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6/01 chlRC/DateDuopGS—als

The depth of blow-up rings of ideals
By

Laura Ghezzi

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2001

ABSTRACT

The depth of blow-up rings of ideals
By

Laura Ghezzi

Let R be a local Cohen-Macaulay ring and let I be an R-ideal. The Rees algebra
R, the associated graded ring g and the fiber cone .7: are graded algebras that
reflect various algebraic and geometric properties of the ideal I . The Cohen-Macaulay
property of R and g has been extensively studied by many authors, but not much
is known about the Cohen-Macaulayness of .7: . We give an estimate for the depth of
R and 9 when these rings fail to be Cohen-Macaulay. We assume that I has small
reduction number, sufficiently good residual intersection properties, and satisfies local
conditions on the depth of some powers. We also study the Serre properties of R and
g and how they are related. In particular the SI property for 9 leads to criteria for
when I " = I (n), where I W is the n-th symbolic power of I. We prove a quite general
theorem on the Cohen-Macaulayness of .7: that unifies and generalizes several known
results. We also relate the Cohen-Macaulay property of .7: to the Cohen-Macaulay

property of R and g.

TABLE OF CONTENTS

1 Introduction

2 Preliminaries

2.1 Basic Notation ................................
2.2 Blow-up Rings and Reductions .......................
2.3 Strongly Cohen-Macaulay Ideals .......................
2.4 Artin-Nagata Properties ...........................

3 The Depth of Blow-Up Rings of Ideals

3.1 Preliminary Results ..............................
3.2 An Estimate of the Depth of the Associated Graded Ring .........
3.3 Examples ...................................

4 The Serre Properties of Blow-Up Rings of Ideals
5 Cohen-Macaulayness of the Fiber Cone

BIBLIOGRAPHY

iii

13
13
14
16
16

20
21
3O
52

56

74

90

 

CHAPTER 1

Introduction

Let R be a N oetherian local ring with maximal ideal m and residue field
1:, and let I be an R-ideal. The Rees algebra R : R[It] § 69.201" and
the associated graded ring 9 : gr1(R) 2 R (8);; R/I E“ 69,-20Ii/Ii+1
are two graded algebras that reflect 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 (Q) corresponds to the exceptional fiber
of the blow-up.

Many authors have extensively studied the Cohen-Macaulay property
of R and Q , since besides being interesting in its own right, it greatly
facilitates the study of several other properties of these algebras, such
as their normality ([6]), the depth of their graded pieces ([16]), the
Castelnuovo-Mumford regularity ([41]) or the number and degrees of
their defining equations ([3], [39]).

A very useful tool in the study of blow-up rings is the notion of

 

reduction of an ideal, with the reduction number measuring how closely
the two ideals are related. This approach is due to Northcott an Rees
([44]). An ideal J C I is called a reduction of I if the morphism
R[J t] ¢—> R[I t] is finite. When investigating R or 9 one tries to simplify
I by passing to a minimal reduction J; one of the big advantages is
that J contains a lot of information about I, but often requires fewer
generators. More precisely, if k is infinite, every minimal reduction
of I is generated by K elements, where 8 is the analytic spread of
I. (We refer to Chapter 2 for the terminology and the definitions.)
Using the finiteness of the morphism above one can recover some of the
properties of R[I t] from those of R[Jt], expecting even better results
on R[It] when J has a “nice” structure and when the reduction is
relatively small. This approach was first used by Huckaba and Huneke
([27], [28]) to prove the Cohen-Macaulayness of R and g for ideals with
reduction number one and analytic deviation at most two. Inspired by
this work, there have been several other results in the literature giving
sufficient conditions for R and g to be Cohen-Macaulay, when the
ideal I has small reduction number and small analytic deviation (see
for example [55], [51], [21], [22], [2], [1], [49], [50]). Johnson and Ulrich

proved a theorem that unifies and generalizes those results.

Theorem ([39, Theorem 3.1]) Let R be a local Cohen-Macaulay ring of

 

dimension d with infinite residue field, let I be an R-ideal of grade g,
analytic spread it, and reduction number r, let k 2 1 be an integer with
r S k, assume that I satisfies 0'3 and AN;3 locally in codimension
€- 1, that I satisfies AN€:max{2,k} and that depth(R/Ij) _>_ d—€+k—j
for 1 g j S h. Then Q is Cohen-Macaulay, and if g 2 2, R is Cohen-

Macaulay.

The Gg property is a local condition on the number of generators of
I up to codimension Z — 1, and we say that I satisfies the condition
AN: if certain residual intersections of I are Cohen-Macaulay. We
study these so-called Artin—Nagata properties in Section 2.4. Although
the definition is rather technical, these properties are satisfied by a large
class of ideals that include for instance perfect ideals of height two and
perfect Gorenstein ideals of height three.

The Artin-Nagata properties play an important role in the proof
of the above theorem as well as in the proof of the main results of
this work, since, together with the G’g property, they guarantee the
existence of a “nice” generating set for a minimal reduction of I. This
facilitates the computations of several intersections and ideal quotients
(see Lemma 3.1.1 and the other technical results presented in Section
3.1 for details). Indeed, most of the previous works also made use of

these residual intersection techniques, but only implicitly.

 

A similar result has been shown by Goto, Nakamura and Nishida in
[23, Theorem 1.1]. They weakened the AN,“ assumptions and substi-
tuted the G3 property with a local condition on reduction numbers.

.Another interesting issue is to estimate the depth of blow-up rings
when they fail to be Cohen-Macaulay. Recently, Cortadellas and
Zarzuela have come up with such formulas in [12], but only in very
special cases. They proved that if I is an equimultiple ideal with
reduction number at most one, then depth g = depth R/I + 9.
They also showed that if I has analytic deviation one and reduc-
tion number at most two and if some additional assumptions on 10-
cal reduction numbers are satisfied (see Theorem 3.2.8 for the pre-
cise statement), then min{depth R/I , depth R/ I 2} + g g depth 9 g
min{depth R/I, depth R/12}+ g + 1.

It is natural then to look for a general estimate of depth 9 involving
the depth of the powers of I. This is done in Chapter 3. If R is a
Noetherian ring and I an R-ideal with analytic spread Z, we easily
get that depth 9 _<_ inf{depth R/Ij | j 2 1} + i (see Remark 3.2.20).
Our main result is the following theorem, which gives the desired lower

bound for depth g .

Theorem 3.2.10 Let R be a local Cohen-Macaulay ring of dimension

d with infinite residue field, let I be an R-ideal of grade 9, analytic

 

spread 8, and reduction number r, let k 2 0 be an integer with r S k,
assume that I satisfies Ge, AN[_,C_1 and that for every p 6 V(I),
depth(R/Ij)p _>_ min{dime—€+k—j,k—j} whenever 1 g j 3 16—1.

Then depth 9 Z min({d} U {depth R/Ij +6 — k +j|1§ j S k}).

In the setup of the theorem we do not have any restriction on the
analytic deviation, but the assumption on the depth of the powers
forces the reduction number to be “sufficiently small”, namely r S
E — g + 1. Section 3.2 is devoted to the proof of the theorem. The
main tool is a combination of the residual intersection techniques used
by Johnson and Ulrich in [39] and of the local cohomology techniques
used by Goto, Nakamura and Nishida in [23]. In particular the theorem
generalizes their results, as well as the formulas obtained by Cortadellas
and Zarzuela. Theorem 3.2.10 also gives an estimate for depth R, since
if g is not Cohen—Macaulay, we have that depth R : depth g +1 ([29,
3.10]).

The theorem is particularly interesting when the reduction number
is small. In this case the assumptions are much simpler and we get a

precise formula for depth 9 .

Corollary 3.2.23 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 reduction number r g 1. Assume that I satisfies 03 and

AN;_2. Then depth 9 = min{d, depth R/I + E}.

Corollary 3.2.25 Let R be a local Cohen-Macaulay ring of dimension
d with infinite residue field, let I be an R-ideal with grade 9, analytic
spread t 2 g + 1, and reduction number r g 2. Further assume that
I satisfies Ge, AN€Z3 and that R/I is Cohen-Macaulay. Then depth

9 = min{d, depth R/I2 + 6}.

Since in our setup R and Q are not necessarily Cohen-Macaulay, it
is natural to ask which Serre properties are still satisfied in this case. In
Chapter 4 we obtain such results. The main idea is to use the estimates
of depth R and of depth 9 from Theorem 3.2.10. For example, under
assumptions similar to those of Theorem 3.2.10 we get the following

result for Q .

Theorem 4.0.1 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, analytic spread 13, and
reduction number r, let k 2 0 be an integer with r _<_ k, assume that I
satisfies Ge, 141N€"_k_1 and that for some integert Z 1, depth(R/Ij)p Z
min{dime — 6+ k —j,/s —j +t} wheneverp E V(I) and 1 g j S k.

Then Q is St.

The SI property for Q is particularly important because it leads to
criteria for when the power I n of I coincides with its symbolic power
I (7‘). Indeed, if Q is 51, under the additional assumption that locally

6

 

at every non minimal prime p in V(I), the analytic spread is less
than the height of p, we obtain the equality of all the regular and the
symbolic powers of I. This issue has been addressed by many authors,
for example by Huneke ([30]) and Hochster ([26]), especially if the ideal
I is prime.

From Theorem 4.0.1 we obtain that g is 31 in the following cases.

Corollary 4.0.2 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade 9, analytic spread t, and
reduction number r g 1. Further assume that I satisfies Ge, ANZZQ,
and that R/ I has no associated primes of height 2 E + 1. Then 9 is

$1.

Corollary 4.0.3 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, analytic spread t 2 9+1,
and reduction number r g 2. Further assume that I satisfies Cg,
ANIZ3, that R/ I is Cohen-Macaulay, and that R/ I 2 has no associated

primes of height 2 t +1. Then 9 is SI.

Of particular interest is also the 82 prOperty for R, because together
with the condition that R is regular at every prime of height one, it
is equivalent to the normality of R. This property is very important
and therefore has been studied by many authors (see for example [6],

[38] and the literature cited there). For instance, if R is a normal ring,

 

the normality of R is equivalent to the normality of the ideal I, which
means that every power of I is integrally closed. We have the following

result.

Theorem 4.0.8 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade 9 2 2, analytic spread E,
and reduction number r, let k 2 0 be an integer with r S k. Assume
that I satisfies Ge, AN[_k_1 and that depth(R/Ij)p Z min{dim Rp —
f + k — j,k — j + 2} whenever p E V(I) and 1 Sj g k. Furthermore
assume that I is t7 — 2-residually $2 locally up to height 6 + 1. Then

R 7:3 SQ.

Here the residually 32 assumption is a requirement that locally some

residual intersections of I satisfy Sg.

It is natural to ask how the Serre properties for R and Q are related.
Brumatti, Simis and Vasconcelos in [6, Theorem 1.5] related the 52
property for R with the 51 property for g . With the methods of their
proof we generalized their result in Theorem 4.0.10. In particular we

obtain the following equivalence.

Corollary 4.0.11 Let R be a Cohen-Macaulay ring and let I be an
R-ideal with ht I 2 t, for some integer t Z 1. Then R satisfies St if

and only if 9 satisfies St..1.

In the same setup, another interesting blow-up ring is the fiber cone
of I, .7: = .7(I) = 63,-20Ii/mIi 2’ R®Rk 92’ Q®Rk. This graded object
is important from a geometric point of view, since Pro j (7‘ ) corresponds
to the fiber over the closed point of the blow-up of Spec(R) along V(I).
The fiber cone encodes a lot of information about the ideal I. For ex-
ample, its Hilbert function gives the minimal number of generators of
the powers of I and, if the residue field is infinite, its Krull dimension
coincides with the minimal number of generators of any minimal reduc-
tion of I. We assume from now on that k is infinite. It is particularly
interesting to know when .7 is Cohen-Macaulay. If J is a minimal re-
duction of I, then it is well known that .7 (J) is a polynomial ring over
It and that .7: (I ) is a finite extension of .7: (J ) Hence .7 (I ) is a free
T (J )-graded module if and only if .7: (I ) is a Cohen-Macaulay graded
ring. This greatly facilitates the computation of the Hilbert function.
Also, the Cohen-Macaulayness of .7: implies, under some additional as-
sumptions, that the ideal m1" is integrally closed for every n. This
property plays an important role in the study of evolutions (see [33],
[35]). Furthermore, if char k = 0 and f is 32, then the reduction
number of I is less than the multiplicity of .7: ([56]).

If I is generated by a regular sequence, or more generally when I is

generated by analytically independent elements; i.e., when r(I) = 0,

 

then .7 is trivially Cohen-Macaulay. So the first interesting case is
that of ideals with reduction number one. Under this assumption, if
the ring R is Cohen-Macaulay, f was shown to be Cohen-Macaulay
by Huneke and Sally ([34]) when I is m-primary, by Shah ([47], [48])
when I is equimultiple, by Cortadellas and Zarzuela ([10]) when I
has analytic deviation one and is generically a complete intersection.
Later, in [11] Cortadellas and Zarzuela proved a more general theorem
for ideals with reduction number one, which covers all the previous
results (see Chapter 5 for the precise statement). Other cases have
been studied in [43], [17], [15], [14], [35]. Recently Huneke and Hiibl
proved a result ([33, Theorem 2.1]) on the Cohen-Macaulayness of .7:
for ideals having analytic deviation one, but any reduction number
r. They assume that F has homogeneous generating relations only
in degree greater than r. This is the only assumption involving the
reduction number.

We study the Cohen—Macaulayness of f in Chapter 5. The work
of Huneke and Hiibl inspired our main result, Theorem 5.0.3, which
generalizes their theorem, as well as Cortadellas and Zarzuela’s. The
main idea is that good “intersection properties” guarantee that we can
find dim]: elements that are a regular sequence on f.

In particular, if I has good residual intersection properties and suffi-

10

ciently small reduction number, as in the setup of the previous chapters,

we have the following.

Corollary 5.0.10 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade 9, analytic spread E, and
reduction number r. Assume that I satisfies Gg, AN[_,,_1 and that
for every p E V(I), depth(R/Ij)p Z min{dime — 6 + r — j,r — j}
whenever I g j g r — 1. Assume that .7 has at most two homogeneous

generating relations in degree 3 r. Then .7 is Cohen-Macaulay.

In particular the above assumptions are satisfied by strongly Cohen-
Macaulay ideals with the “expected” reduction number, and so we ob-

tain the following result.

Corollary 5.0.13 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be a strongly Cohen-Macaulay ideal of grade 9, an-

alytic spread E, and reduction number r with r S E — g + 1. Assume

that I satisfies Ge. Then .7 is Cohen-Macaulay.

It is natural to ask how the Cohen—Macaulayness of .7 is related to
the Cohen-Macaulayness of R and Q. In general these notions are
independent: An example of D’Anna, Guerrieri and Heinzer (Exam-
ple 5.0.11) shows that the Cohen-Macaulayness of R or 9 does not
imply the same property for .7. On the other hand, it is easy to build
examples in which .7 is Cohen-Macaulay, but R and g are not (see

11

 

for instance Example 5.0.17).
However, under additional assumptions, as a special case of Theo-

rem 5.0.3, we have that g Cohen-Macaulay implies .7 Cohen-Macaulay.

Corollary 5.0.9 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal of grade 9, analytic spread f, and
reduction number r 2 t — 9. Assume that I satisfies 05 and that .7
has at most two homogeneous generating relations of degree 3 r. If g

is Cohen-Macaulay, then .7 is Cohen-Macaulay.

A class of ideals which is particularly interesting in this context is
that of perfect ideals of height two. From Corollary 5.0.13 we easily get

the following.

Corollary 5.0.16 Let R be a local Cohen—Macaulay ring with infinite
residue field, let I be a perfect R—ideal of height 2 and analytic spread

6. Assume that I satisfies Ge. If R is Cohen-Macaulay, then .7 is

Cohen—Macaulay.

In light of [52, Corollary 5.4], it is easy to decide if R is Cohen-
Macaulay, just by checking if the presentation matrix satisfies the so-
called “row condition”. This fact has been very useful in building ex—
amples that show that the converse of Corollary 5.0.16 is not true: even
for perfect ideals of height two the Cohen-Macaulayness of .7 does not
imply the Cohen-Macaulayness of R.

12

CHAPTER 2

Preliminaries

In this chapter we fix the notation and give the basic definitions that
we will use in this work. For undefined terminology or further details

we refer to [7] and [54].

2.1 Basic Notation

Let R be a Noetherian local ring with maximal ideal m and residue field
It and let I be a proper R-ideal. We denote the grade of I by g, the
height of I by ht I, and the minimal number of generators of I by u(I).
We say that I is a complete intersection if fi(I) = g. We say that I
is generically a complete intersection if it is a complete intersection
locally at every associated prime. We say that I is unmixed if every

associated prime has the same height.

13

 

2.2 Blow-up Rings and Reductions

The Rees algebra of I is
n = R[It] 2 69.2012
the associated graded ring of I is
Q = gr1(R) = R®RR/I E” EBiZOIi/IHI,
and the fiber cone of I is
J: = EBQOIi/mlf "—3 Rst 9: gagk.

The analytic spread t of I is defined to be the Krull dimension of
7,

€(I) :2 dim.7 = dimR®Rk = dim Q®Rk.
We have that
ht I g [(I) g min{dim R,u(I)}.

The analytic deviation of I is defined as €(I) -— ht I. An ideal with
analytic deviation zero; i.e, with €(I) = ht I, is called equimultiple.

The second analytic deviation is the difference p(I) —— €(I).

The symmetric algebra of I is

S(I) = €Bi208i(l)a

14

 

where 3,-(I) denotes the jth symmetric power of I. There is a natural
surjective homomorhism a : S (I) —% R. If or is an isomorphism, we
say that I is of linear type. If grade I > 0, this is equivalent to S (I)

being R-torsionfree.

A very useful tool in the study of blow-up rings is the notion of
reduction of an ideal. An ideal J C I is called a reduction of I if the

morphism
R[Jt] c——> R[I t]

is module finite, or equivalently if I"+1 = J I r for some r 2
0. The least such r is denoted by r J(I ) and it is called
the reduction number of I with respect to J. We have that €(I) g
u(J) for every reduction J of I. A reduction is minimal if it is mini-
mal with respect to inclusion. If the residue field It is infinite, we have
that €(I) = n(J) for every minimal reduction J of I. In this case we

define the reduction number r of I by
r(I) : min{rJ(I) | J a minimal reduction of I}.

Notice that r(I) = 0 if and only if €(I) = fi(I).

15

2.3 Strongly Cohen-Macaulay Ideals

Let I :2 (51:1,...,:rn) be an ideal of a local Cohen-Macaulay ring
R. By H,(:1:1, . . . ,:rmR) we denote the ith Koszul homology of the
Koszul complex built on 231, . . . ,rn. We say that I is strongly Cohen-
Macaulay (SCM), if H,(:r1, . . . ,xn, R) are Cohen-Macaulay R-modules
for every i, and that I satisfies sliding depth (SD) if depth
Hi(:1:1,...,a:n,R) Z dimR — n +i for every i. Since R is Cohen-
Macaulay, the property of being SCM or SD is independent of the cho-
sen generating set of I. Also, we have that either H,(a:1, . . . ,crn, R) = 0
or dim H,(:L'1, . . . ,rn, R) = dimR/I, and that H,(:1:1, . . . ,ccn, R) = 0 for
every i > n — grade I. Hence, if I is strongly Cohen-Macaulay, then I
satisfies sliding depth. By [31] we have that if I is an ideal in the link-
age class of a complete intersection, then I is strongly Cohen-Macaulay.
So, in particular, if R is Gorenstein and I is perfect of grade two or
perfect of grade three with R/ I Gorenstein, then I is strongly Cohen-
Macaulay. We recall that I is perfect if R/ I is Cohen-Macaulay and

has finite projective dimension.

2.4 Artin—Nagata Properties

The notions presented in this section will play a very important role

throughout this work.

16

Definition 2.4.1 Let R be a local Cohen-Macaulay ring, let I be an

R-ideal of height g, and let 3 2 g be an integer.

1. An s-residual intersection of I is a proper R-ideal K = a: I

where a C I with ”((1) S s S ht K.

2. An s-residual intersection K of I is called a geometric s-residual

intersection if ht(I + K) 2 s + 1.

If R is Gorenstein and I is unmixed, then K is a g-residual in-
tersection of I if and only if K is linked to I, and K is a geometric
g-residual intersection of I if and only if K is geometrically linked to
I. Hence the notion of residual intersections, essentially introduced by
Artin and Nagata in [4], generalizes the concept of linkage to the case
where the two linked ideals need not have the same height. Unlike for
linkage, it is not clear when residual intersections are Cohen-Macaulay.
Ulrich in [51] introduced the term “Artin-Nagata” (AN) for the Cohen-

Macaulayness of residual intersections up to a given height.

Definition 2.4.2 Let R be a local Cohen-Macaulay ring, let I be an

R-ideal of height 9, and let 3 2 g be an integer.

1. We say that I satisfies ANS if for every 9 S i S s and every

i-residual intersection K of I, R/ K is Cohen-Macaulay.

17

2. We say that I satisfies AN; if for every g S i S s and every

geometric i-residual intersection K of I, R/ K is Cohen-Macaulay.

To guarantee the existence of s-residual intersections and geometric

(s —1)-residual intersections, one usually assumes the condition C, (see

[4])-

Definition 2.4.3 Let R be a local Cohen-Macaulay ring, let I be an
R-ideal, and let 5 be an integer.

We say that I satisfies property G3, if u(Ip) S dime for any prime
ideal p E V(I) with dim Rp S s — 1. We say that I satisfies Goo, if I

satisfies G3 for every 3.

Now we recall two important results, the first due to Her-
zog,Vasconcelos and Villareal, and the second due to Ulrich, that guar-

antee that an ideal satisfies the Artin-Nagata properties.

Theorem 2.4.4 ([25]) Let R be a local Cohen-Macaulay ring and let

I be an R-ideal satisfying GS and sliding depth. Then I satisfies ANS.

Theorem 2.4.5 ([51]) Let R be a local Gorenstein ring of dimension
d, let I be an R-ideal of grade 9, and assume that I satisfies Gs and
thatdepth R/Ij Z d—g—j+1 for 1 Sj S s—g+1. Then I satisfies
AN3.

18

Notice that if I is strongly Cohen-Macaulay and satisfies 0,, then
by Theorem 2.4.4 I satisfies ANS. This result provides a large class of
ideals satisfying the Artin-Nagata properties.

19

CHAPTER 3

The Depth of Blow-Up Rings of
Ideals

In this chapter we study the depth of the associated graded ring and of
the Rees algebra of ideals having good residual intersection properties
and sufficiently small reduction number. In Section 1 we present some
technical results that will play a crucial role in the rest of the chapter.
Section 2 is devoted to the proof of the main result, Theorem 3.2.10.
We then give several corollaries.

Throughout this chapter we use the following notation: R is a local
Cohen-Macaulay ring of dimension d with infinite residue field, I is a
proper R-ideal of grade 9, analytic spread t and reduction number r,

R and 9 denote the Rees algebra and the associated graded ring of I.

20

3. 1 Preliminary Results

We begin with a lemma, due to Ulrich, which contains some basics facts

about Artin-Nagata properties.

Lemma 3.1.1 ([39, Lemma 2.3]) Let R be a local Cohen-Macaulay
ring with infinite residue field, let a C I be ( not necessarily distinct)

R-ideals with ,u(a) S s S ht a : I and assume that I satisfies 0,.

(a) There exists a generating sequence a1,...,a3 of a such that for
every 0 S i S s—1andfor every subset {V1, . . .,1/,-} of{1, . . . ,s},

ht (aV1,...,a,,,):IZi andht I+(a,,1,...,a,,,):IZi+1.

(b) Assume that I satisfies AN,‘ for some t S s — 1 and that a 75 I,
write a,- = (a1, . . . ,a,-), K,- = a, : I and let ‘-’ denote images in

R/Ki. Thenfor 0 S i S t+ 1:

(i) K, 2a,- : (a,+1) and a,- =IflK,-, ifi S s— 1.
(ii) depth R/ai = d — i.
(iii) K,- is unmixed of height i.

(iv) ht Izl,ifiSs—1.

Remark 3.1.2 The above result is very useful when applied to reduc-
tions. Let I be an R-ideal with analytic spread E. If s 2 t, then 3 gen-
eral elements in I generate a reduction and, if I satisfies 0,, 3 general

21

elements a1, . . . , as in I have the property that ht (a1, . . . ,as) : I Z 3.
So Lemma 3.1.1 says that we can choose the generators of the reduction
so that they have “good properties”. In particular a1, . . . ,ag form a

regular sequence in R.

A similar lemma, in the sense of being able to choose a convenient

set of generators for a reduction, has been proven by Goto.

Lemma 3.1.3 ([23, Lemma 2.1]) Let R be a local Cohen-Macaulay ring
with infinite residue field, let I be an R-ideal, and let J be a reduction
of I, generated by 3 elements. Then there exists a system of generators

a1, . . . ,as for J satisfying:
(a) (a1, . . . ,a,),, is a reduction of Ip, ifp E V(I) and i: ht p S s.

(b) a,- ¢p ipr Ass(R/(a1,...,a,-_1))\V(I) for any 1 S i S 3.

Definition 3.1.4 ([23]) In the setup of the previous lemma, let J,- =
(a1, . . .,a,-). We define r,- : {maer,p(Ip) | p E V(I) and ht p = i} for

9313s

The ideals J, defined by Goto, and the ideals a,- of Lemma 3.1.1

enjoy similar properties.

22

Lemma 3.1.5 ([23, Lemma 2.4, Lemma 2.5, Corollary 2.6, Lemma 6.2
and their proof]) In the setup of Lemma 3.1.3, let k 2 0 be an integer
with rJ(I) S k, assume that I satisfies AN," for some t S s — 1, and
that r,- S max{0,i —- t -— 1} for all g S i < 3. Write J1: (a1,...,a,-)

andKinz-zI. ThenforOSiSt+1:

(a) ht JizIZi.

(b) ht (JizI)+IZi+1,ifiSs—1.

(c) Ki=J,-:(a,-+1) and Ji=IflK,-,ifiSs—1.

(d) depth R/Jz- Z d — i.
The following lemma is a generalization of [39, Lemma 2.5].

Lemma 3.1.6 Let R be a local Cohen-Macaulay ring of dimension d
with infinite residue field, let a C I be R-ideals with u(a) S s S
V ht a : I, let t be an integer with t S s — 1. Assume that I satisfies
G3 and AN; and that [a,- : (ai+1)] flIj = ailj‘l for 0 S i S s — 1,
j Z max{1,i—t}, where a1, . . . ,aS and a,- are as defined in Lemma 3.1.]

(a). Then
depth R/am' 2 min({d — 2'} u {depth R/Ij‘” -— n | 0 g n _<_ i — 1}),
forOSiSs andemax{0,i—t—1}.

23

Proof. We use induction on i. The assertion is trivial for i = 0; so we
may assume that 0 S i S s — 1. We need to show that the inequality
holds for i + 1. For j = 0 (which can only occur if i + 1 S t-l— 1), we
have that depth R/aHl = d — i — 1 by Lemma 3.1.1 (b) (ii), and so
our assertion follows. Thus we may suppose that j Z 1. But then by

assumption,

(1in fl 0.14.11]- : ai+1[(ailj 3 (9124-1))fl [j]
C ai+1l(ai 3 (02%)) n [j]
= ai+1ai1j_l

C (1in fl ai+le.
Hence we obtain an exact sequence
0 —> ai+1ain_1 —> £1in69 ai+le —> a,+119' —+ 0. (1)
On the other hand, by the assumption for i = 0,
[0 : (a,+1)] n ain‘1 c [0 : (it-.010 Ij = 0,

and therefore ai+1a,-Ij‘1 ’-‘_-’ ain‘l, a,+1Ij g If. The required depth

estimate for R/ €1,111 j follows from (1) and the induction hypothesisa

The next lemma gives conditions that imply the intersections in the

assumptions of Lemma 3.1.6. It is a generalization of [39, Lemma 2.8].

24

Lemma 3.1.7 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, let J be a reduction
ofI generated by 3 elements with ht J: I Z s, r 2 rJ(I), let k 2 0 be
an integer with r S k, assume that I satisfies G3, AN 8"_k_1 and that
for every p E V(I), depth(R/Ij)p _>_ min{dim Rp -- s + k —- j,k — 3'}
whenever I S j S k — 1. Write g = gr1(R), for a E I let a’ denote
the image of a in [g]1, and for a = J, let a1, . . . ,as and a,- be defined

as in Lemma 3.1.1 (a). Then:

(a) For every p E Spec(R), depth(R/a,-Ij)p 2 min{dim R,D —- s + k —

j,k—j}, wheneverO S i S s—1 and max{0,i—3+k} S j S k—l.

(b) [ai : (a,+1)] fl Ij = ain‘1 whenever 0 S i S s —1 andj Z

max{1,i—s+k+1}.
(c) aiflIjzain‘l wheneverOSiSs andei—s+k+1.

(d) a’1,...,a’g form a Q-regular sequence and [(a’1,... ,a[) :g (0:41)]3' :

[(a’1,.. . ,a[)]j wheneverg S i S s—1 andj Z max{1,i—s+k+1}.

Proof. First we show (a), (b) and (c) by induction on j. Notice that
for j = 0, (which can only occur ifi S s—k), (a) holds by Lemma 3.1.1
(b) (ii) with t = s — k — 1. Also, (b) holds for j = 1 by Lemma 3.1.1
(b) (i) and (c) trivially holds for j = 1.

CLAIM 1. If (b) holds for j, then so does (a).

25

Proof. This follows from Lemma 3.1.6, with t = s — k — 1, using the
assumption on the depth of the powers of I.

Now we can assume that (a), (b) and (c) hold for j Z 1 and we show
that they hold for j + 1.

CLAIM 2. If (a) holds for j, then (b) and (c) hold for j+ 1 and the
maximal value of i; namely i = s — k + j.

Proof. If j 2 k, then i = s— k+j Z s; so there is nothing to show in
(b) and I].+1 :2 JIj = ast, which implies (c). Let j S [9—1. In order to
show (b), let p E Ass(R/a,-Ij). By (a), min{dim Rp—s+k—j,k—j} S
0; hence dim Rp S s — k+j = i. If Ip : RP, then a,- C p, since p E
Ass(R/ain). Since ht a,“ : I _>_ i+ 1 by Lemma 3.1.1 (a), we have
that (a,+1)p = R1, and (b) follows. If Ip 75 Rp, since ht I+ai : I Z i+1
by Lemma 3.1.1 (a), we have that Ip = (a,)p and again we get the

desired equality of (b). Since
C11 0 Ij+1 C [C152 (a,+1)]fl Ij+1 2 £1in,

(0) holds.

CLAIM 3. If (b) and (c) hold for j + l and maximal i, then (b) and
(c) hold for j + 1 and any i.

Proof. Take i < s — k + j + 1. By decreasing induction on i, a,“ O

I3.+1 2' ai+1Ij. Hence

26

a.- n 13'+1 = a.- 0 am fl 1’“
= “in “i+11j
: a, D ((1in + ai+11jl
= (1in + 02' O ai+11j
= ailj + ai+1[(ai I (Gull) fl Ijl
2 £1in + Chump—1

= ail].

This shows (c).
Since

[02' 3 (ai+1)] O NH C [11,- 3 (0241)] n [j C at,

we have that
[(1, : (a,+1)] fl I].+1 C (ii 0 Ij+1 : ailj;

hence (b) holds.

The first claim of ((1) follows from [53, Corollary 2.7] and part (c)

with i = . Now let u E a’,...,a’- : a’. -. Pickin an element
9 1 2 2+1 J g

x E Ij with x + I"+1 = u, we have ai+1x E 02- + N”, and so by part

(6).

27

(124.117 6 ai+1 fl (ai + Ij+2>
= ai+ai+1fllj+2
= ai+ai+11j+1
: (11+ azi+11j+1.
Thus ai+1(x—y) E (12- for some y E I9“. Since x—y+Ij'Jr1 : 33+Ij+12

u, we may replace x by x — y to assume that ai+1x E at}. But then

x E [a, : (a,+1)] fl Ij = Clin—1 by (b), which implies u E (a’1,... a’-). [:1

I 2

Now we are ready for the last technical result.

Lemma 3.1.8 Let s and t be integers. Assume [(a’1,.. .,a’) : a§+1]j =

i

[(a’1,...,a[)]j whenever 0 S i S s—1andei—t, where a’1,...,af9

are defined as in Lemma 3.1.7. Then
depth [Q/(a'1,...,a;)]j Z

min({depth R/I”+n—j—l [j-—-i+1 : n s j+1}U {depth R/Ij‘i-Hlll,

wheneverOSiSs andei—t.
Proof. We show by induction on i that

depth [Q/(a’1,...,a:-)]j 2 min{depth In/I"+1 + n —j | j —i S n S j},

28

whenever 0 S i S s and j Z i— t. The assertion being trivial for
i = 0, we may assume that 0 S i S s — 1. We need to show that the
inequality holds for i + 1.

By assumption we have an exact sequence
0 —> [g/(allv - - . Milli —+ [9/(0'1. . - - .02)le —>

[Q/(a'l, - - - ,a§+1)lj+1 -+ 0 (2)

whenever 0SiSs—1 andei—t.
Applying a depth chase and the induction hypothesis to (2) the

conclusion follows. [:1

Remark 3.1.9 ([23, Lemma 2.7, Lemma 3.1, Corollary 3.2, Corol-
lary 3.3 and their proof]) Lemma 3.1.6 and Lemma 3.1.7 are still
satisfied if the assumption “I satisfies Gs” is replaced by “r,- S
max{0,i—t—1} for all g S i < s” (with t = s—k—l in Lemma 3.1.7),
where ri’s are as in Definition 3.1.4. In this context a1, . . . ,as are de-
fined as in Lemma 3.1.3. In fact, in the proofs of Lemma 3.1.6 and
Lemma 3.1.7, the condition “I satisfies GS” is used only to apply
Lemma 3.1.1. With the new assumption we use Lemma 3.1.5 instead.
Also, Lemma 3.1.8 [still holds if a’1,. . . ,a’s are the images of a1, . . . ,as,

with al, . . . ,as as in Lemma 3.1.3.

29

3.2 An Estimate of the Depth of the Associated

Graded Ring

Many authors have extensively studied the Cohen-Macaulay property
of R and Q. One of the most general results for ideals with good
residual intersection properties and sufficiently small reduction number

is the following theorem of Johnson and Ulrich.

Theorem 3.2.1 ([39, Theorem 3.1]) Let R be a local Cohen-Macaulay
ring of dimension d with infinite residue field, let I be an R-ideal of
grade g, analytic spread t, and reduction number r, let k 2 1 be an
integer with r S k, assume that I satisfies Ge and AN;3 locally in

codimension 6—1 , that I satisfies AN [— and that depth(R/ I j ) Z

max{2,k}
d—€+k—j for 1 Sj S k. Then 9 is Cohen-Macaulay, and ifg 2 2,

R is Cohen-Macaulay.

Goto, Nakamura and Nishida proved a similar result, weakening the
Artin-Nagata assumptions. The condition Cg is replaced by a local

condition on the reduction numbers.

Theorem 3.2.2 ([23, Theorem 1.1]) Let R be a local Cohen—Macaulay
ring of dimension d with infinite residue field, let I be an R-ideal of

grade g, analytic spread t, and reduction number r, let k _>_ 0 be an

30

integer with r S k. Assume that r,- S max{0,i—0th} for all g S i < t,
where r,- ’s are defined as in Definition 3.1.4. Assume that I satisfies
AN,,“_,C_1 and that depth(R/Ij) 2 d— 5+ k -—j for 1 S j S h. Then
Q is Cohen-Macaulay. Morover, R is Cohen-Macaulay if g > 0 and

r<€.

Remark 3.2.3 There are many ideals satisfying the assumptions on
the depth of the powers in Theorem 3.2.1 and in Theorem 3.2.2. For
example, if I is strongly Cohen-Macaulay and satisfies G’s, it can be
seen from the Approximation Complex ([24, the proof of 5.1]) that

depth(R/Ij) Zd—g—j-i—l whenever 1Ssz—g+1.

Theorem 3.2.1 and Theorem 3.2.2 generalize several previous results
about the Cohen-Macaulyness of blow-up rings.

The relationship between the Cohen-Macaulay property of R and
Q is well understood, for example by the following results of Huneke,

Lipman, and Simis, Ulrich and Vasconcelos.

Theorem 3.2.4 ([30, Proposition 1.1]) Let R be a Cohen-Macaulay
ring and let I be an R-ideal with positive height. If R is Cohen-

Macaulay, then Q is Cohen—Macaulay.

31

Remark 3.2.5 Lipman showed in [42] that if R is a regular ring, the
converse of the above theorem holds. In this case, if I has positive

height, R is Cohen-Macaulay if and only if Q is Cohen-Macaulay.

Theorem 3.2.6 ([49, Corollary 3.6]) Let R be a local Cohen-Macaulay
ring with infinite residue field, let I be an R-ideal of grade 9 > 0,
analytic spread t, and reduction number r. Assume that I satisfies

Ge. Then the following are equivalent:

(a) R is Cohen-Macaulay.

(b) Q is Cohen-Macaulay and r < 6.

Another interesting issue is to estimate the depth of blow-up rings
when they fail to be Cohen-Macaulay, weakening the assumptions of
Theorem 3.2.1 and Theorem 3.2.2. We can focus our attention on the
study of depth Q, because an estimate of depth R follows by this result
of Huckaba and Marley.

Theorem 3.2.7 ([29, Theorem 3.10]) Let R be a local ring and let I
be an R-ideal. Suppose depth Q < depth R. Then depth R = depth

Q+1.

Cortadellas and Zarzuela have come up with formulas for the depth
of Q in [12], in the case of ideals with reduction number at most two

and analytic deviation at most one. More precisely:

32

Theorem 3.2.8 ([12, Theorem 4.1]) Let R be a local Cohen-Macaulay
ring and let I be an equimultiple ideal with reduction number S 1.
Then

depth Q = depth R/I + 9.

Theorem 3.2.9 ([12, Theorem 5.8]) Let R be a local Cohen—Macaulay
ring and let I be an ideal with analytic deviation one and reduction

number S 2. Assume that I is unmixed and that rg(I) S 1, where

717(1): max{TUp) I p E V(I) with ht p = g}. Then
min{depth R/I,depth R/I2} + g S depth Q S

min{depth R/I, depth 12/12} + g + 1.

These results motivate the search for a lower bound and an upper
bound for depth Q in terms of the depth of the powers of I up to the
reduction number, in more general cases. The upper bound is easy to
find because if R is a Noetherian ring and I is an R-ideal with analytic
spread t, we have that depth Q S inf{depth R/Ij | j 2 1} + i (see
Remark 3.2.20).

Theorem 3.2.10, which is the main result of this chapter, gives an
effective lower bound for depth Q. In particular the theorem unifies
and generalizes the above mentioned results by Johnson-Ulrich, Goto-
Nakamura—Nishida and Cortadellas-Zarzuela.

33

The goal of this section is to prove the following assertion.

Theorem 3.2.10 Let R be a local Cohen—Macaulay ring of dimension
d with infinite residue field, let I be an R-ideal with grade g, let J be
a reduction of I generated by 8 elements with ht J : I _>_ s, r = r J(I ),
and let k 2 0 be an integer with r S k. Assume that I satisfies 0,,
ANS—4:4, and that for every p E V(I), depth(R/Ij)p 2 min{dim Rp -—

s+k—j,k—j} wheneverlSjSk—l. Then
depth g 2 min({d}U{depth R/IJ’ +3 — k+j | 1 Sj g k}).

Remark 3.2.11 Notice that the assumption “for every p E V(I),
depth (R/Ij)p Z min{dime—s+k—j,k—j} whenever 1 S j S k—l”
in Theorem 3.2.10 implies that k S s — g + 1. This is trivial if k S 1.
If k 2 2, let p be a minimal prime of I with ht p = g. One has 0 =
depth(R/I)p Z min{g — s + k — 1,k — 1} and so k S s — 9+ 1. Hence

the reduction number is forced to be “sufficiently small”.

We will make strong use of the technical results proven in the previ-

ous section. The proof of the theorem consists of three main steps:
STEP 1: We show that we can reduce the problem to the case where
g = 0.
STEP 2: We prove the theorem in the case where g :— 0 and s S

k. This is the crucial part of the proof. We will need some more

34

preliminary results.

STEP 3: We prove the general case. It follows rather easily from the

previous step.
Proof of the Theorem.

Let J = (a1, . . . ,as), where a1, . . .,as are defined as in Lemma 3.1.1

(a), and let a’1,...,a; be their images in [Q]1.

STEP 1. Reduction to the case g = 0.

The following lemma shows that we may assume g z 0 in order to

prove the theorem.

Lemma 3.2.12 Using the notation of Theorem 3.2.10, suppose that
g > 0 and let R“ = R/ag, I* = I/ag, J* = J/ag, Q* = Q(I*). Then

we have the following:

(a) R* is a local Cohen-Macaulay ring of dimension d — g and I * is

an R* -ideal of grade 0.

(b) J* is a reduction of 1* generated by s —-— g elements, ht J* : I* 2

s — g, and TJ*(I*) S rJ(I), so that k may be taken unchanged.
(c) 1* satisfies Gs_g.

(d) I* satisfies AN3'_g_k_1.

35

(e) For every p* E V(I*), depth(R*/I*j)p. 2 min{dim R1,. - (s — g) +

k—j,k—j}whenever1SjSk—1.

(f) depth Q: depth Q* + g, and if depth Q* 2 min({d — 9} U
{depth R*/I*J’ + s — g — k +j | 1 g j < 1.)), then depth

mein({d}U{depth R/Ij+s—k+J]1§JS kl)-

Proof. Parts (a) and (b) are clear, and ((1) holds by [36, Lemma 1.1.6].
In order to prove (c), let p* E V(I*) with ht p* S s — g — 1. Then
p* = p/ag, where p E V(I) and ht p S s — 1. By Lemma 3.1.1 (a),
[,0 = (a1,...,a,-)p for all p E V(I) with ht p S i S s — 1; so that
u(I*)p. S ht p*.
In order to show (e), notice that since R*/I*j E” R/ag + 1’, by

Lemma 3.1.7 (c) for every 3' _>_ 1 we have an exact sequence
0 —+ R/ang—l —> R/ag a R/IJ’ —> 12* /I*j —> 0. (3)

By (3), Lemma 3.1.7 (b), and Lemma 3.1.6 with t = s — k — 1, we
have that depth(R*/I*j)p. 2 min{depth(R/Ij’")p — n | o g n g 9}
whenever j > 1. Since the inequality holds also when j = 1, we get
the desired condition on the depth of the powers.

Finally, we have that Q* = Q / (a’l, . . . , a’ ) and that depth Q: depth

9

9* + g by Lemma 3.1.7 (d). If

depth Q* 2 min({d—g}U{depth R*/I*j+s—g—k+j | 1 S j S k}),

36

again using the fact that
depth(R*/I*j) Z min{depth R/Ij—n — n | 0 S n S g}
for j 2 1, we conclude that
depth 9 2 min({d} u {depth R/IJ’ + s — k +j | 1 _<_ j g k})

and this finishes the proof of the lemma and the reduction to the case

920. E]

STEP 2. The proof in the case s S k.

In the setup of Theorem 3.2.10, we now assume g = O and s S k.
We are going to prove the theorem in this special case. Since k S s + 1
by Remark 3.2.11, either k = s or k = 3 +1. If s = 0, then I is
nilpotent and k S 1; so we have either Q = R or Q : R/I 69 I and
depth Q = depth R/ I . In any case the theorem holds; hence we can
assume 3 > 0.

The main idea of the proof is to use suitable truncations of the graded
Q-modules Q / (a’1,...,a’-) in order to have “convenient” short exact

2

sequences. More precisely, for 0 S i S 3 consider the graded Q —modules
M“) : [GAG/1" - - Milly—“Hi = Qi‘s+k+1/(a’1,,,,,a;) i—s+k

and
N . _ gi-8+k/( I I )gi—s+k—1 + Igi—s+k
(z)— + ala'Haai—l + at + -

37

Notice that Mb’) can be obtained as a truncation of N(,-), namely

Mn) = [Molar—3+1.“-

In addition M(,-__1) coincides with N(.,-) in degree i — s + k; i.e.,
[Mali—3+1. = [G/(a'i, - - - wai—1)li—S+k°
Hence for 0 S i S s we have exact sequences
0 —> MU) —> N(,-) —> [G/(a’1,...,a;__1)],-_s..,c —> O. (4)
On the other hand, if 0 S i S s — 1, then
N(i+1) = M(i)/ai+1M(i)

and by Lemma 3.1.7 ((1) we have that 0 :Mc) (a;+1) = 0. Thus, in the

range 0 S i S s — 1 we have exact sequences
0 -—> M(i)(—1)—-> Mm --> N041) —> O, (5)

where the first map is given by multiplication by a;- +1- Furthermore

M(s) = 0, since 1"“+1 = JI’“.

The exact sequences (4) and (5) are an essential tool for the proof
of the theorem. We are going to apply a depth chase and the 10-
cal cohomology functor to these sequences, to get an estimate for
depth MU): starting from i = s and using decreasing induction on
i (Lemma 3.2.15). Eventually we will obtain an inequality for depth

38

Mm). Since M(0) : [Q]21 If k =3 S, 01' M(0) Z [Q]22 If k 2 8+1, WC Will

get the required estimate for depth Q.

First we need a lemma, that we are going to prove in a more general

context.

Let S be a homogeneous Noetherian ring with So local and homo-
geneous maximal ideal 93?, let H '(—) denote local cohomology with
support in 931.

For a graded S -module N and an integer j we put aj(N) : max{n I

[Hj(N)]n 74 0} and we call it the j-th a-invariant of N.

Lemma 3.2.13 Let 0 ——> A —) B —> C —> 0 be an exact sequence of

graded S -modules, let n and j be integers.
(a) If aJ-(A) S n and aj(C) S n, then a,-(B) S n.
(b) (i) If Hj(A) :2 0, then aj(C) Z a,-(B).

(ii) If Hj(B) 2 0, then aj+1(A) Z aj(C).

(iii) If Hj(C) 2: 0, then aj+1(B) Z aj+1(A).

Proof. The assertions follow from the long exact sequence of local

cohomology
—> Hj(A) —> HJ'(B) —> 111(0) —> Hj+1(A) —> Hj+1(B) —+ ..

and the definition of a,-(—). C]

39

Since we will use the following fact several times, we recall it here

for convenience.

Lemma 3.2.14 ([20, Lemma 2.2]) Let S = 63,205,, be a homogeneous
Noetherian ring with (50, mg) a local ring. Let N be a finitely generated
graded S -module with Nn = 0 for all n >> 0. Then for any integers
i,n we have an isomorphism [H34(N)]n E“ H§,0(Nn) of 50—modules,

where M is the maximal homogeneous ideal of S .
Let

A :min({d}U{depth R/IJ' +s—k+j | 1 Sj g k}).

Recall that we want to show that depth Q 2 A. The next lemma gives

an estimate of depth Mm in terms of A.

Lemma 3.2.15 In addition to the assumptions of Theorem 3.2.10, as-

sume that g = 0 and s S k. Let [VIM be defined as above. Then:
(a) a,-(1VI(,-)) S i — s + k for any integerj and 0 S i S s.

(b) depth Mb) 2 A — i — 1, and if depth Mb) 2 /\ — i — 1 then

a,\_,-_1(M(,-)) = ’l — 8 + k.

Proof. First we prove (a) by decreasing induction on i. If i = s,
the assertion is obvious since M(s) = 0. Suppose 0 S i S s — 1 and

40

assume aj(M(,-+1)) S i+ 1 — s + k for any integer j. By Lemma 3.2.14
we have that, for any integer j, [Hj([Q/(a’1,... ,a§)],-+1_s+k)]n = 0, if

n 7t i+ 1 — s + k. Hence, by Lemma 3.2.13 (a) applied to the sequence
0 —> M(i+1) —+ N(i+1) —> [g/(all, . - - aa;)]i+1—s+k —t 0 (6)

we have that a,-(N(,+1)) S i + 1 — s + k for any j. Applying the local

cohomology functor to the sequence
0 —> M(,-)(—1)—+ MU) —> N(,-+1)—> O (7)
we get the exact sequence
—+ Hj‘1(N(,-+1))—> HJ'(M(,,(—1))—+ Hi(M(,-,) —> Hj(N(,-+1)) —>
Now, ifn>i—s+k,we have
0 Z [Hj—1(N(i+1))]n+l —> [Hj(M(i))ln —>
[Hj(M(,-))],,+1 —* [Hj(N(,-+1))]n+1 : 0~

It follows that for any j, [Hj(M(,-))]n = 0 whenever n > i— s + 1:.
Hence aj(M(,-)) S i — s + k and the proof of (a) is complete.

Now we prove (b), again using decreasing induction on i. If
i = s the assertion is trivial since ”((3) = 0. Assume now that
0 S i S s — 1, that depth My“) 2 A —— i — 2, and that if depth
M(i+1) = A — i — 2, then a,\_,-2(M(,-+1)) = i + 1 — s + It. By
Lemma 3.1.7 ((1) and Lemma 3.1.8 with t : s — k — 1, we have that

41

depth [Q/(a’1,... ,a[-)],-+1_S+;c Z min({depth R/Ij +j — i + s — k — 2 |
k—s+2 S j S i—s+k+2}U {depth R/I“‘3+1—i+1}). If

0SiSs—2,theankandso
depth lg/(a’l’ - - - vai>li+1—s+k Z
min{depthR/Ij+j—i+s—k—2|lSjSk}
ZA—i—Z.

If i = s — 1, then j S k + 1. By Lemma 3.1.7 (b) and Lemma 3.1.6

with t = s — k— 1, we have that

depth R/Ik“ = depth R/Jrk
Z min({d — s}
U {depth R/Ik-n —n | 0 S n S s — 1})

> A—s.

So we have that depth [Q/(a’1,.. .,a’s_1)];c _>_ A — s — 1. Hence in any

case it follows that
depth [Q/(a'1,. - - 7a;)]i+1—s+k _>_ A ‘1'" 2,

and so by (6) we have that depth N(i+1) Z A — i —- 2. Since N(,~+1) :
M(,-)/a;+1M(,-) and a; +1 is [Wm-regular, we conclude that depth Aim 2

A —— i -— 1 and this proves the first assertion of (b).

42

If depth Mb) 2 A — i — 1, then depth N(,-+1) = A — i — 2 and so
H A‘i‘2(N(,-+1)) 76 0. Applying the local cohomology functor to the

sequence (6) we get the exact sequence
—> HA—i—2(M(i+1)) —* HA_i_2(N(i+1)) —*

HA_i—2[g/(a’1, . . . ,a;)]i+1_3+k —) . . .

If depth M(,-+1) > A — i — 2, then HA‘i“2(.M(,-+1)) = 0; hence
Hl‘i‘2(N(.-+1)) ’——‘—’ [HA—i_2(N(i+1))li+1—s+k and so aA—i—2(N(i+1)) =
i + 1 — s + k. If depth M011) 2 A — i — 2, then by assumption
a,\_,-_2(M(,-+1)) : i + 1 — s + k. Since depth [Q/(a’1,...,a[)],~+1_3+;c 2
A — i — 2, applying Lemma 3.2.13 (b)(iii) with j = A — i — 3
to (6), we get that a,\_,-_2(N(,-+1)) Z i+ 1 — s + k. In any case
a,\_,-_2(N(,+1)) Z i + 1 — s + Is. Finally, since depth My) 2 A — i — 1,
applying Lemma 3.2.13 (b) (ii) with j = A —i— 2 to the sequence (7),

we conclude that a,\_,-_1(M(,-)) Z i — s + k. 1:]

In the next corollary and lemma we obtain some useful results about
the a-invariants of Q, that we will need to conclude the proof of the

theorem.

Corollary 3.2.16 With the assumptions of Lemma 3.2.15, we have

that aj(Q) S k — s for any integer j.

43

Proof. We have an exact sequence 0 —> M(0) —> Q —> C —> 0, where

R/I if k = s
C :
R/IEBI/I2 ifk=s+1.
Since for any integer j, aj(M(0)) S k—s by Lemma 3.2.15, and aj(C) S
k —- s by Lemma 3.2.14, using Lemma 3.2.13 (a) we conclude that

aj(Q) S k — s for any integer j. [:1

Lemma 3.2.17 If depth Q = t < d, then at(Q) < max{0,at+1(Q)}. In
particular, with the assumptions of Corollary 3.2.16, one has at(Q) <
k — s.

Proof. Suppose t < d. Then we have depth R = t + 1 by Theo-
rem 3.2.7, and so H t(R) = 0. Hence, applying the local cohomology

functor to

0—>R+(1)—>R—>Q—+0
we get the exact sequence
0 —> Ht(g) —+ Ht+1(R+(1)) —> Ht+1(R) —> Ht+1(g). (8)

Let m = max{0,at+1(Q)}. If n > m, then [H‘+1(Q)]n = 0 and so
[H t+1(R+)]n+1 maps onto [H t+1(R)]n. Applying the local cohomology

functor to

O——>R+——>R—>R—>O

44

we get [Ht+1(R+)]n E” [Ht+1(R)]n, since, for every integer j, Hj(R) =
[Hj(R)]0 ’—-‘:’ HMR) by Lemma 3.2.14. Hence [Ht+1(R+)]n = 0 for any

n > m. From (8) we conclude that at(Q) < m. C]

Now we are ready to finish the proof of the theorem.

Conclusion of the case s S k.

Consider the exact sequence
O—>M(0)—+Q——>C—>O.

Recall that

R/I ifk=s
C:
R/IEBI/I2 ifk=s+1

and that
A = min({d} U {depth R/Ij + s — k+j | 1 S j S k}).

Notice that depthRC = depth RC _>_ A — 1. This is obvious if k = s,
and it easily follows from the exact sequence 0 —> I / I 2 ——) R/I2 —>
R/I —> 0 if k = 3 +1. Notice that, if k = 5 +1, we have that depth
R/I > A — 1. Since depth M(0) _>_ A — 1 by Lemma 3.2.15, it follows
that depth Q 2 A — 1. If depth Q = A — 1, then a,\_1(Q) < k — s by

Lemma 3.2.17. The sequence above yields

HH(M(O,) —> H*“1(Q) —> H"“1(C). (9)

45

If depth M(0) > A — 1, then H"‘1(M(0)) = 0. If k = s, then
a,\_1(Q) < 0 and so [HA—1(C)]a,_l(g) = 0, a contradiction. If k = 3 +1,
then a)‘_1(Q) < 1 and so [HA“1(C)]a,_,(g) = [HA—1(R/I)]a,_,(g) = 0,
since depth R/I > A — 1, and again we get a contradiction. If
depth Mm) = A — 1, then a,\_1(M(0)) = k — s by Lemma 3.2.15.
Applying Lemma 3.2.13 (b)(iii) with j = A — 2 to the sequence
0 ——> M(0) —> Q —> C —> 0, we get that a,\_1(Q) Z k — s, a contra—
diction. Hence depth Q 2 A and this proves our result in the case

g=0andsSk.

STEP 3. Proof of the general case.

Let 6 = (5(I) = s — 9 +1 — k and recall that 6 Z 0 by Remark 3.2.11.
We are going to induct on 6. By step 1, we can assume that g = 0;
hence 6 = s + 1 — k. Since by step 2 the theorem holds if 5 = 0 or
6 =1 (i.e., k = s or k : 8+1), we can assume that 6 Z 2 and so
3 — k — 1 Z 0. Hence I satisfies AN0_. Write K = 0 : I and let ‘-’
denote images in R = R/ K .

We will show that our assumptions are preserved, and that (5 de-
creases when passing from R to R. Since I satisfies Cs and s 2 k+1 Z
1, I satisfies G1. Hence [,0 = 0 for every p E V(I) with dim R1, = 0.
It follows that ht I + K > 0, and so K is a geometric 0-residual inter-

section of I. Hence R is Cohen-Macaulay, since I satisfies AN; . By

46

Lemma 3.1.1 (b) I H K = 0 and grade I = 1. Furthermore dimR:
dimR = d, J = (c171,...,cis) is a reduction of I with ht J: I 2 s,
r j(I ) S r J(I ), and thus k may be taken to remain unchanged. Clearly

I satisfies Gs since R is equidimensional of the same dimension as R.

Also, by [39, Lemma 2.4 (b)], I satisfies ANS—4,4. Since IflK = 0 we

have an exact sequence
O—>K—>Q—+grf(R)—>O (10)

where depth K = d since depth R = d.
Now using the degree 0 piece of the sequence (10), we have that
depth R/I 2 min{d — l,depth R/I}. Also, for j 2 1 we have the

isomorphisms
Ij/Ij+1 E” Ij/(Ij‘L1 + If 0 K) 9:“ Ij/IjH.
Using the exact sequences
0 —> iii/ii+1 —> R/Ij“ -—> R/IJ' —+ 0

and
0 -—> Ij/Ij+1—+ R/Ij+1 —> R/Ij —> 0

it follows, by induction on j , that whenever j Z 1,

depth R/Ij Z min({d —- 1} U {depth R/It | 1 S t S j}).

47

Applying this in the ring R 2 RP, we have that whenever 1 S j S k— 1,

for every p E V(I)
depth (fa/I015 2 min({dime — 1}

U {deptth/I; | 1 S t S j})
2 min{dime — s+k —j,k —j}

= {dimRfi—s+k—j,k—j}.

This shows that the condition on the depth of the powers is preserved
when passing from R to R.

NOW

6(I)=s—gradeI+1—k<s+1—k=6(I),

thus we may use our induction hypothesis to conclude that
depth ng-(R) 2 min({d} u {depth R/P‘ + s — k +j | 1 _<_ j g k}).
Then by (10) and using again the inequality
depth R/I—j Z min({d — 1} U {depth R/It | 1 S t S j})
for j 2 1, we have that
depth Q 2 min({d} U {depth R/Ij + s — k +j I 1 S j S k})

and the proof of the theorem is complete. Cl

48

Remark 3.2.18 If in the setting of Theorem 3.2.10 we have stronger
assumptions on the depth of the powers of I; namely, if depth(R/ I j ) Z
d—t+ k —j whenever I S j S k, then depth Q = d; i.e.’ Q is Cohen-

Macaulay. Hence Theorem 3.2.10 generalizes Theorem 3.2.1.

Remark 3.2.19 The theorem still holds if the condition “I satisfies
G3” is replaced by “r,- S max{0,i — s + k} for all g S i < s”, where
ri’s are as in Definition 3.1.4. In this context we choose the reduction
J = (a1,...,a3) with a1,...,as as in Lemma 3.1.3. Notice that all
the technical results of the previous section hold (see Remark 3.1.9).
Furthermore the condition r,- S max{0,i — s + k} for all g S i < s is
still satisfied when we factor out (a1, . . . ,ag) to assume g = 0 (see [23,
Lemma 3.4]), and when we factor out K = 0 : I (see [23, Lemma 5.1]).
Since in the rest of the proof the 0, property is needed only to be able
to use Lemma 3.1.6, Lemma 3.1.7, and Lemma 3.1.8, it follows that

the result still holds. Hence Theorem 3.2.10 recovers Theorem 3.2.2.

The following remark gives an upper bound for depth Q.

Remark 3.2.20 Let R be a Noetherian local ring, and let I be an R-
ideal with analytic spread 6. Then depth Q S inf{depth R/Ij | j Z

1}+t

49

Proof. Since Q is a Noetherian ring, we have that deptlimgQ =
inf{depth Ij/Ij+1 | j. Z 1} = inf{depth R/Ij | j Z 1}. But

depthmgQ Z depth Q — dim Q / mQ = depth Q — t. [3

Remark 3.2.21 Let R be a Noetherian local ring, and let I be an

R-ideal with analytic spread t. Burch’s inequality ([8]) states that
inf{depth R/Ij lj 2 1} + i g dim R.
Hence, if Q is Cohen-Macaulay, we have that

depth 9 = inf{depth R/IJ’ | j 2 1} + i.

We call inf{depth R/Ij | j Z 1} the Burch number of I and we

denote it by B(I).

Corollary 3.2.22 With the assumptions of Theorem 3.2.10 we have
that

min({d—€}U{depth R/Ij—k+j|lSjSk})SB(I)Sd—t.

Proof. By Theorem 3.2.10 and Remark 3.2.20 we have that min ({d}U
{depth R/Ij+€—k+j | 1 Sj S k}) S depth Q S B(I)+t. The
conclusion follows since B (I) : depthmgQ S dimQ —— dim Q /mQ. D

We will mainly use Theorem 3.2.10 when s = E and J is a minimal
reduction of I such that ht J : I 2 I? and r J(I ) = r. The next corol-

laries are special cases of Theorem 3.2.10, for small reduction number.

50

These special cases are particularly interesting because we can get a

precise formula for depth Q.

Corollary 3.2.23 Let R be a local Cohen-Macaulay ring of dimension
d with infinite residue field, let I be an R-ideal with grade 9, analytic
spread E, and reduction number r S 1. Assume that I satisfies Cg and

AN}:2. Then depth Q :2 min{d, depth R/I + 8}.

Proof. The assertion follows from Theorem 3.2.10 with s = t, J a
minimal reduction of I such that ht J : I _>_ 6 and rJ(I) :2 r, k = 1,

and from Remark 3.2.20. [:1

Remark 3.2.24 If we apply the previous corollary with t = g, the
conditions Cg and AN};2 are automatically satisfied. Hence Corol-

lary 3.2.23 covers Theorem 3.2.8.

Corollary 3.2.25 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 t 2 g + 1, and reduction number r S 2. Further assume that
I satisfies Cg, ANe:3 and that R/I is Cohen-Macaulay. Then depth

Q = min{d, depth R/I2 + 3}.

Proof. The assertion follows from Theorem 3.2.10 with s = t, J a
minimal reduction of I such that ht J : I Z t and rJ(I) = r, k = 2,

and from Remark 3.2.20. [1

51

Remark 3.2.26 If we apply Theorem 3.2.10 with s : 9+1 and k = 2,
we get that min{depth R/I + g,depth R/I2 + g + 1} S depth Q S
min{depth R/I + g +1,depth R/I2 + g + 1}. So, by Remark 3.2.19,

we have that Theorem 3.2.10 covers Theorem 3.2.9.
For reduction number 3 we get the following estimate of depth Q.

Corollary 3.2.27 Let R be a local Cohen-Macaulay ring of dimension
d with infinite residue field, let I be an R-ideal with grade 9, analytic
spread t 2 g + 2, and reduction number r S 3. Further assume that I
satisfies Cg, ANEZ4, that R/I is Cohen-Macaulay, and that R/I2 has
no associated primes of height 2 t. Then min{d, depth R/I2 + t —

1,depth R/I3+€} S depth Q S min{depth R/I2 +6, depth R/I3 +5}.

Proof. The assertion follows from Theorem 3.2.10 with s = t, J a
minimal reduction of I such that ht J : I 2 f and rJ(I) = r, k = 3,

and from Remark 3.2.20. [3

3.3 Examples

In this section we give examples of classes of ideals to which Theo-
rem 3.2.10 can be applied in order to compute the depth of the associ—

ated graded ring.

52

Example 3.3.1 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with analytic spread 6, satisfying Cg
and ANEZ2, and let J be a minimal reduction of I . Since I satisfies
Cg and ANef_2, by [51, Proposition 1.11] we have that ht J : I Z 6,
and therefore by [51, Remark 1.12] J satisfies Cg and AN[_2. Clearly
r(J) = 0. Let Q(J) be the associated graded ring of J. Then by

Corollary 3.2.23, we have that

depth Q(J) = min{d, depth R/J + 6}.

Now we present a class of ideals whose associated graded ring is not

Cohen-Macaulay and we can use our results to compute its depth.

Example 3.3.2 Let R be a local Gorenstein ring with infinite residue
field, let I be an R-ideal with grade 9, analytic spread 6 S g + 2
and reduction number r 7é 0. Assume that I satisfies Gg+1 and that
R/ I is Cohen-Macaulay. Let J be a minimal reduction of I. By [51,
Proposition 1.11] we have that ht J : I 2 6+ 1. As J : I # R it follows
that some associated prime of R/ J has height at least 6+1. Therefore
depth R/J S d — 6 — 1. Let Q(J) be the associated graded ring of J.

By Example 3.3.1, we have that
depth Q(J) = min{d, depth R/J + 6} S d — 1.

In particular Q (J) is not Cohen-Macaulay.

53

As a special case of the previous example, we have the following:

Example 3.3.3 Let R = k[[x1,...,x8]], where k is an infinite field.

Let

331 1132 $133 5134

$5 1136 1137 £138

and let I be the ideal generated by the 2 by 2 minors of d,
I = ($1556 — $2175, 1131337 - 11331135, 331138 - 1134335, 332337 — 1133336,

xzxg — x4x5, x3238 — x4x7).

The ideal I has grade 3 and analytic spread 5. Since I is a complete
intersection on the punctured spectrum of R, I satisfies C8. Further-

more, I satisfies AN3— , since R/ I is Cohen-Macaulay. The ideal
J = (331336 - 3321135, 331337 — 3333357331338 — 3341135 + 5132337 — 333556,

3321108 — 3343767 $35138 - 2341137)

is a minimal reduction of I. Since I and J coincide on the punc-
tured spectrum of R, we have that m = (x1,x2,x3, x4,x5,x5, x7,x8) E
Ass(R/J) and so depth R/J = 0. By Example 3.3.2, we have that

depth Q(J) = 5.

Now we present a class of equimultiple ideals of reduction number

one whose associated graded ring is not Cohen-Macaulay.

54

Example 3.3.4 Let R be a local Gorenstein ring with infinite residue
field, let p be a prime ideal of height 9 Z 2 such that R10 is regular, and
let t 2 1 be an integer. Let al, . . . ,ag be a regular sequence contained
in p“), where p“) denotes the t-th symbolic power of p; i.e., p“) :
ptRpflR. Write J = ((11,...,ozg) and set I = J : p“) = J : pt. If either
9 = 2 or t = I, assume that at least 2 of the ag’s are contained in pa“).
By [46, Corollary 4.2] we have that I 2 = J I . Hence I is equimultiple
of reduction number one. Assume that R/pm is not Cohen-Macaulay.
Since I is linked to p“), it follows that R/I is not Cohen-Macaulay.
Let Q be the associated graded ring of I. By Corollary 3.2.23, we have

that
depth Q = min{d, depth R/I + g} = depth R/I + g < d.

In particular Q is not Cohen-Macaulay.
As a special case of the previous example, we have the following.

Example 3.3.5 Let R = k[[x1, x2, x3, x4]], where k is an infinite field,

and let p be the defining ideal of k[[t4, t38, ts3, 84]],

2 3 2 2 3 2
p = (x1x4 — x2333, x1x3 —— x2, x1233 — x4x2, x3 — x4x2).

The ideal p is prime of grade 2 and R/ p is not Cohen-Macaulay. Let

J = ((131174 — $2$3l2a ($3 — 3735132?) C P2

55

andlet
I=sz=

2 3 2 2 5 3 3 2 4 7
((x1x4 — x2x3) ,(x3 — x4x2) ,x1x4x2 — x1x4x3 — x4x2x3 + x3,

4 3 2 2 2 4 5 3 2 2 2 3 5
x1x4x2x3 — x4x2x3 — x1x4x3 +x4x2x3, x1x4x2x3 — x4x2x3 — $1$4I133 +x2x6).

The ideal I is equimultiple of reduction number one. By Example 3.3.4,

we have that depth Q (I) = 3.

56

CHAPTER 4

The Serre Properties of Blow-Up

Rings of Ideals

In this chapter we study the Serre properties of the associated graded
ring and of the Rees algebra of ideals having good residual intersection
properties and sufficiently small reduction number.

Recall that a ring R satisfies the Serre condition St, where t Z 0 is
an integer, if for every p E Spec R, depth Rp Z min{t, dim R1,}.

Clearly if R is St, then R is S), for every I: S t, and if R is Cohen-
Macaulay, then R is St for every t. Since in the setup of the previous
chapter the associated graded ring Q and the Rees algebra R are not
necessarily Cohen-Macaulay, we ask which Serre properties are still
satisfied in this case. We use the estimates for depth Q and depth R
from Theorem 3.2.10.

We first see how assumptions similar to those of Theorem 3.2.10

57

imply the Serre properties for Q.

Theorem 4.0.1 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, let J be a reduction
of I generated by 3 elements with ht J : I Z s, r = rJ(I), and let
k 2 0 be an integer with r S k. Furthermore assume that I satis-
fies C3, ANS—4c”1 and that for some integer t Z 1, depth(R/I~l)p Z
min{dime — s+k —j,k —j+t} wheneverp E V(I) and 1 S j S h.

Then Q is St.

Proof. We need to show that for every 1’ E Spec Q, depth Q1» 2
min{t, dim Qp}. Let q denote the contraction of ’P to R. By Theo-
rem 3.2.10 we have that
depth Q, 2 min({ht q}
u {depth Rq/Ig+s—k+j | 1 SjSk})
_>_ min{ht q,t + 3}.
If ht q S t+ s, then Q, is Cohen-Macaulay and so Qp is Cohen-

Macaulay. Hence we may assume that ht q > t + 8. Since Qp is a

localization of Qq, we have that
dim Qp — depth Q): S dim Qq — depth Qq.
As Q, is equidimensional and catenary, it follows that

dim Q, = dim Qq/RQq + dim Q73.

58

Hence

depth Qp 2 depth Qq — dim Qq/RQq.
But since ’P contracts to q, we have that
diqu/PQq S dim Qq/qu = 6(Iq) S 6 S 3.
So we conclude that depth Qp Z t + s — s = t. [:1

In particular, if the reduction number is small, we have simpler as—

sumptions that imply the 31 property for Q.

Corollary 4.0.2 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, analytic spread 6, and
reduction number r S 1. Further assume that I satisfies Cg, ANZZ2,
and that R/I has no associated primes of height 2 6 + 1. Then Q is

31.

Proof. The assertion follows from Theorem 4.0.1 with s = 6, J a
minimal reduction of I such that ht J : I 2 6 and r J(I ) = r, k = 1

andt=1. {:1

Corollary 4.0.3 Let R be a local Cohen—Macaulay ring with infinite
residue field, let I be an R-ideal with grade 9, analytic spread 6 2 9+1,
and reduction number r S 2. Further assume that I satisfies Cg,
ANng, that R/ I is Cohen-Macaulay, and that R/ I 2 has no associated
primes of height 2 6 +1. Then Q is 51.

59

Proof. The assertion follows from Theorem 4.0.1 with s = 6, J a
minimal reduction of I such that ht J : I Z 6 and r J(I ) = r, k = 2

andt=1. C]

The 51 property is particularly interesting because it leads to criteria
for when I n = I (n), where I ("l is the n-th symbolic power of I. We
recall that I (”l is the intersection over all isolated primary components

of the ordinary power I ".

Let R be a Cohen—Macaulay ring and let I be an R-ideal. If
p E V(I) is the contraction of a minimal prime 73 of Q, then
6(Ip) Z dim Qp/Ppr = dim Qp. The last equality holds since Q, is

equidimensional. Hence 6(Ip) = ht p.

Remark 4.0.4 Let R be a Cohen-Macaulay ring and let I be an R-
ideal. If Q is $1 and 6(Ip) < ht p for every non minimal prime p in

V(I), then I” = [(7‘) for all n 21.

Proof. Since the inclusion I n C I (7‘) always holds, it suffices to show
that 1;") C I; for every p E Ass(Ii‘l/Ii) and 1 S i S n. Let p E
Ass(Ii‘l/Ii). Then p C R for some ”P E Ass Q. Since Q is SI, ’P is a
minimal prime of Q and so its contraction p to R satisfies 6 (Ip) = ht p.
Hence by assumption p is a minimal prime of I and so I5") = 1;}, as

desired. C1

60

Remark 4.0.5 Let R be a Noetherian ring and let I be an R-ideal.
Recall that I is normally torsion free if Q is R/ I -torsionfree. If I n =
[(7‘) for every n 2 1, then Ass(R/I”) C Ass(R/I) for every n 2 1,
and so if x E R/ I is a zero divisor on Q, then x is a zero divisor on
R/ I ; i.e., I is normally torsionfree. If I is a prime ideal the converse
also holds and, if in addition I is generically a complete intersection,

the two conditions are equivalent to Q being a domain.

Combining Theorem 4.0.1, Corollary 4.0.2 and Corollary 4.0.3 with
Remark 4.0.4 we get the following criterion for the equality of regular

and symbolic powers of I.

Corollary 4.0.6 If in addition to the assumptions of Theorem 4.0.1
with t = 1, or of Corollary 4.0.2, or of Corollary 4.0.3, we have that
6(Ip) < ht p for every non-minimal prime p in V(I), then I" = I(”)

for all n 2 1 and I is normally torsion free.

Now we study the Serre properties of the Rees algebra. First we need

the following definition.

Definition 4.0.7 Let R be a local Cohen-Macaulay ring, let I be an
R-ideal of grade g, and let 3 2 g be an integer.We say that I is
s—residually Sg if for every 9 S i S s and every i—residual intersection
K of I , R/ K satisfies Serre’s condition Sg.

61

To obtain the St property for R, in addition to the assumptions that
imply the St property for Q, we need a local condition on the residual

intersections. Namely:

Theorem 4.0.8 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R—ideal with grade 9 _>_ 2, analytic spread
6, and reduction number r, let k _>_ 0 be an integer with r S Is. As-
sume that I satisfies Cg, AN[_[_1 and that for some integer t 2 1,
depth(R/Ij)p _>_ min{dim Rp — 6 + k — j, k —j + t} whenever p E V(I)
and 1 S j S k. Furthermore assume that I is 6 — 2-residually Sg

locally up to height 6 + t — 1. Then R is St.

Proof. We need to show that for every ’P E Spec R, depth Rp Z
min{t,dime}. Denote by q the contraction of ”P to R. By Theo-
rem 3.2.10 we have that depth Q, 2 min{ht q,t + 6}. If ht q 2 t+ 6,
then depth Rq Z depth Q, 2 t+ 6. Since Rp is a localization of Rq,

we have that
dim R7: — depth Rp S dim Rq — depth Rq.
As Rq is equidimensional and catenary, it follows that
dim Rq = dim Rq/RRq + dim Rp.
But since P contracts to q, we have that

dim reg/19R, g dim nq/qnq = 6(Iq) g 6.

62

It follows that depth Rp Z t.

If ht q S t+ 6 — 1, then Q, is Cohen—Macaulay. We claim that also
Rq is Cohen-Macaulay, which implies the Cohen-Macaulayness of Rp.
By Theorem 3.2.6 to prove our claim we only need to show that r(Iq) <
6(Iq). If 6(Iq) = 6, then by assumption r(Iq) S r S 6 — 9 +1 S 6 —- 1,
and so we are done. If 6(Iq) < 6, then r(Iq) = 0 < 6(Iq) by [9, 2.1(g)].

C]

Now we analyze the relationship between the Serre properties for R
and for Q. Brumatti, Simis and Vasconcelos related the property 52

for R to the property 31 for Q in the following theorem.

Theorem 4.0.9 ([6, Theorem 1.5]) Let R be a Noetherian ring satis-
fying 5'2, and let I be an R-ideal of positive grade. The following two

conditions are equivalent:
(1) R satisfies 5'2.
(2) (i) Q satisfies S1, and
(ii) For every p E Spec(R) with ht p = 1, IP is principal.

The following theorem generalizes their result.

Theorem 4.0.10 Let R be an equidimensional and universally cate-
nary Noetherian ring satisfying St, and let I be an R-ideal of positive
height. The following two conditions are equivalent:

63

(1) R satisfies St.

(2) (i) Q satisfies St-1, and

(ii) If q E V(I) and 6(Iq) = ht q S t — 1, then r(Iq) < 6(Iq).
Furthermore, if I is Cg, then (2)(ii) can be replaced by
(2) (ii’) If q E V(I) and 6 : 6(Iq) = ht q S t — 1, then r(Iq) < 6(Iq).

Proof. (1) :> (2). First we verify that (1) implies (2)(i). Let P be a
prime ideal of Q and denote by 73 its inverse image in R. Localize R
at p = R D P and denote the resulting local ring by R. We want to
show that depth Qp 2 min{t — 1, dim Qp}. Since R is equidimensional
and universally catenary and ht I > 0, we have that ht ’P = ht P + 1.
Furthermore ht ’P S ht p+1. Hence, if dim Qp 2 t—l, then dim Rp _>_ t
and so depth R7; 2 t by (1). Also, depth R7: 2 t — 1 since R satisfies

St. From the exact sequences
0——>(It)R—>R——>R——>O

and

0—>IR—>R—>Q—+O

it follows that depth Qp Z t— 1. If dim Qp < t— 1, then dim Rp < t,
and so Rp is Cohen-Macaulay by (1). It follows that Qp is Cohen-

Macaulay and the proof of (2)(i) is complete.

64

In the setting of (2)(ii) we have that dim R, S t— 1, and so dim Rq S
t. Hence R4 is Cohen-Macaulay by (1). The conclusion follows from
[41, Theorem 2.3].

(2) => (1). We may assume t 2 0. Let ”P be a prime ideal of R and
let p = R O 79. We want to show that depth R71: 2 min{t, dim R72}.
If I C p we have Rp :. Rp[t] which satisfies St since Rp does, and so
Rp satisfies St. Hence we may assume that I C p. If It ¢ P, by the
usual prime avoidance argument there exists a nonzero divisor x E I
such that xt E R. We have that x is a regular element of R. Since

t"1 E R3,}, it follows that
(R/xR)xg = (R[It,t"1]/xR[It,t‘1])xg -—-

(R[It,t‘IJ/t—1R[It,t—1]),t = (g)_,,.

So the assumption (2)(i) implies that depth R7: _>_ min{t, dim Rp}.

If It C ’P, then R, is the irrelevant maximal ideal of Rp. If dim RP >
t, then dime > t— 1 and so depth Qp _>_ t— 1 by (2)(i). If Q, is
Cohen-Macaulay, then depth RP 2 depth Qp > t— 1, if Q, is not
Cohen—Macaulay, then depth RP = depth Q, + 1 by Theorem 3.2.7.
In any case depth Rp 2 t, and so depth Rp 2 t. If dime S t,
then dimRp S t— 1. Hence Rp and Q, are Cohen-Macaulay. In this
case R10 is Cohen-Macaulay by (2)(ii) and [41, Theorem 2.3]. If I is
Cg (2)(ii’ ),[41, Theorem 2.3] and [49, Theorem 2.4] imply that R10 is

65

Cohen-Macaulay and the proof is complete. El

Notice that condition (ii) is empty if t S ht I, and, if I is Cg,
condition (ii’) is empty if t S 6. In these cases we have a simpler

version of the theorem.

Corollary 4.0.11 Let R be an equidimensional and universally cate-
nary Noetherian ring satisfying 5', for some integer t Z 1, and let I be
an R-ideal with ht I Z t. Then R satisfies St if and only if Q satisfies
St_1.

Corollary 4.0.12 Let R be an equidimensional and universally cate-
nary Noetherian ring satisfying St , and let I be an R-ideal with positive
height and analytic spread 6 2 t. Assume that I satisfies Cg. Then R

satisfies St if and only if Q satisfies St_1.

Combining Theorem 4.0.1 with Theorem 4.0.10 we obtain another

result on the Serre properties of R (compare with Theorem 4.0.8).

Theorem 4.0.13 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade 9 > 0, analytic spread 6,
reduction number r, and let k 2 0 be an integer with r S It. Further-

more assume that I satisfies Cg, AN[_,C_1 and that for some integer

66

t _>_ 1, depth (R/Ij)p Z min{dimR? — 6 + k — j,k — j + t — 1} when-
ever p E V(I) and 1 S j S Is. Finally suppose that if q E V(I) and

6 = 6(Iq) = ht q S t — 1, then r(Iq) < 6(Iq). Then R satisfies St.

The 82 property is very important in the study of the normality of
R. By Serre’s normality criterion, a Noetherian ring R is normal if
and only if R satisfies R1 and 32. We recall that a Noetherian ring R

satisfies R1 if R1, is regular for every p E Spec(R) with ht p S 1.

Remark 4.0.14 ([6, Remark 2.3], Theorem 4.0.8). Let R be a poly-
nomial ring in n variables (localized at the maximal irrelevant ideal)
over an infinite perfect field, let I be an R-ideal with grade g 2 2,
analytic spread 6 and reduction number r, and let k 2 0 be an integer
with r S k. Let J be the ideal of the presentation of R; i.e., R E’
R[T1, . . . ,Tm]/J. Let h1,. . . ,hs be a set of generators of J and con-
sider the Jacobian matrix M = 8(h1,.. . ,h3)/6(x1, . . . ,xn,T1,. . .,Tm).
Let N be the ideal generated by all (m — 1) x (m — 1) minors of .M .
If ht(J, N) _>_ m + 1, then R satisfies R1. If further we assume that I
satisfies Cg and ANg:k_1, that I is 6 — 2 residually $2 locally up to
height 6+1, and that depth(R/I~l)p _>_ min{dim Rp—6+k—j, k—j+2}
for every p E V(I) and whenever 1 S j S h, then R satisfies $2 and

hence is normal.

67

 

Recall that B(I), the Burch number of I, is the value inf
{depth R/Ij I j 2 1}. We know that if Q is Cohen-Macaulay, then
B(I) = depth Q, (see Remark 3.2.21). Now we want to see how B(I)

is related to the St property for Q.

Lemma 4.0.15 Let R be a local Cohen-Macaulay ring of dimension d
and let I be an R-ideal with analytic spread 6. If Q satisfies St for

some positive integer t S d — 6, then B(I) Z t.

Proof. Since Q is equidimensional and catenary, we have that ht mQ =

d — 6; hence B(I) = depthmgQ = min{depth Qp I P E V(mQ)} Z t. C]

From Theorem 4.0.1 and Lemma 4.0.15 we get the following result.

Corollary 4.0.16 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, analytic spread 6, reduc-
tion number r, and let k 2 0 be an integer with r S k. Furthermore
assume that I satisfies Cg, AN[_,C_1 and that for some integer t with
1S t S d — 6, depth (R/Ij)p _>_ min{dime — s + k —j,k —j+ t}

wheneverp E V(I) and 1 Sj S h. Then B(I) Z t.

Now we want to use the results obtained in this chapter to present
another theorem on the Cohen-Macaulayness of Q (compare with The-

orem 3.2.1).

68

First we recall that a Noetherian ring R is quasi Corenstein if
R 2:“ am, where to}; denotes the canonical module of R. If in addition
R is Cohen-Macaulay, then R is Corenstein.

We want to use the following criterion for the Cohen-Macaulayness

of a ring R.

Lemma 4.0.17 ([32, Lemma 5.8]) Let R be a quasi Corenstein local
ring, assume that for every p E Spec(R) with ht p _>_ 2, depth Rp Z

1 + 1/2 dim Rp. Then R is Cohen-Macaulay.

Theorem 4.0.18 Let R be a local Corenstein ring of dimension d with
infinite residue field, let I be an R-ideal with grade g, analytic spread
6, reduction number r, and let I: _>_ 0 be an integer with r S k. Further
assume that I is unmixed, generically a complete intersection, that I
satisfies Cg and AN,‘_,€_1, that depth (R/Ij)p Z min{dime — 6 + k —
j, 1/2(dimR1, -— 6+ 1) + k —j} whenever p E V(I) and 1 S j S k,
and that 6(Ip) < ht p for every non-minimal prime p in V(I). Then

Q is Cohen-Macaulay.

Proof. First notice that our assumption on the depth of the powers
implies that for every p E V(I), depth(R/Ij)p _>_ min{dim Rp — 6+ k —

j, k — j + 1} whenever 1 S j S k. Hence I is normally torsion free by

69

Corollary 4.0.6, and so R[I t,t'1] is quasi Corenstein by [40, the proof
of Theorem 3.2].

Next we show that for every 7’ E Spec R[I t, t‘l] with ht ’P 2 2,
depth R[It,t—1]p 2 1 + 1/2 dim R[It,t‘1]p.

If t"1 E ’P, then R[It,t_1]’p = R[t,t’1]p is Cohen-Macaulay and so the
inequality is trivially satisfied. Hence we may assume that t"1 E ’P and
so R[It,t—1]p/(t_1)’p 2 Q7: at 0. Let q denote the contraction of P to
R, and notice that I C q.

By Theorem 3.2.10 we have that depth Qq _>_ min({ht q}U{1/2(ht q+
6+ 1) | 1 S j S k}). If ht q S 6+ 1, then R[It,t"1]q is Cohen-
Macaulay, and so R[I t, t_1]’p is Cohen-Macaulay. If ht q > 6 + 1, then

the inequality
dim R[It, t‘1]p — depth R[It, t‘1]p g dim R[It, 25—1], — depth R[It,t‘1]q
shows that

depth R[Itt‘lip _>_ dime+1—htq+depth g,
2 dime+3/2—1/2 ht q+1/26
: 1/2(dimgr>+1)+1

+ 1/2(dimgp — ht q + 12)

IV

1/2 dim R[It, t'1]p +1,

70

where the last inequality holds since

dim Q7: — dim Q, 2 — dim Qq/PQq Z —6.
It follows from Lemma 4.0.17 that R[I t, t’l] is Cohen-Macaulay; hence
Q is Cohen-Macaulay. [:1

Notice that the depth assumptions of Theorem 4.0.18 are weaker
than those of Theorem 3.2.1, since in the above theorem we assume
depth(R/Ij)p 2 1/2(dime — 6 +1) + k —j for 1 S j S k, if ht

pZ6+1.

Monomial Varieties of Codimension 2

Let k[u1,...,un] be a polynomial ring over an infinite field It.
Consider the semigroup ring kIu‘l“, u‘2‘2, . . . ,ugn, u? . . .uf,", ulfl . . . nan] C

k[u1,...,un], where aj,bj,cj E No, a,- > 0,(b,-,c,~) 2 (0,0) for 1 S j S
n, and, further (b1,...,bn) 7E (0,...,0) and (61,...,Cn) 75 (0,...,0).
Let I C R 2 kal, . . . ,xn, y, 2] denote the defining ideal of this semi-
group ring. Following Giménez, Morales and Simis ( [19]) we say that
I defines a monomial variety of codimension 2. The ideals defining
monomial varieties of codimension 2 are a subset of the toric ideals;
i.e., presentation ideals of semigroup algebras.

We want to apply our results to these ideals. We can prove that Q
is Cohen-Macaulay. Furthermore I is normally torsionfree if and only
if I is a complete intersection locally in codimension 3.

71

The ideal I is prime of height two, and the variety in question is
affine of codimension two. Furthermore, the analytic spread of I is
equal to two if I is a complete intersection and equal to three in all
the remaining cases ([18, Theorem 4.2]). Also, I has reduction number
one ([5, Corollary 3.4]) and depth R/ I 2 n — 1 ([45, Theorem 23]). If
I is a complete intersection, then Q is trivially Cohen-Macaulay and
normally torsionfree, since it is a polynomial ring over R/ I . If I is
not a complete intersection, notice that I satisfies C3, since R is a
regular ring. It follows from Corollary 3.2.23 that depth Q 2 depth
R/ I + 3 _>_ n + 2 2 dimR. Hence Q is Cohen-Macaulay.

If I is normally torsionfree, then by Remark 4.0.5 and by [30, The-
orem 2.2] we have that 6(Ip) < ht p for every non minimal prime p in
V(I). It follows that if p E V(I) and ht p 2 3, then 6(Ip) 2 2. Hence
I1, is equimultiple and generically a complete intersection, and so by
[13] it is a complete intersection.

If I is a complete intersection locally in codimension 3, then 6 (Ip) <
ht p for every non-minimal prime p of I. Since Q satisfies 31, it follows
from Remark 4.0.4 and Remark 4.0.5 that I is normally torsionfree,
that I n 2 I ("I for every n 2 1, and that Q is a domain. Also, by [26,
Proposition], Q is Corenstein.

For example, if I C ka, y, z, w] is the homogeneous ideal of a mono-

72

mial curve in P3 lying on the quadric surface xy — wz, then I is a
complete intersection on the punctured spectrum of R and so I is nor-
mally torsionfree, which recovers [43, Proposition 2.3]. In general a
monomial curve in P3 need not be normally torsionfree. For exam-
ple the ideal I C ka, y, z, w] defining k[t5, t4u, t3u2, u5] is not normally

torsionfree, indeed I 2 2 I (2).

Now let R :2 k[x, y, z,w] and let I C R be the homogeneous ideal

of a projective monomial curve defined by

SE = U101, y = U201, Z = Uiblu2al_b1,w = Ingmar“,

(a1 > b1 > c1). We want to find necessary and sufficient conditions for
I to be normally torsionfree, in terms of the exponents a1, b1 and cl.

Since I is normally torsionfree if and only if I is a complete intersec-
tion locally in codimension 3, take p E V(I ) with ht p 2 3 and denote
by 15 its image in R/I. Since ht p 2 1, either ui“ E p or ugl E p. If
u‘f‘ E p, then

b1

Mal—bl, U161

klulll’ ugliul UQGI—Cllfi : klxala $01—01,$al—Cl]fia,
if ug‘ E p, then
a a b a —b c a —c a b c

k[u11,u2‘,u1 121,2 1 ‘,u1 17.1.21 1],, : k[x ‘,x 1,x 1],,a,

where we denote by pa the dehomogenization of the ideal p. The ideal

73

(I a)133 is a complete intersection if and only if k[xa1,xa1‘b1,xal‘cl] and
k[xa1, 2"“, x61] are Corenstein.

We denote by < v1, . . . ,vn > the semigroup S generated by the in-
tegers v1,...,vn, following Bruns and Herzog ([7, page 178]). The

conductor c 2 C(S) of S is defined by
c=max{aEN|a—1ES}.

We say that the semigroup S is symmetric if, for all i with 0 S i S
c— 1, one has i E S if and only if c—i— 1 E S. By [7, Theorem 4.4.8],
S is symmetric if and only if kIt”1,. . . ,t”"] is Corenstein.

Hence we have that I is normally torsion free if and only if

<a1,b1,c1 > and < a1, a1 — b1, a1 — c1> are symmetric.
Notice that if I is the above mentioned ideal defining
k[t5,t4u,t3u2,u5], then <a1,b1,c1 >2< 5, 4, 3> is not symmetric.

Hence I is not normally torsionfree.

74

 

CHAPTER 5

Cohen-Macaulayness of the Fiber

Cone

Let .7, R and Q denote the fiber cone, the Rees algebra and the
associated graded ring of an ideal I in a local Cohen-Macaulay ring
R. In this chapter we study the Cohen-Macaulay property of .7 and
we relate it to the Cohen-Macaulayness of R and Q. In particular we
will give examples for perfect ideals of height two. For most of the
computations we used MACAULAY.

Recall that if u(I) 2 n, then
R’—‘_—’ R[T1,...,T,,]/Q
where Q is an ideal of R[T1,. . . ,Tn], and
7-" T—z R[T1,. . . ,TnI/(m,Q) g k[T1, . . . 2.1/Q,

where “_” denotes images in k 2 R/m, the residue field of R. In
particular, since [Rh 2 [S (I )]1, all the linear polynomials in Q have

75

 

coefficients in 111. Hence all the homogeneous relations of .7 have degree
at least two.

When r(I) 2 0, then .7 is a polynomial ring 'over k, .7 2
k[T 1, . . . ,T g], where 6 is the analytic spread of I. In this case .7 is
trivially Cohen-Macaulay. Some results have been obtained in the lit-
erature for ideals with reduction number at most one. One of the most

general ones is the following theorem of Cortadellas and Zarzuela.

Theorem 5.0.1 ([11, Theorem 32]) 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 6, and reduction number at most one. Assume
that I satisfies ANKZ2, and that r,- S max{0,i—6+ 1} for all g S i < 6,
where r,- ’s are defined as in Definition 3.1.4. Furthermore assume that

depth R/I 2 d — 6. Then .7 is Cohen-Macaulay.

Recently Huneke and Hiibl proved the following theorem for ideals
with analytic deviation one, which does not have any restriction on the

reduction number.

Theorem 5.0.2 ([33, Theorem 2.1]) Let R be a local Cohen-Macaulay
ring of dimension d, let I be an unmixed R-ideal of height d -- 1 and
analytic spread d. Assume that I is generically a complete intersection
and that for all minimal reductions J of I, .7 has no homogeneous

76

 

 

generating relations in degree S r J(I ) Furthermore assume that the
grade of Q. is d — 1, where Q, is the ideal generated by homogeneous

elements in Q of positive degree. Then .7 is Cohen-Macaulay.

The main result of this chapter unifies and generalizes the above

theorems.

Theorem 5.0.3 Let R be a Noetherian local ring with infinite residue
field, let I be an R-ideal with analytic spread 6, minimal number of
generators n, reduction number r, and let k 2 0 be an integer with
r S 13. Let a1, . . . ,ag be general elements in I and let a, 2 (a1, . . .,a,-).
Assume that [(1ng : ag+1] fl Ij 2 ang‘1 whenever 0 S i S 6 — 1 and
j 2 k. Furthermore assume that .7 has at most two homogeneous
generating relations in degree S k if n —6 Z 2, and that .7 has at most
one homogeneous generating relations in degree S k if n 2 6+ 1. Then

.7 is Cohen-Macaulay.

Proof. Write .7 2 k[T1,...,Tn]/J and let (Jgk) denote the ideal
generated by the forms in J that have degree less than or equal to
k. If k 2 0, then .7 is a polynomial ring over a field, and so it is
Cohen-Macaulay. Hence we may assume that k > 0. If n — 6 Z 2,
then by assumption p(J_<_k) S 2. Hence we have that the projective

dimension of kITl, . . . ,Tn] /(JS;,) is less than or equal to 2 and so depth

77

 

k[T1, . . . ,TnI/(Jgk) Z n — 2 Z 6. Similarly if n 2 6 + 1, we have that
pug.) S 1, and depth k[T1,...,Tn]/(J5k) Z n — 1 2 6. In any case
we may assume that the images a’1,...,a’€ of a1,.. . ,ag in [.7]1 2 I/mI
form a regular sequence in the ring MT 1, . . . ,Tn]/(Jsk).

CLAIM: Clng C 11‘le+1 2 £1,ij whenever 0 S i S 6 and j Z 0.

Proof of the claim by induction on j.

Case 1: j S k — 1.

Let A1a1 + -°-+ Aga, E ijH, with A1,...,A,- E 16. We denote by

’1, . . .,A;- the images of A1, . . . , Ag in Ij/ij. Then A’la’1+~--+Aga; 2 0

in k[T1,...,Tn]/(J5k), since j + 1 S Is. As a'1,...,ag form a regular
sequence in this ring, there is an alternating i x i matrix A with entries
in I?"1 so that I ’1,.. .,AI] 2 [a’1, . . . ,a;]A’. Here A’ denotes the image
of A in Ij‘l/ij‘l. Hence [A1, . . . , Ag] 2 [a1, . . .,a,-]A modulo ij. As
[a1, . . . ,a,]A[a1, . . . ,ag]t 2 0, it follows that A1a1+~~+ Agag E ang.

Case 2: j 2 k.

We use decreasing induction on i. If i 2 6, we are done since j Z r.

78

 

 

If i < 6, then
ang n ij+1 2 (1ng fl ij+1fl ag+11j
2 adj fl £1,“ij
2 €1,ij + ag+1[(a,-Ij : ag+1)fl ij]

C €12"ij + ag+1(a,-Ij_1 fl 1111])

 

: (13'ij + ag+1ij’1 -"
—_— angj,
and this finishes the proof of the claim. I6

In order to prove that .7 is Cohen-Macaulay, since a’1,.. .,a2 are a
regular sequence on the ring kIT 1, . . . , T n] / ( Jgk) , it suffices to show that
forlSiS6andj2k,

[(a’t, . . . ,aZ-_1) If aélj = [(a’t. - - . .a§_1)lj
in .7, or equivalently that
[(ag_1Ij + ij+1):a,-]fl [j C (1i_1Ij_l + ij.
But

[(ag_1Ij +ij+1) I a,] F) [j C [(a,_11j+ Cliij) 2 a,] flIj

2 [(a,_11j + aiij) : a,] n It

 

C (a,_1Ij I ag)flIj+ij
C ag_1Ij‘1+ij.

79

Cl

Notice that if I has second analytic deviation one and k is the small-
est degree of a generating relation in .7 , then r(I) 2 k —- 1. Hence .7
does not have any generating relation in degree S r, and we have a

simpler version of the previous theorem.

Theorem 5.0.4 Let R be a Noetherian local ring with infinite residue
field, let I be an R-ideal with analytic spread 6, minimal number of
generators n with n 2 6 + 1, reduction number r, let a1,...,ag be
general elements in I and let a,- 2 (a1, . . . ,a,). Assume that [€1ng :
ai+1] fl 16 2 adj-1 whenever 0 S i S 6 —1 andj Z r. Then .7 is

Cohen-Macaulay.

Remark 5.0.5 Theorem 5.0.3 covers Theorem 5.0.1 and T heo—
rem 5.0.2. Indeed the assumptions of Theorem 5.0.1 and of Theo-
rem 5.0.2 imply the desired intersections [ang : (144.1] 0 Ij 2 ang"1 (see

[11, Lemma 2.5] and [33, the proof of Theorem 2.1]).

We now recall two lemmas that imply the assumptions of Theo-

rem 5.0.3.

Lemma 5.0.6 ([37, Lemma 2.2]) Let R be a local Cohen-Macaulay
ring with infinite residue field, let I be an R-ideal of grade g, analytic

80

 

 

 

spread 6. Assume that I satisfies Cg, let a,- be the ideals defined in
Theorem 5.0.3, and assume that Q is Cohen-Macaulay. Then [(1, :

ag+1]flIj2ang—1wheneverOSiS6—1andj_>_i—g+1.

Lemma 5.0.7 (Lemma 3.1.7) Let R be a local Cohen-Macaulay ring
with infinite residue field, let I be an R-ideal with grade 9, ana-
lytic spread 6, reduction number r, let k 2 0 be an integer with
r S k, assume that I satisfies Cg, AN€—_k_1 and that for every
p E V(I), depth(R/Ij)p 2 min{dim Rp — 6 + k — j,k — j} whenever
1 S j S k— 1. Let a,- be the ideals defined in Theorem 5. 0.3. Then [(1, :

ag+1]flIj 2 ang‘1 wheneverO S i S 6—1 andj 2 max{1,i—6+k+1}.

Combining Theorem 5.0.3 with Lemma 5.0.6 we get the following
corollary, that relates the Cohen-Macaulayness of Q with the Cohen-

Macaulayness of .7.

Corollary 5.0.8 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, analytic spread 6, min-
imal number of generators n and reduction number r. Assume that
I satisfies Cg. Furthermore assume that .7 has at most two homo-
geneous generating relations in degree S max{r,6 — g} if n — 6 2 2,

and that .7 has at most one homogeneous generating relation in degree

81

 

 

S max{r,6 — 9} if n 2 6 + 1. If Q is Cohen-Macaulay, then .7 is

Cohen-Macaulay.

Proof. The statement follows from Lemma 5.0.6 and Theorem 5.0.3

with k 2 max{r, 6 — g}. [3

Suppose that r 2 6 — g. This is the case for instance if I is equimul-
tiple or has analytic deviation one. Again we point out that if I has
second analytic deviation one, then the assumption “.7 has at most
one homogeneous generating relation in degree S r” is automatically

satisfied. So we get a simpler version of the previous corollary.

Corollary 5.0.9 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, analytic spread 6, and
reduction number r 2 6 — g. Assume that I satisfies Cg and that .7
has at most two homogeneous generating relations in degree S r. If Q

is Cohen-Macaulay, then .7 is Cohen-Macaulay.

Combining Theorem 5.0.3 with Lemma 5.0.7 we get the following

corollary.

Corollary 5.0.10 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be an R-ideal with grade g, analytic spread 6, min-

imal number of generators n, reduction number r, and let I: Z 0 be

82

 

an integer with r S k. Assume that I satisfies Cg, AN€‘_,C_1 and that
for every p E V(I), depth (R/Ij)p Z min{dim RP — 6 + k - j,k — j}
whenever I S j S k — 1. Assume that .7 has at most two homogeneous
generating relations in degree S k if n—6 Z 2, and that .7 has at most
one homogeneous generating relation in degree S k if n 2 6 + 1. Then

.7 is Cohen-Macaulay.

Next, we use an example of D’Anna, Guerrieri and Heinzer to point
out that the assumption “.7 has at most two homogeneous generating
relations in degree S max{r,6 —- g}” in Corollary 5.0.8 can not be

removed or weakened.

Example 5.0.11 ([14, Example 2.3]) Let R 2 k[t6,t11,t15,t3l], where
k is an infinite field, and let I 2 (t6,t11,t31). R is a Cohen-Macaulay
ring, I is an ideal of grade 1, analytic spread 1, reduction number 2,
and second analytic deviation 2. Q is Cohen-Macaulay but .7 is not

Cohen-Macaulay. One has
f : k[T1) T27 T3l/(T23) T1T3) T2T37 T32);

so .7 has 3 generating relations in degree 2.

In particular the above example shows that in general Q Cohen-

Macaulay does not imply .7 Cohen-Macaulay. In Example 5.0.11 R

83

 

 

is not Cohen-Macaulay; so it is natural to ask if in general R Cohen-
Macaulay implies .7 Cohen-Macaulay. We get a negative answer to

this question.

Example 5.0.12 Let R and I be as in Example 5.0.11. By adding
two variables x and y we obtain the ideal I’ 2 (I ,x,y) C R[x,y].
Now I’ has grade 3, analytic spread 3, reduction number 2, and Q (I ’ )
is a polynomial ring over Q (I); thus Cohen-Macaulay. Hence R(I’) is
Cohen-Macaulay by Theorem 3.2.6, but .7 (I ’ ) is a polynomial ring over

.7 (I ), and so it is not Cohen-Macaulay.

Let R be a local Cohen-Macaulay ring and let I be a strongly Cohen-
Macaulay ideal with grade g and analytic spread 6, satisfying Cg. By
[24, the proof of Theorem 4.6] the first 6 — g + 1 symmetric powers
of I have no torsion; i.e., .7 does not have any relation in degree less
than or equal to 6 — g + 1. Hence we obtain better results for strongly
Cohen-Macaulay ideals, and in particular for perfect ideals of height

two. We have the following corollary.

Corollary 5.0.13 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be a strongly Cohen—Macaulay ideal of grade g, an-

alytic spread 6, and reduction number r with r S 6 — g + 1. Assume

that I satisfies Cg. Then .7 is Cohen-Macaulay.

84

 

 

Proof. The statement follows from Theorem 2.4.4, Remark 3.2.3 and

Corollary 5.0.10 with k 2 6 — g + 1. D

The next example shows that the above result is not true even for
perfect ideals of height two with second analytic deviation one, if the

reduction number is not the “expected” one.

Example 5.0.14 Let

{x3 0 0A

y20yz

03/2 22

\0 22 x2)

be a matrix with entries in k[[x, y, 2]], where k is an infinite field. Let I

 

 

be the ideal generated by the 3 by 3 minors of d. I is perfect of height
two, it has analytic spread three, and reduction number five. Also, I

satisfies C3. The fiber cone
7 = k[T1, T2, T3,T4]/(T,5T4, T24T42)

is not Cohen-Macaulay.

Next, we recall an important result for perfect ideals of height two,

that has been very useful in building examples.

85

 

 

Theorem 5.0.15 ([52, Corollary 5.4]) Let R be a local Corenstein ring
with infinite residue field, let I be a perfect R-ideal of height 2 with
analytic spread 6, reduction number r, let ()5 be an n by n — 1 matrix
presenting I, and let (b’ be the n — 6 by n — 1 matrix consisting of the
last n — 6 rows of gt. Assume that I satisfies Cg. The following are

equivalent:

(a) R is Cohen-Macaulay.
(b) r<6 (inwhich caser20 orr26—1).

(c) After elementary row operations on (15, In-g(q§’) 2 n-g(q§).

We refer to the condition (c) of Theorem 5.0.15 as the “row condi-
tion”.

In particular, by Corollary 5.0.13 and Theorem 3.2.6 we have.

Corollary 5.0.16 Let R be a local Cohen-Macaulay ring with infinite
residue field, let I be a perfect R-ideal of grade 2 and analytic spread
6. Assume that I satisfies Cg. If R is Cohen-Macaulay, then .7 is

Cohen-Macaulay.

The converse of the above result is not true; i.e., for perfect ideals of
height two satisfying Cg, .7 Cohen-Macaulay does not imply that R
is Cohen-Macaulay. It is easy to build counterexamples for ideals with

86

second analytic deviation one, because in this case, if I is generated by
homogeneous polynomials of the same degree in a power series ring over
a field, .7 is an hypersurface ring and so it is always Cohen-Macaulay.
However, R is not Cohen—Macaulay if the row condition is not satisfied
(see Example 5.0.17).

We recall that since in the following examples we work in power series
rings over a field, which are regular, the Cohen-Macaulayness of R and

Q are equivalent (see Remark 3.2.5).

Example 5.0.17 Let I C k[[x,y]], where k is an infinite field, be the

ideal generated by the 2 by 2 minors of

(0 y?)

45:. y2 $2

v: .y,

 

 

Then
j: = k[T1,T2,T3]/(T13T2 — T24 + 2T1T§T3 — T1271?)

is Cohen-Macaulay, but R and Q are not Cohen-Macaulay, since the

row condition of Theorem 5.0.15 is not satisfied.

However, for perfect ideals of grade two satisfying Cg, .7 Cohen-
Macaulay does not imply that R is Cohen-Macaulay even if the second
analytic deviation is greater than one.

87

 

 

Example 5.0.18 Let I C k[[x, y]], where k is an infinite field, be the

ideal generated by the 3 by 3 minors of

(x2 0 xy\

y2 x2 0

0y2 £132

\0 o w

 

 

Then
J: = kIT1,T2, T3, T4I/(T32 — T2T4, T23 — 2T1T2T3 + T 3T4 - Tan)

is Cohen-Macaulay, but R and Q are not Cohen-Macaulay, because the

row condition of Theorem 5.0.15 is not satisfied, since ,u(12(q§)) > 3.

Notice that in the previous example .7 is a complete intersection. So
it is natural to ask what happens if .7 is not a complete intersection.

Still, .7 Cohen-Macaulay does not imply that R is Cohen-Macaulay.

Example 5.0.19 Let I C kIIx, y]], where k is an infinite field, be the

ideal generated by the 3 by 3 minors of

{x3 0 x2y\
(b2 y3 x3 0
0 y3 x3

 

 

\0 o ,.,

88

Then .7 2 kITl, T2, T3, T.,«,]/(T32 — T2T4, T24T3 — 3T1T22T4 + 3T12T2T3T4 —
T13T42 + Tng, T25 — 3T1T23T3 + 3T12T22T4 — T13T3T4 + T2T3T43) is Cohen-
Macaulay, not a complete intersection, but R and Q are not Cohen-

Macaulay, because the row condition of Theorem 5.0.15 is not satisfied,

since u(Ig(¢)) > 3.

89

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90

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