaulay if and only if 9 > 0 and r < :.

Proof The assertion follows from Theorems 1.2 and 3.5. E]

This result simplifies some of the arguments in [20, 37, 34], where the Rees algebras o f certain ideals with reduction number one are shown to be Cohen- Macaulay. In the setting of these papers, it actually suffices to verify the Cohen- Macaulayness of the associated graded ring, as can be seen from Corollary 3.6.

Associated graded rings which are Gorenstein

The bound for the reduction number in Theorem 3.5 can be improved if gr:(R) happens to be Gorenstein.

Theorem 3.7 Let R be a Noetherian local ring of dimension d with infinite residue fieM, let I be an R-ideal of height 9 with analytic spread : > g and reduction number r, assume that I satisfies "~-i locally in codimension, l - 1, and that the completion of R/I satisfies Sj-/+l. I f G = gr:(R) is Gorenstein, then r < : - g - 1.

Proof. By Lemma 3.4, the reduction number does not change when passing to the completion, so we may assume that R is complete. Furthermore by [13, ( l l .16)(b)] , R is Cohen-Macaulay. We use the notation of Theorem 3.5 and its proof. Now G being Gorenstein implies that a = - g (cf. the proof of Theorem 2.1), and hence by Theorem 3.5, r < : - g . As in the proof of that theorem, one can reduce to the case where g = 0 and .{ = d > 0. We need to show that [H~ = 0. To this end, we may assume that R is complete and Cohen-Macaulay, that g = 0, and that I satisfies ~ l locally in codimension d - 1. Furthermore, by the latter condition and using the assumption that R/I is Sl, it follows that 1 is generically a complete intersection. Therefore I is height unmixed by Theorem 2.1, hence unmixed. Thus R/I is equidimensional of dimension d without embedded primes.

The assertion follows from Lemma 3.1(a), once we have shown that

(12) [H~t(H~)]d = 0, for 1 _< i < d, and

(133 [H~(Ba-1 )]a = O.

434 A. Simis et al.

As to (12), it suffices to check that for 1 < i < d, dimR[H~]d < i, because in that case

t + l r lq ,+ l (H , ) ]d ..o Hm ( [Hi]d ) ---- 0 tJ. M

(cf. [8, Lemma 2.2]). So let p C V( I ) with dimR~, < d - i - 1; then I v satisfies .~l , and hence by Theorem 3.3, Hi(~RRv is concentrated in degrees < v(lv) + i < d - 1, which gives ([H,]d)~, ~ [Hi@RRv]d = O.

As for (13), let H o r n ( - , - ) stand for the graded Hom-functor and let E denote the injective envelope of the residue field of [G]0 = R/I . Graded duality [13, (36.8), (34.6)(4) and (33.2)] yields an isomorphism of graded G-modules,

HomRII(H~(Bd-1 ), E ) ~- HOmc(Bd_ l, WG),

which reduces us to showing that

[HOmc(Bd_l, COC )]_d = O .

However, Bd-I ~-- (G/0: ( b ' l , . . . , b ' a ) ) ( - d ) and WG --~ G, hence

! [Homc(Bd_i ,WG)]_ d ~_ [0- (0 �9 (brl . . . . . bd))]0 .

Now 0 �9 (0 �9 (b' i . . . . . b~l)) C 0 " (0 �9 G+) = G+, where the latter equality follows from the fact that G is Gorenstein and G/G+ ~- R / I is equidimensional of dimension d without embedded primes. Thus

[ 0 : ( 0 " ( b ' , , . . . , b ~ ) ) ] 0 C [G+]0 = 0 .

Our next result indicates that the Gorenstein property of grs(R ) imposes severe restrictions on strongly Cohen-Macaulay ideals.

Theorem 3.8 Le t R be a Noetherian local ring, let I be an R-ideal o f height g and analytic spread {, which is generically a complete intersection, and assume

that the Koszul homology modules H~ o f I are Cohen-Macaulay modules f o r 0 < i < { - g - 1. I f grl(R) is Gorenstein, then I is ~ l and strongly Cohen-Macaulay.

P r o o f First notice that R is necessarily Gorenstein [13, the proof o f (11.16)(b)]. After adjoining variables and localizing, we may assume that the residue field o f R is infinite and that g > 2. Furthermore, by [7, Theorem] we are reduced to the case f > g + 1. Inducting on d = direR, we may suppose that locally on the punctured spectrum, I is .~l and strongly Cohen-Macaulay.

By Theorem 3.7, the reduction number r of I is at most { - g - 1. It suffices to show that v(1) = {, or equivalently, that r = 0. Write I = ( f l . . . . . J'~), let Zo denote the cycles o f the Koszul complex Ko = K ~ . . . . . f, ,) , and consider the approximation complexes

~ , . �9 0 ~ Z~ -~ . . . --+ Zj_ s | Ss(R ~) . . . . ---+ Zo | Si(R ~) --+ O .


Locally on the punctured spectrum of R, ~'~,. are acyclic and I is of linear type, since 1 is ,~-I and strongly Cohen-Macaulay [14, Theorem 5.3 and Theorem 9.1]. On the other hand, by our assumption on the homology of K., depth Z, > d - 9 + 2 for 0 < i < { - g. Hence the acyclicity lemma implies that ~ , . are acyclic in the same range. Now using these complexes, one sees that for 0 < i <_ { - g, Ho(~, . ) ~- S,(I) are torsionfree and therefore S,(1) ~- I i. Thus r = 0 or r > { - g . But since r < ( - g - 1, we conclude that r = 0 .

Example 3.9 (cf. also [9, Theorem 1.3]) Let R be a Gorenstein local ring, and let I be a Cohen-Macaulay R-ideal of analytic deviation at most one (i.e., {(I ) < height I + 1 ) , which is generically a complete intersection. Then grl(R ) is Gorenstein if and only if I is an almost complete intersection.

Remark 3.10 Using Theorem 3.5 instead of Theorem 3.7, one can show the following. Let R be a Noetherian local ring, let I be an ideal of height g and analytic spread {, and assume that H,(I) are Cohen-Macaulay for 0 <- i <_ { - g and that G = gr/(R) is Cohen-Macaulay with a(G) = a. If a < -g , then I is c ~ 1 and strongly Cohen-Macaulay. (Conversely, if I satisfies ~ l , then a : - g by Theorem 2.4.)

Corollary 3.11 Let R be a Gorenstein local ring and let I be an R-ideal that satisfies sliding depth and is generically a complete intersection. Then any two o f the jollowing conditions imply the third one:

(a) I satisfies Y l . (b) I is strongly Cohen-Macaulay. (c) grl(R) is Gorenstein.

Proof By Theorem 3.8 and [14, Theorem 9.1], it suffices to show that (a) and (c) imply (b). However, since I has no embedded prime by our assumption and is height unmixed by Theorem 2.1, I is unmixed. Thus by [17, Lemma 3.6] and [30, Proposition 2.1 and Proposition 1.3], the sliding depth condition implies that R/I is Cohen-Macaulay. But then [16, Theorem 1.3] shows that (b) is a consequence of (a) and (c). []

4 Degrees of polynomial relations

In this section we extend several results of [35] dealing with the role of the reduction number in the Cohen-Macaulayness of R[It], give affirmative answers to some questions raised in [35], and describe several new kinds of ideals with Cohen-Macaulay Rees algebras.

Expected reduction number

We start by introducing the ideals that will be studied in this section.

Assumption and Notation 4.1 Let (R, m) be a Gorenstein local ring of dimension d with infinite residue field, let I be a strongly Cohen-Macaulay R-ideal of

436 A. Simis et al.

height g >= 2, and write n = v(I), { = f ( I ) , r = r(1). Assume that { < n and that I satisfies ~1 locally in codimension n - 2 . Further, let q~ stand for a minimal presentation matrix of I , present S = S(1) as B/J , where B = R[Tt . . . . . Tn] and J is generated by the entries of [T1 . . . . . T,] �9 q~, and let ~r be the S-ideal considered in (7).

We refer to the introduction for classes of ideals satisfying the above conditions. Notice that with Assumption 4.1, { = n - 1 [35, Theorem 4.1], height 1l(~o) = n - 1 [14, Theorem 9.1], I~, is of linear type with Gorenstein associated graded ring for p C V(I ) \V( l l (~o) ) [14, Theorem 9.1], S/II((p)S ~- B/II(q~)B, S is Cohen-Macaulay [14, Theorem 10.1 ], necessarily of dimension d + 1, and Asss(S) = A s s s ( S / ~ ) U {pS[p C V(Ii(~o)) with dimR~, = n - 1}. The next result is a slight generalization of [35, Theorem 2.5].

Theorem 4.2 With Assumption 4.1, min{ i [ [~ ' ] i+0} = n - g + 1, and [d]n_y+L(~RR~, _~ Ext~-I(R/II(~o),R)~RR~, for every p E V( I ) with dimR~, = n - 1. In particular, r > n - 9.

Proo f Since I is strongly Cohen-Macaulay and o~0, the approximation complex ~ o o f / is acyclic with H 0 ( ~ ~ S [14, Theorem 5.3]. Thus again, the strong Cohen-Macaulayness of I implies that for i < n - g, S,(I) is torsion free and hence [.~r = 0. The remaining assertions follow from [35, Theorem 2.5]. []

Motivated by the above result (in the special case n = d + 1), n - g was called the expected reduction number and it was asked in [35, Conjecture 3.1] whether the equality r = n - g implies that R[It] is Cohen-Macaulay. Before giving an affirmative answer to this question, we need to make some general remarks.

Remark 4.3 With Assumption 4.1, the following hold: (a) If r(l~,) < n - g for every p C V(Ii(q~)) with dimR v = n - 1, then

d N I~(~o)S = O. (b) If It(~o) �9 ~r = 0, then ~r = O, ~1 = 0 :sll(~p)S, and I~(q~)S

= O : s ~ .

Proof. (a) For every p E V(II(q~)) with dimRp = n - 1, It(cp) �9 [ d ] , - y + l @RRp = 0 by Theorem 4.2, where [-~]n-g+t ~ pS because r(Ip) = n - g . Thus (ll(r = O. Since on the other hand, ~4,V = 0 for every ~ E A s s s ( S / d ) , we conclude that ~ AlI ( (p)S is zero locally at every associated prime of S.

(b) The assertions follow because ll(r does not lie in any associated prime of ~ , and conversely, the prime ideals of Asss(S/I i(cp)S) = Asss(B/Ii(cp)B) being contained in mS, ,~ cannot lie in any associated prime of ll(~o)S. []

Lemma 4.4 With Assumption 4.1, the following are equivalent: (a) l~(r �9 ~r = O. (b) After 'sui table elementary row operations, the entries o f one row o f q9

generate 11 (~o).


Proo f Condition (a) is equivalent to 11 (q)S,,,s = 0, which means that I1 (~o)B,,,~ = JB,,,B. Since k is infinite, the latter condition holds i f and only i f ll(qO)B~. = JBc., where ~ = (m, Ti - a l . . . . . T , - a~) with (al . . . . . a , ) C R ~ and some a i a unit in R. Specializing modulo the B o - i d e a l generated by the sequence TI - a l . . . . . T , - a , , which is regular on Bo and B ~ / l l ( q ) B c . , one sees that the equality II(qo)Bc. = JB~ holds i f and only if Ii(q~) is generated by the row [al . . . . . a , ] �9 ~0, with some a i a unit.

Codimension two

Proposi t ion 4.5 In addition to Assumption 4.1, suppose that g = 2. The following are equivalent:

(a) r _< n - 2. (b) r(lv) < n - 2 Jbr every p c V(Ii(qo)) with dimR v = n - 1. (c) ,~ is generated by one element. (d) After suitable elementary' row operations, the entries o f one row o f go

generate Ii (q). (e) R[It] is Cohen-Macaulay.

Proo f (a) =~ (b) One uses the fact that : ( I v) = n - 1 = : (cf. [35, Theorem 4.1]).

(b) ~ (d) This follows from Remark 4.3(a) and Lemma 4.4. (d) =~ (c) and (e) Since g = 2, we know from [15, Theorem 6.6] that S

is Gorenstein. Furthermore, by Lemma 4.4, l l ( (p) �9 ~r = 0 and hence l l(q)S,, ,s = 0. Thus,

B, , ,B / I I ((,0)B,,,B ~ S,,,s/ll ( ~o )S,,~s ~-- S,,~s

is Gorenstein, which implies the Gorensteinness of R/Ii(~o) and hence of S/Ii(~o)S ~- B/Ii(~o)B. But then, again since S is Gorenstein and since .C = 0 : Ii(q))S by Remark 4.3(b), it follows from linkage theory [30] that ~v/ is generated by one element and that R[It] ~_ S / . ~ is Cohen-Macaulay.

(c) ==> (a) This is clear by Theorem 4.2, since l = n - 1. (e) =~ (a) Write a = a(grl-(R)). By Corollary 3.6, r < : - 1 < n - 2. ~

Example 4.6 Let k be a field of characteristic zero and let I be the ideal in k[x, y,z] defined by the maximal minors o f the matrix

y - z x - y x y + 3 y 2 - 2 x z - 3 y z "~ z x 3 yz - z 2

J q) = y + 2z - 3 z x y + 2y 2 + 2 x z + 6 y z + 3 z 2 ' x - 3y 4y - 9 y 2 - 4yz

The ideal I does not satisfy condition (d) o f Proposition 4.5, so that R[/t] is not Cohen-Macaulay (in fact, r = 4 and ~ ' is generated by two elements, of degrees 3 and 5). On the other hand, it is proved in [35, Corollary 3.11] (cf. also Corollary 4.11) that i f the presentation matrix of a codimension two ideal has only linear entries then the Rees algebra is Cohen-Macaulay.

438 A. Simis et al.

Higher codimension

For what follows, we need some information about the canonical module COs of the symmetric algebra.

Lemma 4.7 With Assumption 4.1, for every i,

depthR[cos], > max{d - i + 1,d - n + 1}.

Proof By [15, p. 675], S has a homogeneous minimal B-resolution F . with F/ = (~i_0B(-i)/~,' (indeed, using the ~e-complex, one readily shows that T o ~ ( S , R / ~ ) is concentrated in degrees _< j ) . Now, applying the functor _ v = Hom~(--,coB) --~ H o m s ( - , B ( - n ) ) , we obtain a complex F v of homogeneous free B-modules whose only non vanishing cohomology is cos -~ Ext~-l(B, toe). Thus COs ~ C/D, with C and D fitting into exact sequences

0 ~ C --+ FV_l -'+ . . . . FnV+d_2 , (14)

and

(15)

where

(16)

O--+Fff --+...--~F,V_2--~ D - + O ,

J v v O B ( - n +

t :O

Looking at graded pieces of the above complexes, we see from (14), (16) that depthR[C]i > m a x { d - i + 1,d}, and from (15), (16) that projdimR[D ], < min{i - 2, n - 2}, which gives depthn[D]i > max{d - i + 2, d - n + 2}. Thus depthR[cos] , = depthR[C]i/[D]i >= max{d - i + 1,d - n + 1}. []

Our next lemma will provide the crucial step in dealing with ideals of arbitrary height.

Lemma 4.8 In addition to Assumption 4.1, suppose that ll(r �9 ~ = O. Then the fractional R[It]-ideal (1, t) y-2 is a maximal Cohen-Macaulay module.

Proof In addition to the notation of 4.1, write N = R[It], and present N = B/K. Further choose L ~ J to be a B-ideal generated by a regular sequence of n - 1 linear forms such that L@RR v = J@RR~, for every p E Ass(R). Now consider the B-module E = L :J/L : ll(q~)B.

We claim that E is a maximal Cohen-Macaulay N-module. First, E is an .~-module because l l ( q ~ ) - K C J . Next, notice that B,,~B/II(~p)BmB ~- S,,,s/Ii(q~)S,,s ~- S,,,s, where the latter module is Cohen-Macaulay. Therefore R/ll(tp) is Cohen-Macaulay, and hence S/ll(q))S ~ B/Ii(qo)B is a maximal Cohen-Macaulay S-module. Thus, applying _ v __ Homs(- ,cos) to a minimal free S-resolution Fo of S/ll(qOS, one obtains an exact sequence

O---, (S/Ii(q)))v ~o:,oslt(~o)S --~ sV ... cos --, F~/ ---~ ---, F v . . . . d + l ,

which shows that r is a maximal Cohen-Macaulay S-module. However


(17) COs(1 ) ~- L " J/L ,

and therefore (~os/O :,,s II(go)S)(1) ~_ L �9 J/L �9 ll(q~)B ~-- E. Thus E is a maximal Cohen-Macaulay R-module.

Now the assertion of our lemma will follow once we have shown that E x-,g-2 is isomorphic to the R-ideal ( f , f t ) y-2 --- z..,i=0 f y -2 t iR for some R-regular

element f E I. To establish the isomorphism, first notice that the natural map from B onto

R factors through B/L �9 Ii(~o)B, because the image of L in R is zero and the image of Ii(go)B in R has positive grade. Thus we obtain a homomorphism of graded R-modules tp " E -+ R. For every ~ C Ass~(E), ~13 E Asse(.N) and hence p = ~2[~ C3 R E Ass(R). Now by our choice of L, ~P@RR~, is an isomorphism, and thus O@~R,I.~ is an isomorphism as well. Therefore ~9 is injective, and it can be used to identify E with a homogeneous ~-ideal.

In order to describe this ideal, notice that COB/1,(~o)e(1) "" L �9 Ii(q~)B/L and a(B/Ii(go)B) = - n , which shows that [L �9 II(qo)B/L]~ = 0 for i =< n - 2. But then by (17),

(18) [E], _~ [O)s],+l,

On the other hand, it has been shown

f o r i < n - 2 .

in [15, Corollary 6.6 and its proof] that g--2

(19) ( oas/S + ~os )(1) ~- R O ( R/I ) ( - i ) . I=l

Now since Cos(l) maps homogeneously onto E, is generated by homogeneous elements of degrees < 9 - 2 (cf. (19)), and coincides with E in degrees < g - 2 (cf. (18)), it follows that (Cos/S+:os)(1) ~ E/R+E. Thus by (19),

g - 2

(20) E / ~ + E ~_ R O ( R / I ) ( - i ) . t=l

V,~/-2 From (20) we conclude that E ~ z--,,=0 a, t 'R with a, E I ' , and

t--I (21) a , t ' I C ~ a f R + , for 1 < i_< g - 2 .

j=0

Also, by (20), [E]0 ~ R and hence a0 has to be a non zero divisor on R. Thus, after multiplying with a suitable R-regular element, we may assume that a0 = f ~ - 2 for some R-regular element f E I. We are going to prove by induction on k, 0 < k < g - 2, that we can choose ak to be f g - 2 as well.

f g - 2 = f g - 2 Since a0 = . , , we may suppose that 1 < k < g - 2 and that a, = for 0 -< i < k - 1. But then by (21), akI C f~-ZR, and hence ak E f~/-2R,

f g - 2 ) because f~/-2 is R-regular and grade I > 1. Write ak = : , 2 E R. We are done once we have shown that ). is a unit in R.

So suppose that 2 E m; then depth R/(I, 2) < dimR/(I , 2) < d - g, and hence depth(/, 2) < d - g + 1. On the other hand, [E]k --~ fY-2( l , s k ~-- (I, 2), which implies that depthR[E]~ < d - g + 1. But this is impossible, because by (18) and Lemma 4.7, depthR[E]k => d - k > d - g + 2. D~

440 A. Simis et al.

We will also use the following fact, which is well known (see for instance [ 19, Proposition 2.1 ]).

L e m m a 4.9 Let R be a Noetherian local ring, I an R-ideal, f c I an R-regular element, k > O, let . . . . . denote images in k = R / ( f k ) , write G = gr/(R), and

let f ' and f k ' stand for the images o f f , f k in [G]l and [G]k, respectively. The following are equivalent:

(a) R[/t] -~ R[I t] / ( f , f t ) k. (b) grf(k) _~ gr i (R) / ( f k ' ) .

(c) f k ' is G-regular. (d) f ' is G-regular.

Proo/J Parts (a) and (b) are equivalent to saying that for i > k, I ' N ( f k ) = l i - k f ~, which in turn means that f k ' is G-regular. The equivalence of (c) and (d) holds because in G, j .k' = ( f , )~ .

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

Theorem 4.10 With Assumption 4.1, consider the following conditions: (a) r <= n - g (in which case r = n - 9). (b) r(l~,) < n - 9 f o r every p E V(II((p)) with dimR v = n - 1

(in which case r ( I~ , )= n - 9). (c) The relation type o f l is at most n - 9 + 1 (in which case the relation

type equals n - g + 1 ). (d) ~ is generated by one element (in which case .~' is generated by one

f o r m o f degree n - g + 1). (e) Af ter suitable elementary row operations, the entries q f one row o f ~o

generate Ii ((p). (f) R[It] is Cohen-Macaulay. Then (a), (b), (c), (d), (e) are equivalent, and either condition implies (f).

Proof. The parenthetical remarks in parts (a), (b), (c), (d) follow from The- orem 4.2, the implication (d) ~ (c) is trivially true, (c) ~ (a) is clear since E(I) = n - 1 [35, Theorem 4.1], (a) ~ (b) follows because E ( l p ) = n - 1 = E(I), and the implication (b) ~ (e) has been shown in Remark 4.3(a) and Lemma 4.4.

We are going to prove by induction on 9 > 2, that (e) implies (d) and (f). The case 9 = 2 being covered by Proposition 4.5, we may now assume that g > 2 and that the_ implication holds for g - 1.

So suppose that 1 satisfies (e), and write ~ = R[It], G = grI(R ). Pick an R-regular element f C I \ m I such that the image f ' of f in [G]I is not contained in any minimal prime of I i ( q ) G or of raG, and is not contained in any associated prime ~]3 of G with G+ ~ ~1,~. By the first condition,

(22) height(ll(~0).~ + f t M ) > 1 ,

and


(23) : ( I / ( f ) ) < : ( I ) ,

whereas by the second assumption,

(24) f ' is regular on G@RR ~, for every p C V ( I ) \ V(Ii(q))).

The latter assertion follows from the fact that for every ta c V ( I ) \ V(II(q~)), G@RR p is Cohen-Macaulay and therefore has no associated primes containing G@RRv +.

We claim that f ' is actually G-regular. To show this, let . . . . . denote images in R = R/( fY-2) , and consider the natural epimorphism

O" R[It]/(f, f t ) y-2 --* R[/t] .

By Lemma 4.4, Il(q~) �9 o~ = 0, and hence by Lemma 4.8, depth(f, f t ) "-2 = d + 1. Since ~ satisfies &, we may then conclude that for every associated prime 93 of ( f , f t ) y-2, dim~,l~ = 1. On the other hand, by (22), height( l l ( rp)~ + ( f , f t ) y-2) > l, which implies that no associated prime of ( f , f t ) y-2 contains Ii(qo). But for p E V ( I ) \ V(II(~o)), 02, is an isomorphism by (24) and Lemma 4.9. Therefore ~ is an isomorphism, which again by Lemma 4.9, shows that f t is G-regular.

Now, changing our original notation, let " - " denote images in/~ = R / ( f ) . Then [ is strongly Cohen-Macaulay [22, Lemma 1.6], height I = g - 1 > 2, v([) = n - 1, and : ([) < n - 1 (cf. (23)). Furthermore, by Lemma 4.9, gr:(R) _~ G / ( f ' ) and k[[t] ~_ .~/( f , f t ) . This last isomorphism shows that locally in codimension n - 3, [ is of linear type and hence :~t . Thus [ satisfies Assumption 4.1. Moreover, assumption (e) passes from I to [, as can be easily seen using the equivalent condition (a) of Lemma 4.4 and the isomorphism

R[[t] ~-- ~ / ( f , i t ) . By our induction hypothesis, [ satisfies (d) and (f), so it suffices to prove

that these conditions lift from [ back to I . As for (f), recall that gr:(/~) -~ G/( f ' ) , with f ' being G-regular. Now G is Cohen-Macaulay and a(G) = a ( g r : ( R ) ) - 1 < - 1 , where the latter inequality follows from Corollary 1.3. Now again by this same corollary, R[It] is Cohen-Macaulay.

In order to prove that (d) lifts, consider the natural exact sequence

(25) 0 ~ ~ --~ S@RR/I -~ G --~ O,

and let F denote the element f in I = [S]1. Now, F being regular on G with G/FG ~_ gr:(/~), (25) yields an exact sequence

0 --+ ~ @ s S / F S --~ S ([ )@~k/[ --+ gr:(R) --+ 0 .

I f (d) holds for [, then JS(~sS/FS is generated by one homogeneous element of degree n - g + 1, and hence ~ has the same property. Now [14, the proof of Theorem 3.1] shows that ,re is minimally generated by forms of degrees < n - g + 1, with at most one form in degree n - g + 1. But then by Theorem 4.2, I satisfies (d).

442 A. Simis et al.

Linear presentation

Corollary 4.11 Let R : k[xl . . . . . Xd] be a polynomial ring over a field, let I be an R-ideal o f height g ~ 2 having a linear presentation matrix, write n : v(1), and assume that I is strongly Cohen-Macaulay and satisfies ~1 locally in codimension n - 2. Then R[lt] is Cohen-Macaulay and I is generated

n - 1 by forms o f degree -g~_ 1"

Proo f One may assume that k is infinite and g + 1 < n < d + 1. We first prove the Cohen-Macaulayness of R[lt]. I f height ll(~p) > n, then I satisfies ~ l and our assertion follows from [14, Theorem 9.1]. I f however, height 11 (q~) < n - 1 , then without loss of generality, ll(~p) = (xl . . . . , x , - i ). Thus I is extended from an ideal in the subring k[xl . . . . . x , - l ] , which reduces us to the case n = d + 1. But then I has reduction number n - g by [35, Theorem 3.2], and hence R[It] is Cohen-Macaulay by Theorem 4.10.

In order to show that 1 is generated by forms of degree n - 1 g - 1' we may

factor out d + 1 - n general linear forms to reduce once more to the case n : d + 1. Now by [35, Theorem 3.10], the Cohen-Macaulayness of R[It] implies that I is generated by forms of the asserted degree. [~

Generic Gorensteinness and canonical module

Proposition 4.12 With Assumption 4.1, S,,,s is Gorenstein and S is generically Gorenstein,

Proo f We need to show that S,,,s is Gorenstein. To this end, let { f l . . . . . fn} be generators o f I corresponding to the presentation matrix q~, and write [ b l . . . . . bm] = [Tl . . . . , Tn] " qo. Further define R' = B,,,B = R(T1 . . . . . Tn), h, = T n f l - T,f~ for 1 < i < n - 1, h, = fn , F = IR' = (hi . . . . ,hn)R', K; = (hi . . . . , hn- l )R', and J ' = JR' = (h i , . . . , bm)R t. Then [bl . . . . . bm] is the last row of a presentation matrix o f / ' with respect to the generators h i , . . . , h,, and therefore (bt . . . . . bm)R; = (hi . . . . , h , - t ) R ' : (h,)R' , or equivalently, J ' = K ' �9 F . Thus S,,s ~- R ' /K ' :I ' .

But now R' is a Gorenstein local ring, I ~ is a strongly Cohen-Macaulay R;- ideal satisfying "~1 locally in codimension n - 2 , v(K') < n - 1 < height K/" I/, and v( l ' /K/ ) = 1. In this situation, [34, Theorem 2.9 and Remark 2.10] implies that R' /K t �9 I ' is Gorenstein (in the case n - 1 > g; for n - 1 = g, one simply uses linkage theory, cf. [30]).

Corollary 4.13 If, with Assumption 4.1, r < n - y , then R/ll(~p) is Gorenstein.

Proo f By Remark 4.3(a), ll(~p)S,,s = 0, and by Proposition 4.12, Sins is Gorenstein. Thus B,,is/ll(q))B,lte is Gorenstein, which yields our assertion (cf. also the proof o f Proposition 4.5).

Corollary 4.14 If, with Assumption 4.1, r < n - g, then for the canonical module oJ o f ~ = R[lt],


(i) co/.~+co --~ (I i((p))(--1) #7 case 9 = 2, and (ii) co/.r R ( - 1 ) (~]~22(R/I ) ( - i ) �9 ( I1(~o) / I ) ( -9 + 1) in case 9 > 2.

P r o o f By Theorem 4.10 and Remark 4.3, we have an exact sequence

0 ~ ( S / I i ( q ) ) g ) ( - n + g - 1) -~ S -~ ~ ~ 0

which, upon dualizing, gives

(26) 0 -4 co ~ COs --~ (cos/t,(~o)s)(n - g + 1 ) ~ O.

Furthermore, by Corollary 4.13

COs/h(~)s ~ cos/It(q~)~ -~ ( B / I t ( q ~ ) B ) ( - n ) .

Since TI . . . . . Tn form a regular sequence on the latter module, tensoring (26) with @~B/B+ yields an exact sequence of B-modules,

0 --~ co/.~+co ---, cos/S+cos ---+ ( R / I i ( q o ) ) ( - 9 + 1) ~ 0 .

Now our assertion follows because cos/S+COs ~ - R ( - 1 ) ( ~ ] _ ~ l ( R / I ) ( - - i ) by (19).

We finish by mentioning the following open problem about prime ideals [36, Conjecture 3.1.6]:

Conjecture 4.15 Let R be a regular local rin9 o f dhnension three and let 1 be a pr ime ideal 9enerated by f our elements. I f I is a normal ideal then R[It] is Cohen-Macaulay.

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Cohen-Macaulay Rees algebras and degrees of polynomial relations

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