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‘ Graded Local Cohomology
and Its Associated Primes

presented by
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GRADED LOCAL COHOMOLOGY AND ITS ASSOCIATED PRIMES

By

Chia Sien Lim

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JOSOPHY

:hematics

ABSTRACT
GRADED LOCAL COHOMOLOGY AND ITS ASSOCIATED PRIMES
By

Chia Sien Lim

It is known that the i-th local cohomology of a finitely generated R—module M over
a positively graded commutative Noetherian ring R, with respect to the irrelevant
ideal R+ is graded. Furthermore, for every integer n, the n—th component H; (M )n
of this local cohomology module H2“ (M) is finitely generated over R0 and vanishes
for n >> 0. We want to understand the behavior of H}; + (M )n for n << 0.

We will specialise the study of Hfi+(M) to the case where R is a Cohen-Macaulay
ring and M is a Cohen-Macaulay R-module. When dim R0 = 1, we will show that
AssR0 (Hfh(M)n) becomes constant when n becomes negatively large. When R0 is
local, equidimensional and dim R0 = 2, we will show that there exists an integer N
such that either HEJM)” = (0) for all n < N or, H§+(M)n 75 (O) for all n < N.

These results will extend the work of Brodmann and Hellus in [2].

ACKNOWLEDGEMENTS

The author would like to thank the members of his thesis committee for their time
and participation, especially:
His advisor, Dr. Christel Rotthaus, for the generosity of her patience and energy. She
has painstakingly read the earlier manuscripts and provided the key observation to
remove the ”equidimensionality” condition on R0 in Theorem (2.0.1). The readability
of this version of the thesis is largely due to her advice.
Dr. William Brown for always being there for consultation on a variety of topics.

The author would also like to thank Dr. Brodmann and Dr. Hellus for sharing
their manuscript in [2], where this thesis is based on.

Last but not least, the author is grateful to the mathematics department at Michi-
gan State University for providing research fellowships for two semesters which make

the completion of this project possible.

iii

TABLE OF CONTENTS

INTRODUCTION ............................................................... 1

CHAPTER 1

Auxiliary tools ................................................................... 4
1.1 Results from Brodmann-Hellus in [2] ...................................... 5
1.2 Lemmas for calculations in Theorem (2.0.1) and (4.0.1) .................... 6

CHAPTER 2

Asymptotic stability when dim R0 = 1 .......................................... 18
2.1 Auxiliary Lemmas for the special case, (a*): R0 is local .................. 20
2.2 R0 is local ............................................................... 21
2.3 Auxiliary Lemmas for the special case, (b‘): |Min(Ro)| = 1 ............... 26
2.4 Proof of Theorem (2.0.1) special case, (b'): |Min(Ro)| = 1 ................ 28
2.5 Proof of Theorem (2.0.1) ................................................. 30
2.6 Corollaries to Thereom (2.0.1) ........................................... 38

CHAPTER 3

Asymptotically gap free if dim R0 = 2 .......................................... 41
3.1 When is I‘moR (le+(M)) Artinian? ...................................... 42
3.2 When is (HA, (M )) asymptotically gap free? ............................. 47

CHAPTER 4

Applications .................................................................... 51
4.1 Reduction to the case: ”31%| is infinite .................................... 52
4.2 Auxiliary tools for Theorem (4.0.1) ....................................... 53
4.3 The support of (H§+(M)) ............................................... 57
4.4 Proof of Theorem (4.0.1) ................................................. 64
4.5 Criterion for (H;2+ (M )) to be non-vanishing ............................. 67

REFERENCES ................................................................. 70

iv

Introduction

Let R be a commutative Noetherian ring with unit, a an ideal of R and M a finitely
generated R-module. We want to study H“,(M), the i-th order local cohomologt of
M with respect to the ideal a. Since H§(M) is not finitely generated as a R-module
in general, a great effort has been expanded to understand its finiteness properties
(cf. [2], [4], [8]). A question raised by C. Huneke regarding the finiteness of the set
of associated primes of H 3(M ) has prompted many researchers to work in this very
active area of mathematics. The most notable finiteness result in this direction is that
if R is a regular local ring containing a field, then for every integer i, AssR (H;(R))
is a finite set for every ideal, a, of R. The characteristic p case was proved by C.
Huneke and R. Sharp (cf. [11]). G. Lyubeznik settled the characteristc 0 case using
D-modules (cf. [13], [14] and [15]).

However, there are examples of local cohomology modules whose set of associated
primes is not finite. The first example is provided by A. Singh in [18]. In this example,
M is a graded hypersurface over R, R is a positively graded algebra over Ra and R0 is
a polynomial ring over the integers. Subsequently, M. Katzman gave another example
where R0 is a polynomial ring over any field (of. [12]). This result effectively shows
that the set of associated primes of a local cohomology module over a local ring can
be infinite.

In this paper, we wish to investigate the local cohomology module H§(M) in the
graded case. Throughout the paper, we will assume that R = 69:30 R" is a positively
graded homogeneous Noetherian ring of finite Krull dimension; a is the irrelevant
ideal, @220 R", denoted by R+; M is a finitely generated graded R—module. The
R-module H§+(M) has a natural grading and Hfi+(M)n, the n—th component of
H§2+(M), is a finitely generated Ro-module. Moreover, for n >> 0, H§+(M)n = (0).

This raises the following question. What about H};a + (M )n and AssR0 (Hjh(M)n) for
n << 0 and their relations with Ass 3 (Hg+ (M ))'7 This work is motivated by a recent
paper of Brodmann and Hellus in [2]. They showed the following:

Proposition 1. (cf. [2], 4.2) If R0 is semi-local and dim R0 3 1, then for every
i E No, H§+(M) is asymptotically gap free.

Hit, (M) is called asymptotically gap free if there exists an integer N such that either
H§+(M)n = (0) for all n < N or, H}'2+(M)n 7e (0) for all n < N.

Proposition 2. (cf. [2], 5.6) Let t E No. If H}.2+ (M) is finitely generated over R for
all i < t, then {Ass};0 (Hfi+(M)n)}n€z is asymptotically stable.

{Ass};0 (qu+(M)n)}nez is called asymptotically stable if there exists an integer N
such that Ass];o (H13)? (M )n) becomes constant for all n < N.

It was noted by Brodmann and Hellus that asymptotic stability impr that Hf;+ (M)
is asymptotically gap free and Ass R (H A (M )) is finite. However, the converse is false
according to the joint paper by M. Brodmann, M. Katzman and R. Sharp (cf. [3]).

Brodmann and Hellus also showed that for a local base ring (R0, mo), mo is the
maximal ideal of R0, if H§Z+(M) aé (0), then i is bounded above by dim 3%} More
precisely,

Proposition 3. (cf. [2], 3.4) Suppose that (R0, mg) is local and M is finitely generated
graded R-module with dim EMA? : d. If M 75 I‘R+(M), then H§+(M) yé (0) and
H§+(M) for alli > d.

In this project, we will apply and extend Brodmann and Hellus results to the
case where R is a Cohen-Macaulay ring, M is a Cohen-Macaulay R module and dim
R0 is either 1 or 2. In this situation, the grade of M with respect to R+ provides
a lower bound for the order in which the local cohomology does not vanish, while
Proposition (3) of Brodmann and Hellus gives an upper bound. One of our main

ideas is to localise R at a prime ideal of R0 so that our base ring can be local. This

allows us to use Proposition (3) of Brodmann and Hellus to get a better handle on
the support of Hi},+ (M). More precisely, we will prove:

Theorem 2.0.1. Suppose that R is a positively graded homogeneous Cohen-Macaulay
ring and M is a finitely generated graded Cohen-Macaulay R-module. Assume also

that dim R0 = 1. Then, for all i,

{AssRo(H]q+(M)n)}nez

is asymptotically stable.

Theorem 4.0.1. Suppose that R is a positively graded homogeneous Cohen-Macaulay
ring and M is a finitely generated graded Cohen-Macaulay R-module. Assume also
that R0 is local, equidimensional and dim R0 = 2. Then for all i, Hfh (M) is asymp-

totically gap free.

Due to the known results of Brodmann and Hellus in PrOpositions (1) and (2),
the interest of our work will be in the case, Hf;+ (M) for i > grade(R+, M). In fact,
for Theorem (4.0.1), we can simply focus on the situation where ht (I ' 0 R0) = 0
(I’ = annR(M)), because if ht (I’ 0 R0) 2 1, H§+(M) is a %-modu1e and dim
(%)0 _<_ 1; this situation is dealt with by Brodmann and Hellus’s Proposition (1)
above.

Here is an outline of our project: In Chapter One, we will gather the tools that
will be applicable in the proof of Theorems (2.0.1) and (4.0.1); in Chapter Two, we
will give a proof of Theorem (2.0.1) and its Corollaries; in Chapter Three, we will give
some sufficient condition for Hh+(M) to be asymptotically gap free if dim R0 = 2;
in the final Chapter, we will apply the results in Chapter Two and Three to prove

Theorem (4.0.1) and its Corollaries.

1 Auxiliary tools

We begin by recalling some definitions and observations made in [2]. Then, we will
state three results that we will constantly utilised from the paper, [2], by Brodmann
and Hellus. Then, we will prove several Lemmas which will be applicable in the cases
where dim R0 = 1 and dim R0 = 2.

For any unexplained terminology, the reader can refer to [6] and [16]. The reference
we use for background information on local cohomology is the textbook by Brodmann
and Sharp (cf. [5])

We will assume throughout this paper that R = $320 Rn is a positively graded
homogeneous Noetherian ring of finite Krull dimension. This means that R is gen-
erated by degree 1 elements. Assume also that M is a finitely generated graded
R-module. We will refer to these two assumptions as the standard hypotheses on R
and M. We will denote I = W and R+ = 632:0 Rn, the irrelevant ideal of
R. We note that I is graded.

For a q E Spec(R0), Mq (resp. R9) is the localisation of M (resp. R) at the
multiplicative set, R0 — q. We note that Mg and R, are both graded and Hg is graded
local. Furthermore, (Rq)o '5 (R), as rings.

Definitions. Let T be a graded R-module. Then,

(a) T is asymptotically gap free if the set,
{n E Z<o [Tn 7é (0) and Tn+1 = (0)}

is finite.
(b) {AssR0 (Tn)}nez is asymptotically stable if there exists N E Z such that for all
n S N,

ASS};o (Tn) = ASS};O (TN).

We note that (b) implies (a) because any Ro-module, L, is (0) if and only if Ass];0 (L) =
0 (cf. [2]).

The interest in (b) is its relations to AssR (H}2+(M)). There is a bijection of sets:
AssR (HR+(M)) +—> Unez AssRO(HR+(M),,)
p + R+ ('—) 1)

(cf. [2], 5.5). Consequently, AssR (H}°{+(M)) is a finite set if and only if

Unez ASSROHR+ (M)n

is a finite set. Furthermore, since H§+(M)n = (0) for n >> 0 and for each n,

Ass Ro (Hfh(M)n) is a finite set, so if we have asymptotic stability, then

UnEZ ASSRo (Hlt+ (Mln)

is finite. Hence, asymptotic stability of {Ass};O (H}'{+(M)n)}nez implies the finiteness
of A333 (HA (M)).
We also note that the same correspondence gives us a bijection of sets between

the support of H}; (M) and the support of H§+(M)n:
SuppR (Hji+ (M)) <—> Unez SuppRo(H]q+ (M)n)
p + R+ <—> p
1.1 Results from Brodmann-Hellus in [2]

Proposition 1.1.1. (Brodmann-Hellus) Suppose that dim R0 S 1, and R0 is semi-
local. Fix i E No. Then, for every finitely generated graded R—module M, le+ (M) is

asymptotically gap free.

Proof. (cf. [2], 4.2) D

Proposition 1.1.2. (Brodmann-Hellus) Suppose that (R0, mo) is local and M is a
finitely generated graded R—module with dim 31% = d. If M ¢ I‘R+(M), then

(a) HAW) 7s (0) and

(b) H§+(M) for alli > d.

Proof. (cf. [2], 3.4) Cl

Proposition 1.1.3. Suppose that R+M # M. Let g = grade(R+, M). Then:
(a) For alli < g, H§+(M) = (0) ;

(b) f0” = 9, HinM) ¢ (0);

(c) (Brodmann-Hellus) if HEJM) is finitely generated over R for alli < t, then

{ASSRO (HR... (M)n)}nEZ
is asymptotically stable.
Proof. For (a) and (b), (cf. [5], 6.2.7). For (c), (cf. [2], 5.6). CI

We will very often use the preceding two Propositions in the following way. Given
a q E Spec(R0). Suppose that Mg 75 PR+Rq(Mq) and R+M 75 M. Then, if q E
SUPPRO (Hiz+(M)n), then

M
d < '< ' —".
gra e(R+Rq, Mq) _ i _ dlquq

We will give an upper bound for dimajfi— in terms of grade(R+, M). This is because

grade<R+, M) s graduate. M.).
This basic setup can help us understand the support of H§+(M).

1.2 Preliminary Lemmas in the Cohen-Macaulay case

Given a q E Spec(Ro). This subsection is devoted to the understanding of diquA-Jil;

and grade(R+, M), when both R and M are Cohen-Macaulay. Our first goal is to

give an upper bound of dimfigfi in terms of ht (q + IL.) and ht (q + I )R., where,
I = W. This is the content of Lemma (1.2.6). Then, we will use the fact
that R0 is equidimensional to write ht (q + R+) in terms of ht (R+). This will be
accompanied by a calculation of ht (q + I )Rq in terms of ht (1). Finally, in Lemmas
(1.2.9) and (1.2.10), we will relate grade(R+, M) to ht (I) and ht (R+). This will give
us a more concrete link between dimq—MA—f; and grade(R+, M).

Some notations and conventions: Min(Ro) will represent the set of minimal
primes of R0. Max(Ro) will represent the set of maximal primes of R0. For i E N,
Spec‘(Ro) will represent the set of primes of R0 whose height equals i. For an ideal J

in R, Min(J) will represent the set of minimal primes over J.

Lemma 1.2.1. Suppose that (R0, m0) is local. Let m 2 mo + R+. Then,
dim M = dim Mm

Proof. We may assume that M 75 (0). Otherwise, dim M is undefined. Let P E
Supp (M) and P* be the ideal generated by all the homogeneous elements in P. By
([6], 1.5.6), P“ E Supp (M). This implies that Mm 75 (0) and hence, dim Mm is
defined.
If P = P“, then dim Mp S dim Mm, by the definition of dimension. Suppose that
P # P‘. Then,

dim Mp = dim (Mp.) +1

(cf. [6], 1.5.8). Since m is the graded maximal ideal of R, so P“ C m and thus
dim (M p-) < dim Mm. Therefore, dim Mp S dim Mm.
Since dim M = dim MQ for some Q E Supp (M) and dim M Z dim Mm, we have

our contention. Cl

Lemma 1.2.2. Suppose that both R and M are Cohen-Macaulay. Assume also that
(R0, m0) is local. As before, I = t/annp (M). If P, Q E ASSR (M), then

ht P = ht Q.
In particular, all minimal primes of I have the same height.

Proof. Let m = mg + R+. Let P and Q 6 Assn (M). They are both graded and
hence, contained in m ([6], 1.5.6(b)(ii)). Since Mm is Cohen—Macaulay, by ([6], 2.1.2),

dim gun; = dim Mm. (1.1)
Since Rm is also Cohen-Macaulay and P (_I m,
ht PR". + dim £1"— = dim R... (1.2)
PR".
(cf. [6], 2.1.4). Hence, combining equations (1.1) and (1.2), we get
ht PR... + dim Mm = dim Rm.
Similar equation holds for Q implies that
ht PR4n = ht QRm.
Since P and Q are prime ideals, and are contained in m,

htP=th.

Since M is a finitely generated R-module, V(I) = Suppp (M) and Ass}; (M) has
the same minimal elements as Suppp (M). Therefore, all the minimal primes of I
are in A333 (M). More importantly, by the argument above, they all have the same

height. Cl

Lemma 1.2.3. Suppose that both R and M are Cohen-Macaulay. Assume also that

(R0, m0) is local. I = t/annR(M). Let p E Spec(RO) andp Z_> (I 0 R0).
If there exists W E Min(I) such that W (1 R0 = p, then

ht (p+1)R,, = ht 1.

Proof. Since all minimal primes of I are graded ([6], 1.5.6), W Q (p+ R+). Note that
(p + R+) 6 Spec(R) and (p + Rt) 0 (R0 - p) = 0. Therefore, ht WRP = ht W.
Since all minimal primes of I have the same height (cf. Lemma (1.2.2)), ht W = ht I.

By the definition of height,
ht WRp 2 ht (p+I)Rp 2 ht I.

Hence, ht (p+ I)Rp = ht I. C]

The following will only be used in Section (2.4). It is essentially Lemma (1.2.2)

without assuming that R; is local.

Lemma 1.2.4. Suppose that both R and M are Cohen-Macaulay. I = m.
Assume also that dim R0 = 1, Min(Ro) = {p} and for all W E Min(I), W (1 R0 = p.
Then,

(a) all minimal primes of R+ have the same height;

(b) all minimal primes of I have the same height.

Proof. For (a), note that R+ has only 1 minimal prime over it, namely p + R+.

For (b), the hypothesis implies that ht (I 0 R0) = 0 and I 9 (p + R+). Let Q E
Min(I). Then, Q 0 R0 = p by assumption. By ([6], 1.5.6(b)), Q is graded and hence,
Q E (p + R+). In particular, ht Q = ht QRp.

Note that, (Rp)o is local as well as equidimensional, and both RI, and Mp are Cohen-

Macaulay. Therefore, by Lemma (1.2.2), all minimal primes of I R10 have the same

height. Since QR,D E Min(IRp),

ht QR, = ht 13,.

More importantly,
ht Q = ht IR,

and the latter is independent of Q. Therefore, all minimal primes of I have the same

height. Cl

The fact that all the minimal primes of R+ have the same height will make the

calculations of grade(R+, M) easier in Lemmas (1.2.9) and (1.2.10).

Lemma 1.2.5. Suppose that R is Cohen—Macaulay and (R0, mo) is local. I = t/annR(M).
Ifq E Spec(Ro) and q 2 (I 0 R0), then

. R. _
dlmm'l-ht (q+I)Rq—-ht (q+R+).

In particular,

, R
dim m-l-ht (mo+I) —ht (mo+R+).

Proof. Let q E Spec(Ro) and q 2 (I 0R0). Put Q = q+R+. Note that, Q E Spec(R).
Since I is also graded, (q+ I) g Q and hence, (q+ I)RQ 75 RQ. Then, by ([6], 2.1.4),

. RQ _ .
d1m (q—+T)R_Q ‘1” hi) (q + I)RQ — dlm RQ. (1.3)

We will make three observations:

(0
R,

d' RQ _—
(q+1)Rq

1m (q+I)Ro

= dim

because

10

as Rq-module and since QRq is the graded maximal ideal of Rq,

(cf. Lemma (1.21)). Note that, the dimension is defined here because every minimal
prime of q + I is contained in Q.
(ii) ht Q = dim Rq because

Bo g (12,)an

as rings and since QRq is the graded maximal ideal of Rq,

dim (Rq)QRq = dim R9
(of. Lemma (1.2.1)); dim RQ = ht Q since Q E Spec(R).
(iii) ht (q + I)Rq = ht (q + I)RQ because (q + I)Rq is graded implies that all the
minimal primes of (q + I )R., are graded ([6], 1.5.6) and hence, they all have the form
PRq, where P is a graded prime ideal of R and P g Q.

From these observations, the following can be deduced from equation (1.3),

d1m—+ht +I =ht +R . 1.4
Toshowthat
dim—L+ht(m +I)—ht(m +R)
(m0+I) 0 _ 0 + t

we simply replace q and Q by mo and m respectively, where m 2 mo +R+, in equation

(1.4). Then, we have

 

. R... _
dlm (mo+I)Rmo +1113 (7110 + I)Rmo — ht ("to + R+).

Since (R0 — mo) consists of units of R, we can identify RmO with R. In particular,

the preceding equation becomes

R
' —— h = .
d1m(m0+1)+ t(mo+I) ht (mg+R+)

11

For q E Spec(Ro), we will use the nice formula in Lemma (1.2.5) to give a bound

. M
on dim —1—.
‘11qu

Lemma 1.2.6. Suppose that both R and M are Cohen-Macaulay and (R0, mg) is
local. I = t/annp(M). Ifq E Spec(Ro) and q 2 (I (1 R0), then

M
dim —q S ht (q+ R+) — ht (q + I)Rq.
qu

In particular,

 

dim S ht (mo+R+) -ht (mo-l-I).

mo
Proof. Let q E Spec(Ro) and q 2 (I n R0). Then, qu 75 Mg. Otherwise, Mg 2 (0)
(cf. [6], 1.5.24(a)).

By the definition of dimension,

. M. . R,
By Lemma (1.2.5),
dim —Rq— + ht (q + I)Rq -—- ht (q + R+). (1.6)
(q+ 1qu

Combining inequality (1.5) and equation ( 1.6), we get

dim L42- S ht (q + R+) — ht (q + I)Rq. (1.7)
qu

Since (R0 - m0) consists of units of R, we can identify Mm0 with M and Rmo with

R. In particular, inequality ( 1.7) becomes

 

dim MM S ht (mo + R+) — ht (mg + I),

mo

after we substitute mo for q. C]

The object of the next few Lemmas is to facilitate the calculations of grade(R+, M).

12

Lemma 1.2.7. Suppose that R is Cohen-Macaulay and (R0, m0) is local.
Then, R0 is equidimensional if and only if all the minimal prime ideals of R+ have

the same height.

Proof. Let P E Min(R+) and m = mo+R+. Then, P = p+R+ for some p E Min(RO).
Furthermore, for every q E Min(Ro), (q + R+) E Min(R+).
Since Rm is Cohen-Macaulay, by ([6], 2.1.4),

. R. _.
htPR’"+dlmPR;n_—dlmRm°

Then, P E Spec(R) and P g m implies that ht PR”, = ht P.

Since
. Rm _ mRm
dim PR". —- ht PRvn,
we have
ht P+ht % =ht m.
Note that
m 7710
h — =h —
t P t p

because, every prime containing P is of the form q + R+ where q E Spec(Ro). More
importantly,
dim & = ht E.
P P
Hence, dim 559 is the same for all p E Min(RO) if and only if all minimal primes of

R+ have the same height. El

Lemma 1.2.8. Suppose that R is Cohen-Macaulay. Assume also that (R0, m0) is

local and equidimensional. Let i E No.

Ifq E Spec(RO) and ht q = i, then (q + R+) E Spec(R) and

13

Proof. Let q E Spec(Ro) and ht q = i. Then, (q + R+) E Spec(R) because

R a
9+R+ q

HZ

 

as rings and the latter is a domain. There also exists p E Min(Ro) such that p Q q
andht % =i. LetP=p+R+ andQ=q+R+.

Since RQ is Cohen-Macaulay,

dim RQ = ht (PRQ) + dim 51%; (1.8)
(cf. [6], 2.1.4). Since P E Spec(R), equation (1.8) can be written as
_ Q
th—ht P+ht P' (1.9)

Since all primes of R containing R+ must be of the form, u+R+, where u E Spec(Ro),

m%=m%. (um
Since all the minimal primes of R+ have the same height (cf. Lemma (12.7)),
ht (P) = ht R+. (1.11)
Hence, combining equations (1.9), (1.10) and (1.11), we have
ht (q + R+) = ht (R+) +i.
[:1

Lemma 1.2.9. Suppose that both R and M are Cohen-Macaulay. Assume also that
(R0, mg) is local and equidimensional. I = Mann}; (M). Let i E No. If ht (I 0 R0)

= i, then

grade (R+, M) = ht (R+) +i—ht I.

14

Proof. By ([6], 12.10(a)),
grade (R+, M) = inf {depth Mp | P E V(R+) }.

Since depth (0) 2: 00, we have

grade (R+, M) = inf {depth Mp | P E V(R+ + I) }.

Since M is Cohen—Macaulay,
grade (R+, M) = inf {dim Mp | P E V(R+ + I) }.
For any P E V(R+ + I),

Since Rp is Cohen-Macaulay,

. R .
dim 71%); = dim (Rp) -- ht IRp

(1.12)

(1.13)

(1.14)

(cf. [6], 2.1.4). By Lemma (1.2.2), all minimal primes of I have the same height.

Hence, combining (1.12), (1.13) and (1.14), we have

grade (R+, M) = dim RQ — ht I

(1.15)

for some Q E V(R+ + I) such that dim RQ is the smallest. Since ht (I 0 R0) = i, Q

will be of the form q + R+ such that q E Spec(Ro) and ht q = i. By Lemma (1.2.8),

ht (q + 12+): ht (R+) + i.
Therefore, combining equations (1.15) and (1.16), we get

grade (R+, M) = ht (R+) + i — ht I.

15

(1.16)

The following will only be used in Section (2.4). It is essentially Lemma (1.2.9)

in the case where R0 is not necessarily local.

Lemma 1.2.10. Suppose that both R and M are Cohen-Macaulay. I = W.
Assume also that dim R0 = 1, Min(RO) = {p} and for all W E Min(I), W 0 R0 = p.
Then,

grade(R+, M) = ht (R+) — ht I.

Proof. By ([6], l.2.10(a)),
grade (R+, M) = inf {depth MQ I Q E V(R+) }.
Since depth (0) :2 00, we have
grade (12,, M) = inf {depth MQ I Q 6 V(R+ + 1)}.
Since M is Cohen—Macaulay,
grade (12,, M) = inf {dim Me I Q 6 V(R+ + 1)}. (1.17)

For any Q E V(R+ + I),

dim MQ = dim 133$. (1.18)
Since RQ is Cohen-Macaulay,
dim & + ht IRQ = dim RQ (1.19)
112d

(cf. [6], 2.1.4). By Lemma (1.2.4)(b), all minimal primes of I have the same height.

Hence, putting the equations (1.17), (1.18) and (1.19), we have

grade (12., M) =-. dim HQ — ht I (1.20)

16

for some Q E V(R+ + I) such that dim RQ is the smallest. Since ht (I (1 R0) = 0,
Q = p + R+. ht (I (1 R0) = 0 because Min(Ro) = {p} and (I 0 R0) Q p. Now, by
Lemma (1.2.4)(a),

ht Q = ht R+. (1.21)

Note that, dim HQ 2 ht Q. Therefore, combining equations (1.20) and (1.21), we get
grade (R+, M) = ht R+ — ht I.

Cl

Lemma 1.2.11. Suppose that both R and M are Cohen-Macaulay. Assume also that

(R0, mo) is local and equidimensional. I = {/annp(M).
Ifq E Spec(Ro) and q 2 (I (1 R0), then

grade(R+. M) S grade(R+Rq, Mg).
Proof. Given q E Spec(Ro) such that q Z_> (I (1 R0). By ([6], l.2.10(a)),
grade (R+, M) = inf {depth Mp | P E V(R+)}
and
grade (R+Rq, Mq) = inf {depth (Mq)qu | PR, E V(R+Rq) }.
Note that, for PR., E V(R+Rq),

(Mq)PR., '5 MP

as Rp-modules where, P has the form p + R+ for some p E Spec(Ro) and q g p.

Consequently,
depthmnpa, (Mm. = deptth (Mp)
In particular,

grade(R+, M) S grade(R+Rq, Mq)-

17

2 Asymptotic stability when dim R0 = 1

The object of this chapter is to Show the following Theorem.

Theorem 2.0.1. Suppose that both R and M are Cohen-Macaulay. Assume also
that dim R0 = 1. Then, for all i,

{ASS};0 (HR+(M)71) }nEZ
is asymptotically stable.

We will first prove Theorem (2.0.1) in two special cases namely,
(a*) Suppose that R and M are Cohen-Macaulay. Assume also that R0 is local and
dim R0 S l.
(b‘) Suppose that R and M are Cohen-Macaulay. Assume also that |Min(R0)| = 1
and for all W E Min(I), W 0 R0 E Min(Ro).

In Section (2.5), we will prove Theorem (2.0.1) by first decomposing Spec(Ro) into
a finite union of localisation, where each localisation of R0 will give us the hypothesis
of Theorem (2.0.1)(a*) and Theorem (2.0.1)(b"‘).

We begin by proving the following Proposition that deals with the case
ht(I 0 R0) = l, where I = W. This is essentially due to Brodmann and

Hellus’s result, Proposition (1.1.1).

Proposition 2.0.1. Assume the standard hypotheses on R and M. Suppose that
dim R0 =1. I = t/annp(M). Ifht (In R0) = 1, then for all i,

{ASS};0 (13};+ (M)n) }nEZ

is asymptotically stable.

Proof. Let i E No. First, we show that for all n E Z,

Ass};O (H§3+(M),,) = Supp};O (H]2+(M),,).

18

The containment, ” g ”, and both sets having the same minimal elements are common
knowledge. For the other containment, let p E Supp Ro (H j; +(M )n) for some n. Then,
p Q (I 0R0). Hence, p E Max(Ro) because, ht (I 0R0) = 1. Therefore, all the elements

of Suppp0 (H§+(M)n) are maximal. In particular, for all n E Z,

ASSRo (HR+(M)n) 2 SUPPRO (Hlt+(M)1l) -

Finally, we will Show that

{SUPPRO (HR+(M)n) }nEZ
is asymptotically stable. We note that Suppp (Hi2+ (M )) is a finite set because
Suppn (Hie,(M)) E Suppn (M) 0 V(R+)

and the latter is a finite set. Since, ht (I (1 R0) = l and dim R0 = 1.
Let q + R+ E Suppp (H};+(M)) and q E Specl(R0). Since dim (R0)q = 1,

H}; 12.,(Mq) is asymptotically gap free (cf. Pr0position (111)). Therefore,

q 6 Supple. (Hiz+(M)n)

for n << 0 or,

q ¢ SUPPRO (Hiz+(M)n)
for n << 0. Since Suppp (HR. (M )) is also a finite set, there exists N such that for
all n S N,

811131312. (Hi2..(M)a) = Suppna (Hi2.(M)~).

Remark:
(i) During the preparation of this paper, we realise that Theorem (2.0.1)(a‘) is proven

by M. Brodmann, ST. Fumasoli and R. Tajarod without assuming either R or M is

19

Cohen-Macaulay (cf. [1], Theorem 3.5(e)). Therefore, if we wish to prove asymptotic
stability for dim R0 = 1, the case (b‘) is at the heart of the matter.

(ii) Our proof of Theorem (2.0.1)(a’) is different from the one in [1].

2.1 Auxiliary Lemmas for the special case, (a*): R0 is local

We recall the hypothesis of Theorem (2.0.1)(a“) .

Suppose that both R and M are Cohen-Macaulay. Assume also that (R0, m0) is local
and dim RD S 1.

Note that the assumption on R0 imply that R0 is equidimensional. Hence, all the

minimal prime divisors of R+ have the same height (cf. Lemma (12.7)).

Lemma 2.1.1. Assumptions as in Theorem (2.0.1)(a‘). I = ‘/311IIR (M).

Then, all minimal primes of I have the same height.
Proof. We refer to Lemma (1.2.2). C]

Lemma 2.1.2. Assumptions as in Theorem (2.0.1)(a“). I = m.
Letp E Spec(Ro) andp 2 (IflRo). If there exists W E Min(I) such that WflRo = p,
then

ht (p+I)Rp = ht 1.

Proof. We refer to Lemma (1.2.3). Cl

Lemma 2.1.3. Assumptions as in Theorem (2.0.1){a*). As before, mo is the maxi-

mal ideal ofRo and I = {/annR(M).
Ifq E Spec(Ro) and q 2 (I (1 R0), then

, M,
__ < _ .
dim qu __ ht (q + R+) ht (q + I)Rq

In particular,

 

dim MM S ht (mo + R+) — ht (mo + 1).

mo

20

Proof. We refer to Lemma (1.2.6). C]

Lemma 2.1.4. Assumptions as in Theorem (2.0.1)(a*) except that dim R0 = 1.
Then,
ht (mg + 12,.) = ht (12,.) + 1.

Proof. (R0, m0) is equidimensional and htp0 (mo) 2 1. By Lemma (1.2.8),

ht (mg + 12+) = ht (R+) + 1.

El
Lemma 2.1.5. Assumptions as in Theorem (2.0.I)(a*). I = W.
If ht (I 0 R0) = 0, then
grade (R4,, M) = ht R+ — ht I.
Proof. We refer to Lemma (1.2.9). Cl
Lemma 2.1.6. Assumptions as in Theorem (2.0.1)(a‘). I = m.
Ifq E Spec(Ro) and q 2 (I (1 R0), then
grade<R+, M) s grade<R+Ra M.)
Proof. We refer to Lemma (1.2.11). [3

2.2 R0 is local

The main object of this section is to show the following.
Theorem (2.0.1)(a‘) Suppose that both R and M are Cohen—Macaulay. Assume
also that (R0, mg) is local and dim R0 S 1. Then, for all i,

{AssRO (H§+(M)n)}nez

21

is asymptotically stable.

We begin by dealing with the case dim R0 = 0. This is the content of Proposition
(2.2.1). The proof is essentially 3 Corollary of Proposition ( 1.1.1).

After that, in light of Proposition (2.0.1), we will focus on the case
ht (I 0 R0) = 0. We will use Lemmas (2.2.2) and (2.2.3) to describe the ele-
ments in Supppo (H§2+(M)n) . For p E Spec(Ro), we will bound dim pig: in terms of
grade(R+, M). We want to get a handle of dim {73: because by Proposition (1.1.2),
it tells us that for all i > dim %, H§+Rp(Mp) = (0). These effort will be culminated
in Proposition (2.2.4).

Finally, we will prove Theorem (2.0.1)(a*).

Proposition 2.2.1. Assumptions as in Theorem (2. 0. 1)(a* ) except that dim R0 = 0.
Then, for all i,

{ASS};o (IIk+ (M)n) }nEZ

is asymptotically stable.

Proof. Fix i E No. Then, by Proposition (1.1.1), HfiJM) is asymptotically gap free.

Since R0 is also local and Artinian, there exists N such that either
Ass);o (H§+(M)n) = 0

for all n S N or,
Ass];0 (Hfi+(M)n) = {mg}.
for all n S N. 13

Lemma 2.2.2. Assumptions as in Theorem (2.0.1)(a*) except that dim R0 = 1.

I = Vannp (M) Let q E Min(Ro). Then:
(a) Iffor all W E Min(I), W 0 R0 75 q, then Mg 2 (0).

22

(b) If there exists W E Min(I) such that W (1 R0 = q, then

M
d' ——‘—’-< d M.
1m qu _ gra e(R+, )

Proof. For (a), suppose that for all W E Min(I), (W (1 R0) 75 q. Then, q 2 (I 0 R0).
Otherwise, (q + R+) contains a prime W E Min(I) such that W 0 R0 = q because
q E Min(Ro). This is contrary to assumption. In particular, (I 0 R0) 0 (R0 — q) 7é 0.
Therefore, Mg 2 (0).

For (b), suppose that there exists W E Min(I) such that (W 0 R0) = q. Then,
q 2 (I 0 R0). By Lemma (2.1.3),

M
dim q—NI— S ht (q + R+) — ht (q + I)Rq. (2.1)

9

By Lemma (2.1.2),

ht (q+I)Rq =ht I. (2.2)
By the equidimensionality of R0,

ht (q + R+) = ht Re. (2.3)
Putting expressions (2.1), (2.2) and (2.3), we get

M
d' —-"—<ht —htI
lquq_ R+ ,

which is grade(R+, M) by lemma (2.1.5). Note that our assumption in (b) implies
that ht (I 0 R0) = 0. El

Lemma 2.2.3. Assumptions as in Theorem (2.0.1)(a*) except that dim R0 = 1. As
before, mo is the maximal ideal of R0 and I = annp (M).

Suppose that ht (I 0 R0) = 0. Then:

(a) Iffor all W E Min(Ro), W (1 R0 75 mo, then

 

M
moM S grade(R+, M).

dim

23

(b) If there exists W E Min(I) such that W 0 R0 = mg, then

 

dim MM S grade(R+, M) + 1.

mo

Proof. For (a), by Lemma (2.1.5), it suffices to show that

 

 

d' < h — ht I.
1m mOM _ t (12+)
By Lemma (2.1.3),
d' M <ht(m +R)-ht( +1) (24)
1m moM _ 0 + mo . .
By Lemma (2.1.4),
ht (mg + R+) = ht (12,.) + 1. (2.5)
By assumption,
ht (mo+I) th (I)+1. (2.6)

Putting the expressions (2.4), (2.5) and (2.6), we have
M
d' — < ht — h I .
1m mOM _ (12+) t
For (b), we still have expressions (2.4) and (2.5). By Lemma (2.1.2),
ht (mo+I)=ht I

after we identify RmO with R. Therefore, combining the three expressions together,

we get

 

dim MM 5 ht (12+) — ht (1) + 1.

mo

Since grade(R+, M) = ht (R+) — ht (I), we have our contention. El

24

Proposition 2.2.4. Assumptions as in Theorem (2.0.1)(a*) except that dim R0 = 1.
As before, mo is the maximal ideal of R0 and I = W. Let g = grade(R+, M).
If ht (I O R0) = 0, then
(a) H§+(M) = (0) for all i > g +1;
(b) Suppa (Hi:1(M)n) c {mo}-
Proof. Note that g E No since M is Cohen-Macaulay. We may assume that PR+ (M) 75
M. Otherwise, H§+(M) = (0) for all i > 0.

For (a), let d = dim git-1,7. By Proposition (1.1.2), for all i > d, sz+(M) = (0).
Futhermore, by Lemma (2.2.3), d S g+ 1. Therefore, H§+(M) = (0) for alli > 9+1.

For (b), let p E Supp};o (H EKM )n) for some n. Then, by the Flat Base Change
Theorem ([5], 13.1.8, 15.2.2(iv)), HfikJMp) 75 (0). Obviously, Mp ¢ (0). By Propo-
sition (1.1.2),

dim £13:- 2 g + 1.

By Lemma (2.2.2), p E Min(Ro); otherwise, dim £3; S g. This shows that for all
n E Z,

Supp};0 (Hfl:l(M)n) fl Min(Ro) = 0.

Since (R0, m0) is local and dim R0 = 1,

Suppa. (Harms) c {mo}-

We will now prove the main result of this section.
Theorem (2.0.1)(a*) Suppose that both R and M are Cohen-Macaulay. Assume
also that R0 is local and dim R0 S 1. Then, for all i,

{ASS};0 (I'll;+ (Mln) lnez

is asymptotically stable.

25

Proof. Case 1: dim R0 = 0. This is proven in Proposition (2.2.1).

Case 2: dim R0 = 1. We will consider the following 2 situations seperately.

Case 201: ht (I 0 R0) = 1. This is proven in Proposition (2.0.1).

Case 25: ht (I 0 R0) = 0. Let g = grade(R+, M). We may assume that M at
I‘RJM). Otherwise, H§+(M) vanishes for all i > 0 and Hg+(M) = I‘p+(M) is

finitely generated. By Proposition (1.1.3)(c), for i = g,

{ASS};0 (Hi-{*(M)n) }nEZ

is asymptotically stable. By Proposition (2.2.4)(a), for i > g + 1, H§2+(M) = (0).
Therefore, it remains to check for i = g + 1.

By Proposition (1.1.1), H QRM ) is asymptotically gap free. The nontrivial case
is when H§:I(M)n aé (0) for n << 0. Then, for n << 0,

Ass);O (Hfi:1(M)n) 2 {mo}.

This is because:

By Proposition (2.2.4)(b), for all n E Z,

Suppa. (Haws) 9 {mo}
and
Assp0 (Hfil(M)n) Q Suppp0 (Hg1(M)n).
In fact, we have equality in the last two expressions because any Ro-module L is (0)
if and only if Ass);0 (L) 2 (ll. CI

2.3 Auxiliary Lemmas for the special case, (b*): |Min(R0)[ : 1

We will state the hypothesis of Theorem (2.0.1)(b*).
Suppose that both R and M are Cohen-Macaulay. I = t/annp (M). Assume also
that dim R0 = 1, Min(Ro) = {p} and for all W E Min(I), W 0 R0 = p.

26

We will collect some Lemmas from Chapter (1) that will be applicable in the
context of Theorem (2.0.1)(b").

Lemma 2.3.1. Assumptions as in Theorem (2.0.1)(b*). Then,
( a) all minimal primes of R+ have the same height;

( b ) all minimal primes of I have the same height.
Proof. We refer to Lemma (1.2.4). B

Lemma 2.3.2. Assumptions as in Theorem (2.0.1)(b‘). I = {/annp(M).
Ifq E Max(Ro), then
. Mq
dim — S ht (q + R+) — ht (q+ I)Rq.
qu

Proof. Let q E Max(R0). Then, R, is graded local and (Rq)o is equidimensional. Now,
we can apply Lemma (1.2.6) to Mg and R, to get
. Mq
dim — S ht (q + R+)Rq — ht (q + I)Rq.
qu

Note that, q(Rq)o is the maximal ideal of (Rq)0. Since (q + R1,) E Spec(R),

ht (q+ am. = ht (q+R.)

Lemma 2.3.3. Assumptions as in Theorem (2.0.1)(b‘).
Ifq E Max(Ro), then (q + R+) E Spec(R) and

ht (q + 12,.) = ht (R+) + 1.

Proof. Let q E MaX(R0). Then, R., is graded local. (Rq)o is local, equidimensional
and dim (Rq)0 = 1. Then, by Lemma (1.2.8),

ht (q + R+)Rq = ht (R+Rq) + 1. (2.7)

27

Since (q + R+) E Spec(R),

ht (q + R+)Rq = ht (q + R+). (2.8)

By Lemma (2.3.1)(a),
ht (R+Rq) = ht R+ (2.9)
Therefore, combining equations (2.7), (2.8) and (2.9), we have our goal. [3

Lemma 2.3.4. Assume the hypothesis of Theorem (2.0.1)(b‘). Then,
grade(R+, M) = ht R+ — ht I.
Proof. We refer to Lemma (1.2.10) E]

2.4 Proof of Theorem (2.0.1) special case, (b*) : [Min(R0)| = 1

The goal here is to prove the following.
Theorem (2.0.1)(b"‘) Suppose that both R and M are Cohen-Macaulay. I = WAT).
Assume also that dim R0 = 1, Min(Ro) = {p} and for all W E Min(I), W 0 R0 = p.
Then,

HEJM) = (0) if and only ifi 75 grade(R+, M).

In particular, by Proposition (1.1.3)(c), for all i,
{A3330 (Hii+(M)n)}nEZ

is asymptotically stable.

Lemma 2.4.1. Assumptions as in Theorem (2..0.1)(b*)
Then, for all q E Max(R0),

 

, M
d1m (IN; S grade(R+, M).

q

28

Proof. Let q E Max(R0). By Lemma (2.3.4), it suffices to show that

M
d' —l<ht —htI.
1m qu_ R+

We note that both Mg and R, are Cohen Macaulay. (Rq)o (E’ (R0)q) is local and

equidimensional. Therefore, by Lemma (2.3.2),

dim 3113— S ht (q + R+) — ht (q + I)Rq. (2.10)
qu

Since no minimal prime of I R., contains (q + I )Rq

ht (q+I)Rq zht (I)+1. (2.11)
By Lemma (2.3.3),

ht (q + 12+): ht (12+) + 1. (2.12)
Putting the expressions (2.10), (2.11) and (2.12) together, we get,

M
dim-JShtR —htI.
qu +

Cl

Theorem 2.0.1(b*) Suppose that both R and M are Cohen-Macaulay. I = W.
Assume also that dim R0 = 1, Min(Ro) = {p} and for all W E Min(I), W n R0 = p.
Then,

H§+(M) = (0) if and only ifi 75 grade(R+, M).

In particular, by Proposition (1.1.3)(c), for all i,

{Ass};0 (H§+(M)n)}nez

is asymptotically stable.

29

Proof. Note that, R+M 75 M because (p+R+) E Suppp (M). Let g = grade(R+, M).
By Proposition (1.1.3)(a), it remains to show that for all i > g, Hfi+(M) = (0).
Fix i > 9. Suppose that H§+(M) 7é (0). Then, there exists q E Max(R0) and

n E Z such that
Hie+(M)n ®Ro (R0)q 75 (0)-

By the Flat Base Change Theorem ([5], 13.1.8, 15.2.2(iv)), this implies that

HR+Rq(Mq) 25 (0)-

Then, by Proposition (1.1.3),

M
dim—q—Zi,
q q

contrary to Lemma (2.4.1). [I]

2.5 Proof of Theorem (2.0.1)

The object here is to prove Theorem (2.0.1).

In Proposition (2.5.1), we will show that if we can break up Spec(Ro) into smaller
pieces and on each piece, we can answer the questions on asymptotic stability and
aymptotic ”gap freeness”, then we will have a better handle with the same questions
on the whole Spec(Ro).

In Propositions (2.5.2) and (2.5.3), we will show that if dim R0 = 1, then Spec(Ro)
can be decomposed into pieces where either the hypothesis of Theorem (2.0.l)(a“) or
(2.0.1)(b*) is fulfilled. To this end, we recall the hypotheses of the two special cases
of Theorem (2.0.1).

(a‘) Suppose that both R and M are Cohen Macaulay. Assume also that R0 is local
and dim R0 S 1.

(b*) Suppose that both R and M are Cohen Macaulay. Assume also that dim R0 = 1,
[Min(R0)| = l and for all W E Min(I), W 0 R0 E Min(RO).

30

Finally, we will prove Theorem (2.0.1).
Notation: For a closed multiplicative subset, S, of R0, we will identify Spec(S"1R0)
with the set,
{q E Spec(Ro) | q n S = (0}.

Proposition 2.5.1. Suppose that R and M satisfy the standard hypotheses. Let

31,. . . , Sp be closed multiplicative subsets of R0 with
Spec (=R0) U Spec(S

Fixi E No. Then,
(a) Iffor allj,
{AssSj—m.o (Hg?!2+ (SJ.‘1M),.)}nEz

is asymptotically stable, then

{Ass};0 (Hjh (M),,) }n€z

is asymptotically stable.

(b) Iffor allj,
H;;.R+(S;1M)

is asymptotically gap free as a Sj’lR-module, then HgJM) is asymptotically gap free.

Proof. Suppose that

{Asss— ‘Ro (Hg) .11, (SJ—1M)n)}

nEZ

is asymptotically stable for every j, 1 S j S 1:. Then, there exists N such that for all

n < N, and allj,

AssSj—1Ro (Hg- ,R (s,- 1M) ) =AssSj—1Ro (H;;.R+(SJT1M)N).

31

Fix n < N and let p E Ass};0 (Hf2+ (M)n) . Then,

11:
Spec (Re) = U Spec (5,712,)

implies that S;1 p E AssS—iROII1lg.J_( RJ,J'_(S 1M),,) for some j. Fix this j. There-
fore, S;1 p E Asss— 1R0 (IPS;R(SJT1M)N) so that, p E Ass};O (HR (M))v) . Conse-
quently,

AssRO (H24 (M),,) Q Ass);o (Hi;+ (M)N) .
For the other inclusion, let q E Ass);0 (HE (M) N) . Then,
I:
spec (Re) = U Spec (8.71120)
j=1
implies that S; 1q E A5333]?0 (HS-111+
Sflq E AssSt—i]?0 (H271,cz+ (St—1M)n) for all n < N so that, q E Ass};o (Hi;+ (M),,)

for all n < N. Consequently,

(St-1M)N) for some t. Fix this t. Therefore,

Ass)?0 (Hi;+ (M)N) Q ASSRo (Hi2. (Mln)

for all n < N.
For (b), suppose that
H§;1R+ (Si—1M)
is asymptotically gap free for every j, 1 S j S 1:.

Case 1: There exists 3' such that
H;;1R+(S;1M)n 7e (0)

for n << 0. Then, by the Flat Base Change Theorem ([5], 13.1.8, 15.2.2(iv)),
H},+ (M),, 72 (0) for n << 0.
Case 2: For all j,

Heads-1M). = (0)

32

for n << 0. Then, there exists N such that for all n < N,
Hbf‘m (Si-11W)” Z (0)
for all j. This is because we only have finitely many j.
Suppose that H}.2+ (M )n 7A (0) for some n < N. Fix this n. Then, there exists
p E Suppp0 (H}-1.+ (M),,) . Since
I:
Spec (R0) = U Spec (SflRo),
j=1
p E Spec (SJ-_IRO) for some j. Fix this j. Then,
He. (M). ea. 8,4120 s (0)
since p n S,- = 0. By the Flat Base Change Theorem ([5], 13.1.8, 15.2.2(iv)),
Hgflm (SglM)., 7s (0),
a contradiction to assumption. [1

Proposition 2.5.2. Suppose that dim R0 = 1. Then, there are closed multiplicative

subsets 8,, T,- Q R (1 S i S s, 1 S j S t) such that

Spec(Ro) = U Spec(SIIRo) u U Spec(fly‘lao) (2.13)

i=1

where,
1. for all i, [Min(SflROH = 1 and dim 5,7le = 1;
2. for allj, Tj’lRo is local and dim TflRo S 1.

Proof. Let
90 = Min(RO) fl MaX(R0)

and

fl] = {m E Max(R0) Im contains at least two distinct elements of Min(R0)}.

33

Min(Ro) is a finite set since R0 is Noetherian. Since we also have dim R0 = 1, for
any two distinct minimal primes of R0, there are at most finitely many m E Max(Ro)
that contains both. Consequently, 90 U 91 is a finite set.

Put
Min(Ro) - 90 := {p1, . . . ,ps}.

Min(Ro) — (lo # (I) because dim R0 = 1. For every i, 1 S i S s, we choose

1131' E n 193') —Pz'

[’1‘ #Pi

and define S,- :: {23? [n E No}. Then, for all i, |Min(S,-‘1Ro)| = 1 and dim SflRo = 1.

Furthermore, for each i, we put
1V.- := {q E Specl(Ro)| < we. 19.- >9 4}

Since ht};0 (< x,, p, >) = 1 or < 93,-, p,- >= R0, so M: is finite for all i. Therefore, we
can write

UMU90U91={m1,...,mt}.

i=1

Now, for 1 S j S t, put T,- = R0 — mj. Then, Tj‘lRo is local and dim Tj'lRo S 1.

With these choices of Si’s and Tj’s, we would like to show that the equation (2.13)
holds.
It is clear that

s t
Min(Ro) c U Spec(SflRo) u U Spec(Tji‘lRo).
i=1 j=1

Let q E Spec1(R0). Suppose that q E Spec(TJ-‘1R0) for all j. Then, q contains exactly
1 minimal prime, p,, for some p.- E (Min(RO) -— $20) and does not contain xi. Therefore,

q E Spec(S,‘1R0). El

34

Proposition 2.5.3. Suppose that dim R0 = 1 and Min(Ro) = { p }. Let J be a
graded ideal of R with ht (J 0 R0) = 0. Assume that

E 2: {q E Specl(R0) | q = W 0 R0 for some W E Min(J)} ¢ (ll.

Then, there exist closed multiplicative subsets, Q). C R0 (1 S k S l), and x E (R0 -p)

such that 1
Spec(Ro) = Spec(Rolx u U Spec(nzla.)
where L1
1. for all k, (ZERO is local and dim 9,:1R0 = 1.
2. dim (R0),B S 1 and for all W E Min(JRa), (W O (R0)x) = p(R0)x.

Proof. Note that all minimal primes of J are graded ([6], 1.5.6). Since Min(I) is a

finite set, 2 is a finite set. We choose

”(03:)":

Then, dim (Ro)x S 1 and for all W E Min(JRx), (W n (R0)x) = p(Ro),,. We can
(and will) identify (R¢)o with (Ro)z.

Let {q1, . . . , qz} consists of all the prime ideals of R0 that contains 1:. For all k,
1 S k S l, we put

Qk=Ro—Qk-

Then, dim leR0 = 1 and HERO is local for all k. It is now clear that
1
Spec(Ro) = Spec(Ro),, U U Spec(QEIRo)
k=l

because if a prime, q, in R0 contains 2:, then q E Spec(leRo) for some k; if q does

not contain x, then q E Spec(Ro)x. Cl

35

We recall Theorem (2.0.1), Theorem (2.0.1)(a*) and Theorem (2.0.1)(b*).
Theorem 2.0.1 Suppose that both R and M are Cohen-Macaulay.

Assume also that dim R0 :2 1. Then, for all i,

{ASSRO (H};+(M)n) }nEZ

is asymptotically stable.
Theorem 2.0.1(a*) Suppose that R and M are Cohen—Macaulay. Assume also that
R0 is local and dim R0 S 1.Then, for all i,

{ASSRO (HR... (M)n) }nEZ

is asymptotically stable.
Theorem 2.0.1(b") Suppose that R and M are Cohen-Macaulay. Assume also that
dim R0 = 1, Min(Ro) = {p} and for all W E Min(I), W (1 R0 = p. Then, for all i,

{ASS}?O (HAL/14)") }nEZ

is asymptotically stable.

We will now prove Theorem (2.0.1).

Proof. Recall that I = {/annp (M). Fix t E No. We want to show that

{ASS}?0 (Hiu (Mlnl }nEZ

is asymptotically stable.
Case 1: ht (I 0 R0) = 1. This is done in Proposition (2.0.1).
Case 2: ht (I (1 R0) = 0. By Proposition (2.5.2), there are closed multiplicative

subsets 5., T,- 9 R0 (1 S i S s, 1 Sj S t) such that

. z
SpeC(Ro) = U SpeC(S.-‘1Ro) U U SpeCU’j’lRo)
i=1

i=1

36

where
(l) for all i, [Min(SflRoM =1 and dim SflRo =1;
(2) for all j, Tj‘lRo is local and dim TflRo S 1.

We wish to show that for all i, j, both

{AssSi—i}?O (Hg; R+(S,"1M),,) }nEZ and {AssTj 1R0 (Ht;1 R+(Tj‘1M)n)}

1262

are asymptotically stable. Then, by Proposition (2.5.1), we have our contention.
By Theorem (2.0.1)(a"‘), for all j,

{Assay—1130 (HT-"111+(TJ—1M)n)}nez

is asymptotically stable. Note that the Cohen-Macaulay condition on R and M is

preserved under localisation. Hence, it remains to consider the case,

dim R0 = 1 and Min(Ro) = {p}.
Let
2 := {q E Specl(R0) I q = W 0 R0 for some prime W E Min(I)}.

Case 201: E = 0. Then, for all W E Min(I), W 0 R0 = p. Then, by Theorem
(2.0.1)(b"),

{A8330 (H}.+(M),,)}nEz
is asymptotically stable.
Case 20: 2 75 0. By Proposition (2.5.3), there exist multiplicative closed sets {9),};l
in R0 and x E (R0 — p) such that

z
Spec(Ro) = Spec(R0)x U U Spec(leRo)
k=1

where

(1) for all k, leRo is local and dim Q;1R0 = 1;

37

(2) for all W E Min(IRx), (W (i (R0)x) = p(Ro)x and dim (Ro)z S 1.
Recall that, (RAD is identified with (R0)x. Once again, by Theorem (2.0.1)(a‘) (for
the case where dim (R0)x = 0) and Theorem (2.0.1)(b*),

{Ass(Ro), (Hiz+34(Mx)71)}nez

is asymptotically stable. Note that Min((R0)m) = {p(R0)x}.
By Theorem (2.0.1)(a"), for all 1 S k S l,

—1
{A8895 1R0 (Hill 1mm" M)") }nEZ
is asymptotically stable. Therefore, by Proposition (2.5.1),
{Ass};O (Hli+(M)")}nez

is asymptotically stable.

2.6 Corollaries to Theorem (2.0.1)

We recall
Theorem 2.0.1 Suppose that both R and M are Cohen-Macaulay. Assume also that
dim R0 = 1. Then, for all i,

{AssR0 (Hg (M)n)}nEZ
is asymptotically stable.

Corollary 2.6.1. Suppose that both R and M are Cohen-Macaulay. Assume also
that dim R0 = 1.

Then:

(a) For all i, H}2+(M) is asymptotically gap free;

(b) for all i, AssR (H};+(M)) is afinite set.

38

Proof. For i = 0, H}2+(M) -—- FR+(M) which is a submodule of M. It is finitely
generated over R and hence both (a) and (b) are satisfied.
Fix i E N. Then, by Theorem (2.0.1), there exists N such that for all n S N,

ASSRO (H}{+(M)n) 2: A535,»,0 (H}2+(M)N).

Note that,

H}3+(M)n = (0) if and only if AssRO (H}h(M)n) = (ll.

Therefore, H}3+(M) is asymptotically gap free.

As for (b), we recall the bijection between

AssR(H;',+(M)) <—> UnezAssRo (H;,+(M)..).

Since H}{+ (M)n = (0) for n >> 0 (cf. [5], 15.1.5), so by Theorem (2.0.1),

U,Ez Asst». (Hi2.(M)..) is finite.
Therefore, Ass);O (H}{ + (M )) is finite. Cl

Corollary 2.6.2. Suppose that both R and M are Cohen-Macaulay. Assume also
that R0 is semi-local and dim R0 = 2. Then, for all i, AssR (H}z+ (M )) is a finite

set.

Proof. Fix i E No. Let W consists of all height 2 maximal ideals of R0. We choose

I E n m — U pi-
mEW ngMin(Ro)

Let T = {p + R+ I p E Spec(Ro) and x E p}. T is finite because R0 is semilocal,
Noetherian, dim R0 = 2, and htpO (x) = 1.

39

Let Q = {x" | n E No}. Now, we put

Rb = Q—IRO R, : RblR+l
R'+ = Q—1R+ 1W: M (8);; RI

Then, dim R}, = 1. Both M’ and R’ are (graded) Cohen-Macaulay. By Corollary
(2.6.1)(b), AssR: (H}v+(M’)) is a finite set. Since

Assp;(H}2,+(M’)) ={Q'1PIP E Ass}; (H}2+(M)) and P 0 Q = 0},

and

AssR(H}1+(M))§ {PIP E AssR(H}Z+(M)) and Phil = 0} UT

so, AssR (H}2+(M)) is a finite set. E]

40

3 Asymptotically gap free if dim R0 2: 2

Throughout this chapter, in addition to the standard hypotheses on R and M, we
will also assume that (R0, m0) is local and dim R0 = 2. The object of this chapter is
to Show that under some conditions, H}2+(M) is asymptotically gap free in the case
where dim R0 = 2.

The first section will show that under certain conditions, I‘mop (H}z+(M)) is
Artinian. The final section will show that under some restrictions on Assn (H g1 (M )),
H}{+ (M) is asymptotically gap free. This will be an extension of a result of Brodmann
and Hellus, Proposition (1.1.1), to the case dim R0 = 2.

We would like to begin by proving the following Lemma. The Lemma relates
the notion of a R-module being Artinian to it being asymptotically gap free. This
observation was used in the paper by Brodmann and Hellus ([2], 4.1(ii)). We include

a proof here since it will be used in some of our arguments later.

Lemma 3.0.1. Assume the standard hypothesis on R in this paper. Let T be a graded
R-module. Then, if T is either Artinian or Noetherian, then T is asymptotically gap

free.

Proof. If T is Noetherian, then there are homogeneous elements xfis in T such that
T =< x1,...,x,c > over R; since R is positively graded, T, = (0) for all n <
min{degree (x,)}}:’f
Assume that T is Artinian. Suppose that T is not asymptotically gap free. Then,
the set
9 1: {n 6 Z<o I Ta 75 (0) and Tn+1 = (0)}
is infinite.

Since R is positively graded and R+ is generated by degree 1 elements, Tn is a graded

41

R-module (R+T,, = (0)), for all n E (2. For N E Z, we put

T(N) 2: ® Tn

116!) and n<N
Then, T(N) Q T(N — 1) for all N E Z, gives us a decreasing chain of graded R-

submodules in T that will not stablise, contrary to T being Artinian. E]

3.1 When is I‘m}; ( }2+(M)) Artinian?

In this subsection, we would like to give some sufficient condition to guarantee that
I‘moR (H in (M l )

is Artinian. This has some interest in its own right. Since, there are R—modules that
are only supported at a maximal ideal, but are not Artinian (cf. [?]).
The following lemma will be useful in the proof of the main proposition in this

subsection.

Lemma 3.1.1. Assume the standard hypotheses on R, R0 and M in this chapter.

Let x0, yo be a system of parameters for R0. Then,

FiloR (H(R+J:o)(M))

is an Artinian graded R-module for all i E No.

Proof. Fix i E No. We want to Show that

F3103 (H(R+,xo)(M))

is a homomorphic image of an Artinian graded R-module.

Consider the following exact sequence (cf. [5], 13.1.12),

Hi

(Hi-$3013.10)

(M) ——* H(R+,xo)(M) —’ HlR+,zo)(M)yo-

42

This induces the exact sequence of graded R-modules,

Hi

(R+’IO)yO

)(IW) —" I‘yoR (H(R+,:co)(M)) —_) 0-

By ([5], 7.1.4),
R

(3+, 130, yo)

 

is Artinian implies that
HZR+Jo.3/o)(M)
is Artinian. Therefore,

FIJOR (HzR+,20) (M))

is a homormorphic image of an Artinian R—module.

It is clear that the module is graded. El

Proposition 3.1.2. Assume the standard hypotheses on R, R0 and M in this chap-

ter. Fixi E No. Suppose that
(a) AssR (HE1(M)) is finite and
(b) AsaR0 (Hg1(M),,) n Min(RO) : 0 for n << 0.
Then,
I‘molit (Hlt+(M))

is an Artinian graded R-module.

Proof. The idea is to show that I‘mop (H}{+(M)) has a submodule .A such that both

1‘... H‘ M .
A and 0“ AR+( )) are Artinian R—modules.

 

First, we show that we can choose 2 elements x0, yo in R0 such that

FmoR(HR+(M)) = FyaR(1"eoR(Hiz,(M))) -

43

Define

U 2: {pt [pt E ASSRO(H}Z:1(M)n) for some n and htR0 (pt) 2 1} U {m0}.

Note that U is a finite set and we choose

$06 flpj_ U P1-

pJ-EU szMin(Ro)

Then, htp0 (x0) = 1. Since dim R0 = 2, we can find yo E mo such that
(330, yolRo = mo-
We note that for all R-module T, and any ideal a of R,

Ffi(T) : Pa(T)'

Therefore,
an(Hia.(M)) = PW(H;'.+(M)).
Since
PWWMM» = Fae (I‘aa<Ha.<M))).
we have

PntoR(Hl{+ (M)) : I‘yoR (onR(Hi{+(M))) -

Consider the exact sequence of graded R—modules (cf. [5], 13.1.12),

i— ‘1’ i i i
HR+1(M)330 -—-—> H(R+,I0)(M) ——> HR+(M) —_) HR+(M)30'

The sequence (3.5) induces the exact sequence of graded R-modules,

O ——> Image (P ——> H}R+,xo)(M) ———> I‘xop(H}2+(M)) ———) 0.

44

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

We can then apply the left exact functor I‘yop(o) to sequence (3.6) to get another
exact sequence of graded R-modules (cf. [5], 1.2.2, 12.3.3),

. q, i
PyoR(H(R+,xo)(M)) ——> PyoR (FxoR(HR+(A/I))) —-> H503 (Image <I>). (3.7)

By equation (3.4), sequence (3.7) can be written as

r..a<Hza.,..)<M>> —“’—> I‘moa(Hia(M)) —> H3.R(Imaee <2). (3.8)

We wish to show that both Image \II and Coker \II are Artinian.

First, we will show that Image \II is Artinian. With sequence (3.7),

I‘yoR (H{R+,xo)(M))
Ker \II '

 

Image ‘11 91

Moreover, equation (3.3) tells us that $0, yo is a system of parameters for R0, which
implies that FyOR (HlRJ..zo)(M)) is Artinian (cf. Lemma (311)). Therefore, Image \II
is Artinian.

Finally, we will show that Coker \II is Artinian. We begin by proving that it is
sufficient to prove that (HE1(M)IO),. = (0) for all n << 0.

Suppose that (HEI(M)30),, = (0) for n << 0.

Claim: (Coker \II),, = (0) for n << 0.

Proof. By sequence (3.5), (Image (1))" = (0) for n << 0. Therefore, for n << 0,
(H5012 (Image <I>))n = (0). This is because

(H1 R (Image <I>))n E’ H5030 ((Image <I>)n)

110

as Ro—modules (cf. [5], 13.1.10). Then, with sequence (3.8), (Coker \II),, = (0) for

n<<0. C]

Claim: Coker \II is Artinian.

45

Proof. Since H}Z+(M),, is a finitely generated Ro-module and H}Q+(M)n = (0) for
n >> 0 (cf. [5], 15.1.5), (Coker ‘11)” = (0) for n >> 0. This is because Coker \I! is a
graded quotient of I‘m}; (Hl‘a+ (M )) (cf. sequence (3.8)).

With this and the preceding Claim, (Coker ‘11)" at (O) for only finitely many n. Since
for each n, (Coker \II),, is a finitely generated Ro-module, we conclude that Coker \II

is a Noetherian R—module. Furthermore,
Suppp (Coker \Il) Q V(mo + R1,)

Since, Coker \II is a quotient of I‘mop(H}Q+(M)). Consequently, Coker \II is Artinian.
E]

This shows that it is sufficient to prove that (H}:(M)$0 )n = (0) for all n << 0.
We will check that this condition is actually satisfied.
By the definition of U (cf. expression (3.1)), for n << 0,

n p,- Q \/annRO(HjQ-+1(M)n).

ijU

 

Note that U is a finite set by assumption (a) in the hypothesis. Then, for n << 0,

 

x0 E \/annpO (HE1(M)n)

(cf. expression (3.2)). To see this, we consider the Graded Independence Theorem

([5], 13.1.6), it says that for all j,
HR+(M)$0 g HR+Rxo(M)x

0

as graded R-modules and the isomorphism is homogeneous. Furthermore, by ([5],

15.2.2(iii)), for all n E Z and j,

(Hi.a,(M)a)n s H£.(M)n ea. Ra (3.9)

46

as (R0)$0 (and hence R0)-modules. By the choice of 2:0,
HEW). ea. R... = (0)
as a RO-module for n << 0. Hence, for n << 0,
(Hi-.‘(Mtdn = (0)
(cf. isomorphism (3.9)). El
3.2 When is H}Z+(M) asymptotically gap free?
We will give some sufficiency conditions for H};+(M) to be asymptotically gap free.

Proposition 3.2.1. Assume the standard hypothesis on R, R0 and M in this chap-

ter. Fixi E No. Suppose that

(a) AssR (H;1(M)) is finite;
(b) ASSRO (Hg1(M),.) n Min(Ro) = 0 for n << 0,-
(c) Ass};o (H}2+(M),,) Q {m0} forn << 0.
Then,
ng (M) = A + N
where A and N are both graded R-submodules of H}2+ (M); A is Artinian and N is

Noetherian as R-modules.

In particular, by Lemma (3.0.1), H}{+(M) is asymptotically gap free.
Proof. Define
A :2 I‘mop (H}2+(M)).

By Proposition (3.1.2), A is an Artinian graded R-module. According to ([5], 13.1.10),

for all n E Z,
A“ : I‘moRo (Hi2+(M)n) -

47

Then, condition (0) in the hypothesis and the maximality of mo in R0 implies that
for n << 0,

I‘moRo (Hi2+(M)n) Z Hii+(M)n-

Therefore, there exists (2 E Z such that for all n < 52,

An : HRp(M)n
Define
N := 69 HAW),
712!)

Since H}Q+(M)n = (0) for n >> 0 and for all n E Z, H};+(M),, is a finitely generated
Ro-module (cf. [5], 15.1.5), N is a graded Noetherian R—module (R is positively
graded).

It is now clear that we can write H}{+(M) as A + N. C]
We will use Proposition (3.2.1) to prove the last Proposition of this chapter.

Proposition 3.2.2. Assume the standard hypothesis on R, R0 and M in this chap-
ter. Let x E R be a homogeneous element of degree 1 which is also M —regular. Fix

i E No. Suppose that

(a) AssR (HE1(;A1{7)) is afinite set;
(b) Asslq0 (HE1(%),,) fl Min(Ro) = (ll forn << 0;
(c) Ass]; (H}2+(M)) is afinite set.

Then, H}z+(M) is asymptotically gap free.

Proof. We may assume that H}z+(M)n % (0) for infinitely many n < 0. Otherwise,
H}{+(M) is Noetherian (cf. [5], 15.1.5) and hence, by Lemma (3.0.1), it is asymptot-

ically gap free.

48

We may also assume that Ass};O (H}{+(M)n) Q {mo} for n << 0. Otherwise,
there exists a p E Spec(Ro) of height 0 or 1, such that p E Ass);o (H}2+(M)n) for
infinitely many n < 0; by Proposition (1.1.1), H}2+(M)n @130 (Ra),D 76 0 for n << 0
since dimension of (R0),, is 1. Hence, H};+(M) is asymptotically gap free.

Since for all n E Z, H}{+(M)n is a finitely generated Ro-module (cf. [5], 15.1.5);

Ass);O (H;z+(M)n) Q SUPPRO (Hit+(M)n)

and mo is the maximal ideal of R0, we are in fact assuming

Supppo (H}{+(M)n) Q {mo}. (3.10)

We will continue with the proof with this additional assumption.

By our hypothesis, we have the exact sequence of graded R—modules,

O—>M(—1)——x——>M———>f‘,7——->O.

Then, the sequence above induces another exact sequence of graded R-modules,

H}2+(M)(—1) —“”> H},+(M) -—> HMEAAA, . (3.11)
Put .
C: Ha.(M>

Claim: It suffices to show that C is asymptotically gap free.

Proof. If C" 75 (0) for n << 0: Then H}l+(M)n 7E (0) for n << 0.

If C. = (O) for n << 0: Then for n << 0, we have an exact sequence of Ro-modules,
Hg+(M),. —“’—> H}{+(M)n+1 ——+ 0

Hence, H}{+(M)n+1 74 (0) implies that H}{+(M)n 7t (0) for all n << 0. Cl

49

We will now show that C is asymptotically gap free.

, M
AZ: HR+ (m)

Note that with sequence (3.11), C can (and will) be identified as a graded submodule

Define

of A. We also note that Suppp0 (C) Q {mo} for n << 0. This is because
Supper. (Ca) E Suppl:0 (HR+(M)n)
and expression (3.10). Hence, there exists N E Z such that for all n < N,
C. Q (I‘mop (A))n. (3.12)
For this, we recall that for all n E Z,

(P771013 (Alt. = Puma. (An)-

With expression (3.12), we can write
6 = (c n PM. (A)) + 69 c... (3.13)
nZN
By Proposition (3.1.2), I‘mop (A) is Artinian. Therefore, its submodule, C flI‘mo R (A),
is also Artinian. By Lemma (3.0.1), it is asymptotically gap free. More importantly,
by equation (3.13), for all n < N,

Cu 2 (C D I‘moR (A))n
Therefore, C is asymptotically gap free. C]

It will be more meaningful to see that the hypotheses of the two propositions that
we just proved are satisfied by some concrete rings and modules. To accomplish that,
we will apply the propositions to prove one of our main Theorems in the following

chapter.

50

4 Applications

We will appply our results in the previous chapter to prove the second main result in

this project:

Theorem 4.0.1. Suppose that both R and M are Cohen-Macaulay. Assume also
that (R0, m0) is local, equidimensional and dim R0 = 2. Then, for all i, H}{+(M) is

asymptotically gap free.

In Section (4.1), we will show that to prove Theorem (4.0.1), it suffices to prove
the Theorem with the additional hypothesis that the residue field, "51%, is infinite.
This condition will guarantee us a homogeneous M -regular element, x, of degree 1
whenever grade(R+, M) Z 1. In the context of Proposition (3.2.2), such an element
will be useful. We will not be needing this additional assumption till we actually give
a proof of Theorem (4.0.1) in Section (4.4).

In Section (4.2), we will state most of the Lemmas we proved in Chapter (1)
under the hypotheses of Theorem (4.0.1). We will need these Lemmas to perform
some of the calculations in Section (4.3). We will also include the three main results
of Brodmann and Hellus, which we have constantly utilised. The reader can refer to
Chapter (1) and [2] for all the proofs.

In Section (4.3), we will analyze the support of H}1+(M). From Corollary (2.6.2),
we know that Ass}; (H}z+(M)) is a finite set under the assumptions of Theorem

(4.0.1). In fact, we will show that for all i > grade(R+, M) and n E Z,
p E Suppp0 (H}2+(M)n) implies that htR0 (p) 2 1.

This will give a situation where condition (b) of Proposition (3.2.1) is satisfied.
In Section (4.4), we will prove Theorem (4.0.1) using the results in the preceding

two sections.

51

In Section (4.5), we will give a criterion for the vanishing of H}2+(M) using the
analysis made in Section (4.3).

We begin by proving the following Proposition that deals with the case
ht(I 0 R0) 2 1, where I = W. This is essentially due to Brodmann and

Hellus’s result, Pr0position ( 1.1.1).

Proposition 4.0.1. Assume the standard hypothesis on R and M. Suppose that R;
is local and dim R0 = 2. Let I = \/annp(M). If ht (I 0 R0) 2 1, then for all i,

H}; (M) is asymptotically gap free.

Proof. Fix i E No. Let I’ = annp(M) and R’ = %. Then, M is a finitely generated
graded R’ -module. R’ is a positively graded homogeneous N oetherian ring. R}, is
local and dim R}, S 1. By Pr0position (1.1.1), Hi ,+ (M) is asymptotically gap free
as a R’ (and hence, R)-module. By the Graded Independence Theorem ([5], 13.1.6),

H}Q+ (M) is asymptotically gap free. C]

With that, the more interesting part of our work will be in the case,

ht (I 0 R0) = 0. This will be the main focus in section 4.3.

4.1 Reduction to that case: [gig is infinite

We would like to use a standard construction (cf. [5], 15.2.4) to show that to prove
Theorem (4.0.1), we may assume additionally that the residue field, "3%, is infinite.

This construction does not use the Cohen-Macaulay condition on R or M, and the

equidimensionality of R0. We do assume that (R0, mo) is local. Put

Rb : ROlemoRo[Xl

where X is an indeterminate over R. Then, R6 is a faithfully flat local Ro-algebra

with an infinite residue field. Furthermore, both rings have the same Krull dimension.

52

Put

RIZR®R0R6 M’=M®RR’

m 2 mg + R+ m’ = m0R0[x] + (R+ ®po R6).

Then, (R’, m’) is a positively graded homogeneous and graded local N oetherian ring
with R}, as its O-th component. M’ is a finitely generated graded R’ -module. By ([5],
15.2.2(iv)), for all i E No and n E Z,

H},+(M),, 8% RS 2 HR+R'(M’)R (4-1)

as Rg-modules. Furthermore, R}, being faithfully flat over R4) implies that for every
n, H}Z+(M),, = (0) if and only if HARAM’)“ = (0). Therefore, H}q+(M) is asymp—

totically gap free if and only if H in R,(M ’ ) is asymptotically gap free.
Lemma 4.1.1. Assume that we have R, R0, M, R’, R6, M’ as above. Then
gradema M) = grade (R'., M').

Proof. let 9 = grade(R+, M) and g’ = grade (R’+, M’). If R+M = M, then by a
graded version of Nakayama’s Lemma (cf. [6], 1.5.24(a)), M = (0). Then, M’ = (0).
Hence, by definition, g = 00 and g’ = 00.

Consider the case that R+M ¢ M. Then, g E No. By Proposition (1.1.3)(a) and
(b), for all i < g, H}{+(M) = (0) and H}; (M) 76 (0). Hence, by the isomorphism in
expression (4.1), for all i < g, H},,+(M’) = (0) and H9,+(M’) 75 (0). Then, R’+M’ 75
M’. Another application of Proposition (1.1.3)(a) and (b) to R’ and M’, we get

I

gzg. D

Proposition 4.1.2. If both R and M are Cohen-Macaulay, then both R’ and M’ are

Cohen-Macaulay.

53

 

I

Proof. Since R’ is graded local, by ([6], 2.1.27), to show that both R’ and M’ are
Cohen-Macaulay, it suffices to show that both R6,, and M,’,,, are Cohen-Macaulay.
Consider the ring extension Rm ——-) R6,). The extension is local, flat and its fibre

R6,,
mRfin,

 

is a field. Then by ([6]. 1.2.16), for every finitely generated Rn-module, N, we have

mRin,
and

dim)... ,N en," 12;, = dimRmN + dimR: 3' '

I
m’ mRmI

. R’ , . R’ , . R’ , .
Smce $139; IS a field, depthpzn,;§9m—I and dlmR;,;}%:; are both zero. If we substitute

Rm (resp. Mm) for N, we will have that R’m, (resp. M6,.) is Cohen-Macaulay.
Cl

Finally, we would like to show that if R is Cohen-Macaulay and R0 is local, then
R; is equidimensional if and only if R6 is equidimensional. By Lemma (1.2.7), this is

equivalent to the following Proposition.

Proposition 4.1.3. If R is Cohen-Macaulay and R0 is local, then all minimal primes
of R+ have the same height if and only if all minimal primes of RC, have the same

height.

Proof. Let p’ E Min(R6) and p’ D R0 := p. By going-down (fiat extension), p E
Min(Ro). In our case, pR6 = p’ because pR6 is a prime ideal in R6. Note that, if
q E Min(RO), then qR6 :2 q’ is in Min(R6). Hence, there is one-to-one correspondence
between Min(Ro) and Min(R6), via contraction and extension.

We put P = p + R+ and P’ = p’ + R’+. Note that P and P’ are in Min(R+) and

Min(R’+) respectively.

54

Now, consider the extension

This is a local and flat extension whose fibre 7,51%- is a field. By ([6], 1.2.16),
PI

. . . R’
d1m R’ , —— d1m Rp + d1m—ifi
P PRP,

Therefore,

htP’=htP.

4.2 Auxiliary tools for Theorem (4.0.1)

We collect all the preliminary tools we need in order to do the analysis in Section
(4.3). The proof of these Lemmas and Propositions can be found in Chapter ( 1). We
recall the assumptions in Theorem (4.0.1):

Suppose that both R and M are Cohen-Macaulay. Assume also that (R0, mo) is local,

equidimensional and dim R0 = 2.

Lemma 4.2.1. Assumptions as in Theorem (4.0.1). I = {/annp (M).

Then, all minimal primes of I have the same height.
Proof. We refer to Lemma (1.2.2). [:1

Lemma 4.2.2. Assumptions as in Theorem (4.0.1). I = {/annR(M). Let p E
Spec(Ro) and p 2 (I 0 R0). If there exists W E Min(I) such that W {'1 R0 = p, then

ht (p+ I)R,, = ht 1.

Proof. We refer to Lemma (1.2.3). E]

55

Lemma 4.2.3. Assumptions as in Theorem (4.0.1). I = annp(M).

Then, ifq E Spec(Ro) and q 2 (I (1 R0), then

. M
d1m FIE}; S ht (q+ R+) — ht (q+ I)Rq.

In particular,

 

(1’
1m moM

Proof. We refer to Lemma (1.2.6).

Lemma 4.2.4. Assumptions as in Theorem (4.0.1).

Then, all minimal primes of R+ have the same height.

3 ht (mg + 12..) — ht (mg + 1).

Proof. R0 is equidimensional implies that all minimal primes of R+ have the same

height (cf. Lemma (12.7)).

Lemma 4.2.5. Assumptions as in Theorem (4.0.1).

Then, ifq E Spec(Ro) and ht q = i, then

Proof. We refer to Lemma (1.2.8).

Lemma 4.2.6. Assumptions as in Theorem (4.0.1). I = {/annp (M).

If ht (IflRo) = i, then

Proof. We refer to Lemma (1.2.9).

Lemma 4.2.7. Assumptions as in Theorem (4.0.1). I = {/annR(M).

Ifq E Spec(Ro) and q 2 (I 0 R0), then

grade(R+, M) S grade(R+Rq. Mq)-

56

Cl

Proof. We refer to Lemma (1.2.11). E]
Proposition 4.2.8. (Brodmann-Hellus) Suppose that dim R0 S 1, and R0 is semilo-

cal. Then, for every finitely generated graded R-module M, H};+ (M) is asymptotically

gap free.
Proof. (cf. [2], 4.2) 1]

Proposition 4.2.9. (Brodmann-Hellus) Suppose that (R0, mo) is local and M is fi-

nitely generated graded R-module with dim fi = d. If M aé I‘R+(M), then

(a) Hi.(M) a (0) and
(b) H}2+(M) for alli > d.

Proof. (cf. [2], 3.4) C]

Proposition 4.2.10. Suppose that R+M 7é M. Let g = grade(R+, M). Then:
(a) For alli < 9, H}}+(M) = (0) ,'

(b) forz' = g, H3..(M) a (0).-

(c) (Brodmann-Hellus) if H}z+(M) is finitely generated over R for alli < t, then

{ASSRo (Him (Mlnllnel
is asymptotically stable.

Proof. For (a) and (b), (cf. [5], 6.2.7). For (c), (of. [2], 5.6). E]

4.3 The support of H}2+(M)

We recall the assumptions in Theorem (4.0.1).
Suppose that both R and M are Cohen-Macaulay. Assume also that (R0, mo) is local,

equidimensional and dim R0 = 2.

Let I = {/annR(M) as before. For a prime, q E Spec(Ro), containing (I (1 R0), we

begin by analyzing the two dichotomy:

57

(a) q # (W (1 R0) for all W E Min(I);

(b) q = (W 0 R0) for some W E Min(I).

We do this by case study in terms of the height of q. These findings are summarized
in Proposition (4.3.4).
In view of Proposition (4.0.1), the case ht (I 0 R0) = 0 is at the heart of the issue

here.

Lemma 4.3.1. Assumptions as in Theorem (4.0.1). I = {/annR(M).
Let q E Min(Ro).

(a) Iffor all W E Min(I), W (1 R0 # q, then Mg 2 (0).
(b) If there exists W E Min(I) such that W 0 R0 = q, then

M
dim—1S adeR,M.
qu gr (+ )

Proof. For (a), suppose that for all W E Min(I), W F) R, aé q. Then, q ,7) (I (1 R0).
Otherwise, (q + R+) contains a W E Min(I) such that W 0 R0 = q because q E
Min(Ro). This is contrary to assumption. In particular, (I 0 R0) F) (R0 - q) 75 0.
Therefore, M9 = (O).

For (b), suppose that there exists W E Min(I) such that W 0 R0 = q. Then,
ht (I 0 R0) = 0. By Lemma (4.2.6),

grade(R+, M) = ht (R+) — ht I.
According to Lemma (4.2.3),
. Mq
d1m — S ht (q + R+) — ht (q + I)Rq.
qu

By Lemma (4.2.2),
ht (q+ DR, 2 ht I.

58

By Lemma (4.2.5) and q E Min(RO),
Therefore, putting the preceding four expressions together, we get

M
dim-—q< adeR,M.
qu_gr (+ )

Lemma 4.3.2. Assumptions as in Theorem (4.0.1). I = {/annR(M).
Assume also that ht (I 0 R0) = 0. Let p E Specl(Ro) and p D (I 0 R0).

(a) Iffor all W E Min(I), W 0 R0 75 p, then

, M
d1m j S grade(R+, M).

P

(b) If there exists W E Min(I) with W I’i R0 = p, then

M
dim—1S adeR,M +1.
pMp ST (+ )

Proof. For (a), by lemma (4.2.6), it suffices to show that

M
d' ——”—<h — .
1m pMp _ t (R+) ht I

According to Lemma (4.2.3),

dim fl S ht (p + R...) - ht (p + I)Rp. (4.2)
pMp

By assumptions,

ht (p+I)R1D 2 ht (I) + 1.

Since p E Specl(R0),

ht (p + R+) = ht (R1,) + 1. (4.3)

59

(cf. Lemma (42.5)). Therefore, putting the preceding three expressions together, we

get

 

M
dim P<htR —htI.
M. (+)

For (b), we note that we still have inequality (4.2) and equation (4.3). By Lemma

(4.2.2),
ht (p+ I)R1D = ht (I).

Putting these three expressions together, we get

 

M
dim ”shtR —htI+1
pMp (+) ()

Since grade(R+, M) = ht (R+) — ht I (cf. Lemma (42.6)), we have our contention.
1]

Lemma 4.3.3. Assumptions as in Theorem (4.0.1). I = {/annR(M).
Assume also that ht (I (1 R0) = 0.

(a) Iffor all W E Min(I), W 0 R0 74 mo, then

 

M
dim M S grade(R+, M) + 1.

mo

(b) If there exists W E Min(I) with W (1 R0 = me, then

 

M
dim M S grade(R+, M) + 2.

mo

Proof. For (a), by lemma (4.2.6), it suffices to show that

 

dim MM 3 ht (13,) — ht (I) + 1.

mo

According to Lemma (4.2.3),

 

dim S ht (mo + £24.) — ht (m0 + I). (4.4)

mo

60

By assumptions,
ht (mo+I)_>_ht (I)+1. (4.5)
Since mo E Spec2(R0),
ht (mo + 12,.) = ht (R+) + 2. (4.6)

(cf. Lemma (42.5)). Therefore, putting inequality (4.4), equations (4.5) and (4.6)

together, we have

 

' < h — I .
d1m moM" t(R+) ht( )+1
For (b), we note that we still have inequality (4.4) and equation (4.6). By Lemma

(4.2.2),
ht (mo + I)R,,IO = ht (I). (4.7)
Putting these three expressions together, we get
dim—M— < (htR )—ht (I)+2
moM ‘ +

Since grade(R+, M) = ht (R+) — ht I (cf. Lemma (42.6)), we have our contention.
Cl

Assumptions as in Theorem (4.0.1). As before, I = m and
ht (I 0 R0) = 0. Let i E No. According to the three Lemmas just shown, for a prime p
to be in Supp R0 (H}2+ (M )n) , it is important to know whether there exists W E Min(I)
such that W 0 R0 = p. This condition determines an upper bound on dim p113; in
terms of grade(R+, M). For its relevance, we recall that if Hit+R,,(Mp) 71$ (0), then

MP
I’M p

 

grade(R+Rp, Mp) S i S dim

61

“F.
:J

.fih'

(cf. Proposition (42.9)). Moreover, by Lemma (4.2.7),
gradema M) s gTade(R+Re, Mp).

Hence, if H6+Rp(Mp) 7é (0), then

 

Define
A := {p E Specl(R0) I there exists W E Min(I) with W (1 R0 = p}.

If A 74 (ll, we put A := {p1 . . .ps}. We further define

~__ A ifforallWEMin(I),WflRoaém0
._ A U {mo} if there exists W E Min(I) with W n Ro = mg.

Notation: For i E No, we denote

S,- := UnEZ SupppO (H}2+(M)n) .

Proposition 4.3.4. Assumptions as in Theorem (4.0.1). Let g = grade(R+, M).
If ht (I 0 R0) = 0 then,

a H’- M = 0 foralli>g+2andi<g.
R...

(b)
Sg+1 C {V091 n-ans) ifé 75 0;
— V(mo) = {mo} ifA 9 {mo}.

(6)

{mo} if mo E A;
89” Q {0 if m0 E A.

In particular, fori = g + l, g + 2, and for all n E Z,

Suppa. (Hi..(M).) r) Min(Ro) = 0;

<00.

 

[Una SUPPRO (Hi2+ (M)n)

62

Proof. We note that g E No because M is Cohen-Macaulay implies that M 75 (0)
and hence, by a graded version of N akayama Lemma ([6], 1.5.24(a)), R+M 3A M. We
may also assume that M 75 PR+ (M). Otherwise Hg (M) 2 (0) for all z' > 0.

We note that Hfi+(M) = (0) if and only if S,- = (b.

For (a), by Lemma (4.3.3), dim EMA? S g + 2. Moreover, by Proposition (4.2.9),
for all 2' > dim 731117, sz+(M) = (0). Therefore, H§+(M) = (O) for all 2' > g + 2. For
2' < 9, Hlb. (M) = (0) (cf. Proposition (4.2.10)(a)).

For (b) and (c), let q E SuppRo (H§+(M),.) fl Min(Ro). We want to show that
i g 9. Note that, this is clear if i = 0 because 9 E No. We consider the situation
where i 2 1. By the Flat Base Change Theorem ([5], 13.1.8), H§+Rq(Mq) 75 (0).
Then, Mq aé I‘qu(Mq) and M9 75 (0). By the contrapositive statement of Lemma
(4.3.1)(a), there exists W E Min(I) such that W 0 R0 = q. By Lemma (4.3.1)(b),
dim q—MM’; S g. Since we also have i S dim $443; by Proposition (4.2.9). Hence, 2' S 9.

For (b), we will first show that if p E 89“ fl Spec1(Ro), then p E A.

Let p E 89“ fl Specl(Ro). Then, M,D 76 (0) and hence, p 2 (I 0 R0). Furthermore,
g + 1 > 0 implies that Plug, (Mp) 75 Mp. By the Flat Base Change Theorem ([5],
13.1.8) and the choice of p, Hfik(Mp) 75 (0). By Proposition (4.2.9), dim {—14% 2
g + 1. By the contrapositive statement of Lemma (4.3.2)(a), there exists W E Min(I)
such that W 0 R0 = p. Therefore, p E A.

Suppose A = 0. Then, 89“ fl Specl(Ro) = 0. If u E 89“, then u = mo because R0 is
local and dim R0 = 2.

For (c), we want to show that if v E 89”, then 1) = mg.

Let ’0 E 89”. Then, Mv # (O) and hence, v Q (I 0R0). We also have Hfl2(Mv) 75 (0)
by the Flat Base Change Theorem ([5], 13.1.8). By Proposition (4.2.9), dim 31%;: 2
9+2. By Lemma (4.3.1) and (4.3.2), htR0 (v) 2 2. Hence, 1) 2 mo because dim R0 = 2
and R0 is local.

63

Moreover, there exists W E Min(I) such that W 0 R0 = 1). Otherwise, by Lemma
(4.3.3)(a), dim fl”; 5 g + 1. a contradiction to dim 1%}: 2 g + 2.
Therefore, if H g2(M ) 7e (0), then mo E A.

Finally, to see that

 

<00.

lUnEZSuppR. (Hi..(M).)

for all i > 9. We note that both V(p1 fl . . . n p,) and V(mo) are finite because for all
1 S j S s, p, E Spec1(Ro); R0 is local and dim R0 = 2. C]

There is another ramification of this analysis besides an application of Proposition
(3.2.1) to the proof of Theorem (4.0.1). We will explore it in Section (4.5), after the
proof of Theorem (4.0.1).

4.4 Proof of Theorem (4.0.1)

Theorem 4.0.1. Suppose that both R and M are Cohen-Macaulay. Assume also that
R0 local, equidimensional and dim R0 = 2. Then, for all i, H§+(M) is asymptotically
gap free.

Proof. We may assume that FR.» (M) 76 M. Otherwise, Hk+(M) = (O) for all i > 0.
Let g = grade(R+, M). M 74 (0) implies that g E No. As before, I = W.
The case where ht (I 0 R0) 2 1 is dealt with in Proposition (4.0.1). It remains to

consider the case ht (I 0 R0) = 0. By Pr0position (4.3.4), for all i > g and n E Z,
Supp... (Ham) 0 Minot.) = 0 (4.8)
and

< 00. (4.9)

 

lUnez SUPPRO (Hh+(M)n)

64

By Proposition (4.2.10)(c) and (4.3.4)(a), it remains to check for i = g + l and
i=g+2
Case 1: i = g + 2. H192:2(M) is asymptotically gap free because the hypotheses of

Proposition (3.2.1) are satisfied:

(a) Ass}; (Hfil(M)) is finite (cf. expression (4.9)).
(b) AssR0 (Hfil(M)n) fl Min(Ro) = 0 for n << 0 (cf. expression (4.8)).

(c) Ass};0 (Hg2(M)n) 9 {m0} (cf. Proposition (4.3.4)(c)).

Case 2: i = g + 1. We will examine the cases 9 = O and g 2 1 separately.

Case 2(0): 9 = 0. Then 9 + 1 = 1. H}{+(M) is always asymptotically gap free

because it is either finitely generated over R or the first non-finitely generated local

cohomology of M with respect to R+ (cf. Proposition (42.10)).

Case 2(6): 9 2 1. By Section (4.1), we may assume that the residue field, 5%, is

infinite. Then, by ([5], 15.1.4), there exists :1: E R1 that is M -regular. Now z—MM is still

a Cohen-Macaulay graded R-module and grade(R+, %) = g — 1 (cf. [6], 1.2.10(d)).
Let I ’ = ‘ / annR (%). H fiKM ) is asymptotically gap free because the hypotheses

of Proposition (3.2.2) are satisfied:

(a) AssR (Hi(% ) is a finite set and
(b) Asst;0 (H§+(;MM n) n Min(Ro) = 0 for n << 0 because if ht (1’ (1 R0) = O, we
simply apply expressions (4.8) and (4.9) to %; if ht (I’ 0 R0) 2 1, we recall that

M M
SUPP}: (High—4'0 Q SUPPR (EM) 0 V(R+)-

(c) AssR (Hfiil(M)) is a finite set (cf. expression (4.9)).

65

Corollary 4.4.1. Suppose that both R and M are Cohen-Macaulay. Assume also
that R0 is semi-local, dim R0 = 2 and for all p E Spec2(Ro), (110),, is equidimensional.

Then, for all i, H§+(M) is asymptotically gap free.

Proof. Fix i E No. Let Max(R0) 2 {p1, . . . ,pk}. Let pj E Max(R0) and S, = R0 —pj.
If htRo (p,-) S 1, then

thJMp)
is asymptotically gap free by Proposition (4.2.8). If ht};0 (p,-) = 2, then by Theorem
(4.0.1),

112..., (M...)
is asymptotically gap free. Since

Spec(Ro) = U Spec(S:‘Ro)

meMaxmo)

and as shown above,

H;;1,.+(S.“M)
is asymptotically gap free for all t, so H§+(M) is asymptotically gap free (cf. Propo-
sition (2.5.1)(b)). [1

Lemma 4.4.2. Suppose that R is regular and (R0, m0) is local. Then, R0 is a do-

main.

Proof. Let a and b be two elements in mg such that ab = 0. Then, 919 = 0 in Rm,
where m 2 mo + R+. Since Rm is regular, it is a domain. Then, either 515 = O or
% = 0. If % = 0, then there exists 3 E m such that ma = O. The O—th component of a:

is a unit implies that a = 0. Similarly, for % = 0. Consequently, R0 is a domain. Cl

Corollary 4.4.3. Suppose that R is regular and M is Cohen-Macaulay. Assume also
that R0 is semilocal and dim R0 = 2. Then, for all i, H§+(M) is asymptotically gap

free.

66

Proof. R is regular implies that R is Cohen-Macaulay. Furthermore, for all p E
Spec2(Ro), RP is regular. By Lemma (4.4.2), (Rp)o is a domain and thus equidimen-

sional. By Corollary (4.4.1), we have our result. Cl

4.5 Criterion for H§2+(M) to be non-vanishing

Assumptions as in Theorem (4.0.1). As before I = ‘/annR(1l/I) and g = grade(R+, M).
We wish to give a criterion for the vanishing of HEJM).
We first recall the definition of A and A in Section (4.3).

A z: {p E Spec1(R0) |there exists W E Min(I) with W 0 R0 = p}.

and

~._ A ifforallWEMin(I),WnR07émo
_ A u {m0} if there exists W e Min(I) with W n R0 2 m0,

Our goal in this section is to prove Proposition (4.5.2). Before we do that, we

need a lemma.

Lemma 4.5.1. Suppose that R satisfy the standard hypothesis.Let P := (p + P+)
be a graded prime ideal of R, where p = P H R0 and P+ = P n R+. Ifq 2 p and
q E Spec(RO), then q + P+ E Spec(R).

Proof. Let q 2 p and q E Spec(Ro). Put Q := q+ B... Let a, b E R such that ab E Q.
We wish to show that either a E Q or b E Q. Since Q is graded, it suffices to show it
for the case where both a and b are homogeneous. Note that P g Q.

Case 1: deg (a) > 0 and deg (b) > 0. Then, deg (ab) > 0 and ab E P+. Since P is a
prime ideal, a E P or b E P. In other words, a E Q or b E Q.

Case 2: deg (a) = 0 and deg (b) = 0. Then, deg (ab) = O and ab E q. Since q is a
prime ideal, a E q or b E q.

Case 3: deg (a) = 0 and deg (b) > 0. Then, deg (ab) > 0, implies that ab E P+. P

67

is a prime ideal implies that a E P or b E P. Since P C Q, we are done.

Case 4: deg (a) > 0 and deg (b) = O. This is symmetrical to the previous case. [3

Proposition 4.5.2. Assumptions as in theorem (4.0.1). I = annR(M) and g =
grade(R+, M). IfA = a) and ht (I n R0) = 0, then

H§+(M) = (0) if and only ifi 74 9.

Proof. M 7é (0) because M is Cohen-Macaulay. If FR+(M) 2: M, then 9 = O and
Hi (M) = (0) for all i > 0. Therefore, we will consider the case PR+(M) 75 M.
Let dim 'ef—M = d and m = mg + R+. By Propositions (4.2.9) and (4.2.10), it suffices

to show that

 

dim _

<.
7720 9

By Lemma (4.2.6), it suffices to show that

 

dim

M
M g ht (12+) -ht I.

mo

By Lemma (4.2.3),

 

M
moM g ht (mo +R+) — ht (mo+I).

dim
By Lemma (4.2.5),
ht (mo + R4,) = ht (12+) + 2.
All we need to do next is to show that
ht (mo + I) 2 ht (I) +2.

Let Q be a (graded) minimal prime ideal of (mo+I) such that ht (mo+I) = ht Q.
Note that mo 2 (Q 0 R0). Then, Q Q P, a minimal prime over I. Since A = 0 and
ht (I 0 R0) = 0, so p := P 0 R0 E Min(Ro). Now, put P+ :2 (P n R+). Since Rois

equidimensional, there exists u E Spec1(R0) such that

pCquo.

68

By Lemma (4.5.1), we have a chain of primes in Spec(R):

Note that, Q 2 (mo + P+) because (mg + P+) is a prime ideal that contains (mg + I ),
(mo + P+) g Q, and Q is minimal over (mo + I). By Lemma (4.2.1), all minimal

primes of I have the same height. In particular,
ht (p+ R.) = ht I.

Therefore,

th=ht(mo+P+)th(I)+2.

69

References

[1] , M. Brodmann, ST. thasoli and R. Tajarod, Local cohomology over homoge-

neous rings with one-dimensional local base ring, preprint.

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[3] M. Brodmann, M. Katzman and R. Sharp, Associated primes of graded compo-
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Mathematical Society.

[4] M. Brodmann, C. Rotthaus and R. Sharp, 0n annihilators and associated primes

of local cohomology modules, to appear in Journal of Pure and Applied Algebra.

[5] M. Brodmann and R. Sharp, Local cohomology - an algebraic introduction with
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