F-stable submodules of top local cohomology modules of Gorenstein rings

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Katzman, M. (2006)

F-stable submodules of top local cohomology modules of

Gorenstein rings

arXiv.org

arXiv:math/0602550v1 [math.AC] 24 Feb 2006

F-STABLE SUBMODULES OF TOP LOCAL COHOMOLOGY MODULES OF GORENSTEIN RINGS

MORDECHAI KATZMAN

Abstract. This paper applies G. Lyubeznik’s notion ofF-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology

modules over Gorenstein rings. As a consequence we obtain an explicit description of

the test ideal of Gorenstein rings in terms of ideals in a regular ring.

1. Introduction

Throughout this paper (R,m) will denote a regular local ring of characteristic p, and A

will be a surjective image of R. We also denote the injective hull of R/m with E and for

any R-module N we writeHomR(N, E) as N∨. We shall always denote with f : R →R the Frobenius map, for whichf(r) =rp _{for all r∈R and we shall denote the eth iterated}

Frobenius functor overRwith Fe

R(−). AsR is regular,FRe(−) is exact (cf. Theorem 2.1 in [K].)

For any commutative ringS of characteristicp, the skew polynomial ringS[T;f]

associ-ated toS and the Frobenius mapf is a non-commutative ring which as a leftR-module is

freely generated by (Ti₎

i≥0, and so consists of all polynomialsPn_i=0siTi, wheren≥0 and

s0, . . . , sn∈S; however, its multiplication is subject to the rule

T s=f(s)T =sp_{T for alls∈S.}

AnyA[T;f]-moduleMis aR[T;f]-module in a natural way and, asR-modules,Fe R(M)∼=

RTe_⊗ RM.

It has been known for a long time that the local cohomology module HdimmAA(A) has

the structure of an A[T;f]-module and this fact has been employed by many authors to

study problems related to tight closure and to Frobenius closure. Recently R. Y. Sharp has

described in [S] the parameter test ideal of F-injective rings in terms of certain A[T;f

]-submodules of Hdimm_AA(A) and it is mainly this work which inspired us to look further into

the structure of theseA[T;f]-modules.

The main aim of this paper is to produce a description of the A[T;f]-submodules of

complete intersection. TheF-injective case is described by Theorem 3.5 and as a corollary

we obtain a description of the parameter test ideal ofA. Notice that for Gorenstein rings the

test ideal the parameter test ideal coincide (cf. Proposition 8.23(d) in [HH1] and Proposition

4.4(ii) in [Sm1].) We then proceed to describe the parameter test ideal in the non-F-injective

case (Theorem 5.3.) We generalise these results to Gorenstein rings in section 6.

2. Preliminaries: F-finite modules

The main tool used in this paper is the notion of F-modules, and in particularF-finite

modules. These were introduced in G. Lyubeznik’s seminal work [L] and provide a very

fruitful point of view of local cohomology modules in prime characteristicp.

One of the tools introduced in [L] is a functorH_{R,Afrom the category ofA[T;f]-modules}

which are Artinian as A-modules to the category of F-finite modules. For any A[T;f

]-moduleM which is Artinian as anA-module theF-finite structure ofH_{R,A(M) is obtained}

as follows. Letγ:RT⊗RM →M be theR-linear map defined byγ(rT⊗m) =rT m; apply

the functor∨_{to obtainγ}∨_:M∨_→F

R(M)∨. Using the isomorphism betweenFR(M)∨and

FR(M∨) (Lemma 4.1 in [L]) we obtain a map β : M∨ → FR(M∨) which we adopt as a generating morphism ofH_R,A(M).

We shall henceforth assume that the kernel of the surjection R→A is minimally

gen-erated by u = (u1, . . . , un). We shall also assume until section 6 that A is a complete

intersection. We shall write u = u1·. . .·un and for all t ≥ 1 we let utR be the ideal

ut

1R+· · ·+utnR.

To obtain the results in this paper we shall need to understand the F-finite module

structure of

H_R,A

Hdimm_AA(A)

∼

= HdimuR R−dimA(R);

this has generating root

R

uR

up−1

−−−→ R

up_R (cf. Remark 2.4 in [L].)

Definition 2.1. Define I_{(R,u) to be the set of all ideals I ⊆R containing (u1, . . . , un)R}

with the property that

up−1_(I+uR)⊆I[p]_+up_R.

Lemma 2.2. Consider the FR-finiteF-moduleM = HnuR(R)with generating root

R

uR

up−1

−−−→ R

F

(a) For any I∈I_{(R,u) theFR-finite module with generating root}

I+uR

uR

up−1

−−−→ I

[p]_+up_R up_R ∼=FR

I+uR

uR

is anF-submodule ofM and every FR-finiteF-submodule of M arises in this way.

(b) For any I∈I_{(R,u) theFR-finite module with generating morphism}

R I+uR

up−1

−−−→ R

I[p]_+up_R ∼=FR

R I+uR

is an F-module quotient of M and everyFR-finiteF-module quotient ofM arises

in this way.

Proof. (a) For anyI∈I_(R,u), the map

I+uR

uR

up−1

−−−→ I

[p]_+up_R up_R

is well defined and is injective; now the first statement follows from Proposition 2.5(a) in

[L]. IfN is anyFR-finiteF-submodule ofM, the root of N is a submodule of the root of

M, i.e., the root ofN has the form (I+uR)/uRfor some idealI⊆R (cf. [L], Proposition

2.5(b)) and the structure morphism ofN is induced by that ofM, i.e., by multiplication by

up−1_{, so we must haveu}p−1_I⊆I[p]_+up_{R, i.e.,I∈I(R,u).} (b) For anyI∈I_{(R,u), the map}

R I+uR

up−1

−−−→ R

I[p]_+up_R ∼=FR

_R

I+uR

is well defined and we have the following commutative diagram with exact rows

(1) 0 // I+uR

uR

/

/

up−1

R uR / /

up−1

R I+uR

/

/

up−1

0

0 // _F

R

I+uR

uR

/

/

up(p−1)

FR R uR / /

up(p−1)

FR

R I+uR

/

/

up(p−1)

0

..

. ... ...

Taking direct limits of the vertical maps we obtain an exact sequence 0 → M′ _{→ M →}

M′′_{→0 which establishes the first statement of (b).}

Conversely, ifM′′_{is aF-module quotient ofM, say,M}′′∼_=M/M′_{for someF-submodule}

M′ _{ofM use (a) to find a generating root ofM}′ _{of the form}

I+uR

uR

up−1

−−−→ I

for some I ∈ I_{(R,u). Looking again at the direct limits of the vertical maps in (1) we}

establish the second statement of (b).

Definition 2.3. For allI∈I_{(R,u) we define}N_{(I) to be theF-module quotient of H}n_u

R(R) with generating morphism

R I+uR

up−1

−−−→ R

I[p]_+up_R ∼=FR

R I+uR

.

Lemma 2.4. Assume thatR is complete. Let H be an ArtinianA[T;f]-module and write

M =H_{R,A(H). LetN be a homomorphic image ofM with generating morphismN0. Then}

N∨

0 is anA[T;f]-submodule of H andN ∼=HR,A(N0∨).

Proof. Notice thatM (and henceN) are F-finite modules (cf. [L], Theorems 2.8 and 4.2).

Let N0 be root ofN and M0 a root of M so that we have a commutative diagram with

exact rows

M0 // //

µ

N0 //

ν

0

FR(M0) // // FR(N0) // 0

where the vertical arrows are generating morphisms. Apply the functor Hom(−, E) to the

commutative diagram above to obtain the following commutative diagram with exact rows

0 // _F

R(N0)∨ //

ν∨

FR(M0)∨

µ∨

0 // _N∨

0 // M0∨

and recall thatM0is isomorphic toH∨(cf. [L], Theorem 4.2). SinceRis complete, (H∨)∨∼= H and we immediately see thatN∨

0 is a R-submodule of H. We now show that N0∨ is an A[T;f] submodule ofH by showing thatT N∨

0 ⊆N0∨.

The construction of the functorH_{R,A(−) is such that for any h∈ H} ∼_=M∨

0,T his the

image ofT⊗Rhunder the map

FR(M0)∨ µ

∨

−−→M∨

0

and so forh∈N∨

0,T his the image ofT⊗Rhunder the map

FR(N0)∨ ν

∨

−−→N∨

0

and henceT h∈N∨

0.

Now the fact thatN ∼=H_R,A(N∨

F

Notation 2.5. LetM be a leftA[T, f]-module. We shall write ATα_{M for the A-module}

generated byTα_{M. Note thatAT}α_{M is a left A[T, f]-module. We shall also writeM}⋆ =

\

α≥0

ATαM.

Lemma 2.6. Assume thatR is complete. Let H be anA[T;f]-module and assume thatH

isT-torsion-free. LetI, J ⊆Abe ideals. If, for someα≥0,

ATαannHIA[T;f] =ATαannHJA[T;f]

thenannHIA[T;f] = annHJA[T;f].

Proof. BothATα_ann

HIA[T;f] andATαannHJA[T;f] are leftA[T;f]-submodules. Now for everyT-torsion-freeA[T;f]-moduleM, and every ideal K⊆A, if



M

i≥0 KTi



ATαM =



M

i≥0 KTi+α



M

vanishes then so does



M

i≥0

K[pα]Ti+α 

M =



M

i≥0

TαKTi 

M =Tα 

M

i≥0 KTi



M

and sinceM isT-torsion-free, _

M

i≥0 KTi



M = 0.

We deduce that gr-annATα_{M = gr-annM. Now}

gr-annATα _ann

HIA[T;f]

= gr-ann annHIA[T;f],

gr-annATα annHJA[T;f]

= gr-ann annHJA[T;f]

and Lemma 1.7 in [S] shows that annHIA[T;f] = annHJA[T;f].

3. The A[T;f] module structure of top local cohomology modules of

F-injective Gorenstein rings

Definition 3.1. As in [Sm1] we say that an idealI ⊆Ais an F-ideal if ann_Hdim(A)

mA (A)I is

a leftA[T;f]-module, i.e., if ann_Hdim(A)

mA (A)

I= ann_Hdim(A)

mA (A)

IA[T;f].

Theorem 3.2. Assume thatRis complete. Consider theFR-finiteF-moduleM = Hnu_R(R)

with generating root

R

uR

up−1

−−−→ R

up_R

and consider the ArtinianA[T;f]moduleH = Hmdim(A A)(A). LetN be a homomorphic image

(a) M =H_{R,A(−)(H)and has generating root H}∨∼_{=R/uR−−−→}up−1 _R/up_R∼_=F R(H∨). (b) If N has generating morphism

R I+uR

up−1

−−−→ R

I[p]_+up_R

thenIA is an F-ideal, N ∼=H_{R,A annHIA[T;f]. If, in addition, H isT-torsion}

free then gr-ann annHIA[T;f] =IA[T;f]andI is radical.

(c) Assume that H isT-torsion free (i.e.,Hr=H in the terminology of[L]). For any

idealJ ⊂R, the F-finite moduleH_{R,A annHJA[T;f]has generating morphism}

R I+uR

up−1

−−−→ R

I[p]_+up_R

for some idealI∈I_(R,u)with annHIA[T;f] = annHJA[T;f].

Proof. The first statement is a restatement of the discussion at the beginning of section 2.

Notice that Lemma 2.2 implies that N must have a generating morphism of the form

given in (b) for someI∈I_(R,u).

Since Ais Gorenstein, H is an injective hull of A/mA which we denoteE. Lemma 2.4

implies thatN ∼=H_{R,A(L) whereL=}

R I+uR

∨

is aA[T;f]-submodule ofH =E. But

R I+uR

∨

= annE(I+uR)

= ann(annuRE)I

= ann_EI.

ButLis aA[T;f]-submodule ofE and so IAis anF-ideal andL= ann_EIA[T;f]. Also,

(0 :R annEIA[T;f]) = (0 :RannEI)

= (0 :R(R/I)∨) = (0 :R(R/I))

= I

(where the third equality follows from 10.2.2 in [BS]) IfH isT-torsion free, Proposition 1.11

in [S] implies thatI= gr-ann ann_EIA[T;f] and Lemma 1.9 in [S] implies thatI is radical.

To prove part (c) we recall Lemma 2.2 which states that H_{R,A annHJA[T;f] has}

generating morphism

R I+uR

up−1

−−−→ R

I[p]_+up_R

F

Part (b) implies thatH_{R,A annHJA[T;f]=}H_{R,A annHIA[T;f]for someI∈}I_(R,u)

and Theorem 4.2 (iv) in [L] implies

∞

\

i=0

ATi _ann

HJA[T;f]

=

∞

\

i=0

ATi _ann

HIA[T;f]

and sinceHis Artinian there exists anα≥0 for whichATα _ann

HJA[T;f]

=ATα _ann

HIA[T;f]

and the result follows from Lemma 2.6.

Remark 3.3. Theorem 3.2 can provide an easy way to show thatH = Hdim(mA A)(A) is not

T-torsion free. As an example considerR=K[[x, y, a, b]],u=x2_a−y2_{bandA=R/uR. Its}

easy to verify that (x, y, a2_{)R∈I(R, x}2_a−y2_{b) whenKhas characteristic 2, and we deduce}

that H3_(x,y,a,b)A(A) is notT-torsion free.

Theorem 3.4. Assume that Ris complete and that Hdim(mA A)(A) isT-torsion free.

(a) For allA[T;f]-submodulesL of Hdim(mA A)(A),

L⋆ =

∞

\

i=0 ATi_L

has the formATα_{M whereα≥0 andM is a special annihilator submodule in the}

terminology of [S].

(b) The set N_(I)|I∈I_{(R,u) is finite.}

Proof. (a) LetLbe aA[T;f]-submodule of Hmdim(_A A)(A). Pick aI∈I(R,u) such thatN(I) =

H_{R,A(L). Now use part (b) of Theorem 3.2 and deduce that}N_(I)=∼H_{R,A(annHIA[T;f]).}

Now the result follows from Theorem 4.2 (iv) in [L].

(b) Theorem 3.2(b) implies that

_N

(I)|I∈I_{(R,u) =}nH_R,Aann

HdimA

mA (A)IA[T;f]

|I∈I_(R,u)o_;

now Corollary 3.11 and Proposition 1.11 in [S] imply that the set on the right is finite.

The following Theorem reduces the problem of classifying allF-ideals of A(in the

ter-minology of [Sm1]) or all special Hdim(m_A A)(A)-ideals (in the terminology of [S]) in the case

whereAis anF-injective complete intersection, to problem of determining the setI_(R,u).

Theorem 3.5. Assume H := Hmdim(_A A)(A) is T-torsion free and let B be the set of all

H-specialA-ideals (cf. §0 in[S])

(b) There exists a unique minimal elementτ in{I|I∈I_{(R,u), htIA >0}and that τ}

is a parameter-test-ideal forA.

(c) A isF-rational if and only if I_(R,u) ={0, R}.

Proof. (a) Assume first thatRis complete. Theorem 3.2(b) implies that Ψ is well defined,

i.e., Ψ(I) ∈ B for all I ∈ I_(R,u), and, clearly, Ψ is injective. The surjectivity of Ψ is a

consequence of Theorem 3.2(c).

Assume now thatRis not complete, denote completions withband writeHb = Hdim(Ab)

mAb ( b A).

If I is a Hb-special Ab-ideal, i.e., if there exists an Ab[T;f]-submodule N ⊆ Hb such that

gr-annN =IAb[T;f] then I= (0 :Ab N) (cf. Definition 1.10 in [S]). But recall thatHb =H

and N is a A[T;f]-submodule of H; now I = (0 :Ab N) = (0 :A N)Ab. If we let Bb be the

set of Hdim(Ab)

mAb ( b

A)-specialAb-ideals, we have a bijection Υ : B_→Bb _{mapping I to IA}b_{. This}

also shows that all ideals inI_(R,b _{u) are expanded fromR, and now sinceR}b _{is faithfully flat}

overR, we deduce that all ideals inI_(R,b u) have the formIRbfor someI∈I_(R,u). We now

obtain a chain of bijections

I_(R,u)←→I_(R,b _u)←→Bb _←→B_.

(b) This is immediate from (a) and Corollary 4.7 in [S].

(c) IfAisF-rational, Hmdim(_A A)(A) is a simpleA[T;f]-module (cf. Theorem 2.6 in [Sm2]) and

the onlyH-specialA-ideals must be 0 andA. The bijection established in (a) implies now

I_{(R,u) ={0, R}.}

Conversely, ifI_{(R,u) ={0, R}, part (b) of the Theorem implies that 1∈Ais a}

parameter-test-ideal, i.e., for all systems of parametersx = (x1, . . . , xd) ofA, (xA)∗ = (xA)F =xA where the second equality follows from the fact that Hdim(mA A)(A) isT-torsion free.

4. Examples

Throughout this sectionKwill denote a field of prime characteristic.

Example 4.1. Let R be the localization of K[x, y] at (x, y), u = xy and A = R/uR.

Then H1_xyR(R) = H_{R,A H}1

xA+yA(A)

ought to have four proper F-finite F-submodules

corresponding to the elements 0,xR,yRandxR+yRofI_{(R, xy).}

We verify this by giving an explicit description theA[T;f]-module structure of

H := H1xA+yA(A)∼= lim_−→

A

(x−y)A

x−y

−−−→ A

(x−y)2_A

x−y

−−−→ A

(x−y)3_A

x−y

F

First notice that in H, for all n≥1 and 0< α ≤n,xα_{+ (x−y)}n_{A=x+ (x−y)}n−α+1

andyα_{+ (x−y)}n_{A=y+ (x−y)}n−α+1 _{soH is theK-span of{x+ (x−y)A} ∪X∪Y ∪U}

where

X=x+ (x−y)n_{A|n≥2 ,}

Y =y+ (x−y)n_{A|n≥2 ,}

U =1 + (x−y)n_A|n≥1

and notice also that the action of the Frobenius mapfonHis such thatT(xα_{+ (x−y)}n_{A) =}

xαp_{+ (x−y)}np_AandT(yα_{+ (x−y)}n_{A) =y}αp_{+ (x−y)}np_{Afor allα≥0.}

Next notice that any A[T, f]-submodule M of H which contains an element 1 + (x−

y)n_{A ∈ U must coincide with H: for 1 ≤m < n we have (x−y)}n−m_{(1 + (x−y)}n_{A) =} (x−y)n−m_{+ (x−y)}n_{A = 1 + (x−y)}m_{A, whereas for m > n, pick an e ≥ 0 such that}

npe_{> m, write}

Te_{(1 + (x−y)}n_{A) = 1 + (x−y)}npe_A∈M

and use the previous case (m < n) to deduce that 1 + (x−y)m_{A∈M. Since nowU ⊆M,}

we see thatM =H.

We now show that there are only three non-trivial A[T, f]-submodules of H, namely

Span_KX and Span_KY, and Span_K{x+ (x−y)A} ∪X. By symmetry, it is enough to show

that, if M is an A[T, f]-submodule of H and x+ (x−y)n_{A ∈ M for some n ≥ 2, then}

X ⊂M. If 1≤m < n,

xn−m_{(x+ (x−y)}n_{A) =x}n−m+1_{+ (x−y)}n_{A=x+ (x−y)}n−(n−m)_{A=x+ (x−y)}m_A

whereas, ifm > n≥2, pick ane≥0 such thatnpe_−pe_{+ 1> mand write}

Te_{(x+ (x−y)}n_{A) =x}pe

+ (x−y)npe

A=x+ (x−y)npe_−pe₊₁ A∈M

and using the previous case (m < n) we deduce thatx+ (x−y)m_A∈M.

Example 4.2. LetR be the localization ofK[x, y, z] atm= (x, y, z),u=x2y+xyz+z3

andA=R/uR. Fedder’s criterion (cf. Propositon 2.1 in [F]) implies thatAisF-pure, and

Lemma 3.3 in [F] implies that theA[T;f] module H1mA(A) isT-torsion-free.

HereI_{(R, u) contains the ideals 0,xR+zRandxR+yR+zR. We deduce thatAis not}

F-rational and that its parameter-test-ideal isxR+zR. Also, Theorem 3.5(b) implies that

the only proper ideals inI_{(R, u) are the ones listed above.}

Example 4.3. LetR be the localization of K[x, y, z] at m= (x, y, z) and assume thatK

u= (x+y+z)(x2_+y2_+z2_{+xy+xz+yz). Fedder’s criterion implies thatA isF-pure,}

and Lemma 3.3 in [F] implies that theA[T;f] module H1mA(A) isT-torsion-free.

Here

I_{(R, u) ⊇ 0,(x+y+z)R,(x}2_+y2_+z2_+xy+xz+yz)R,

(x+z, y+z)R,(x+y+z, y2+yz+z2)R,

(x, y, z)R .

The images inAof the first three ideals have height zero while the images inAof the fourth

and fifth ideals have height 1. Using 3.5(b) we conclude that that the parameter test-ideal

ofAis a sub-ideal of

J= (x+z, y+z)A∩(x+y+z, y2+yz+z2)A= (x2+yx, y2+xz, z2+xy)A.

But this ideal defines the singular locus of A and Theorem 6.2 in [HH2] implies that the

parameter test-element ofAcontainsJ, soJ is the parameter test-ideal ofA.

5. The non-F-injective case

In this section we extend the results of the previous section to the case where A is not

F-injective. First we produce a criterion for theF-injectivity ofA.

Definition 5.1. Define

I_{0(R,u) =}

n

L∈I_(R,u)|u(p−1)(1+p+···+pe−1)_∈L[pe]₊upeRfor somee≥1o.

Proposition 5.2. (a) For any L∈I_(R,u),N_{(L) = 0 if and only ifL∈}I_0(R,u).

(b) Hdim(mA A)(A)isT-torsion free if and only ifI0(R,u) ={R}.

Proof. (a) Recall that theF-finite moduleN_{(L) has generating morphism}

R L+uR

up−1

−−−→ R

L[p]_+up_R ∼=FR

R L+uR

.

Proposition 2.3 in [L] implies thatN_{(L) = 0 if and only if for somee≥1 the composition}

R L+uR

up−1

−−−→ R

L[p]_+up_R u(p−1)p

−−−−−→ R

L[p]_+up2 R. . .

u(p−1)pe−1

−−−−−−−→ R

L[p]_+upe R

vanishes, i.e., if and only ifu(p−1)(1+p+···+pe−1₎

∈L[pe]_+upe_{Rfor somee≥1.}

(b) WriteH = Hdim(mA A)(A). IfHisT-torsion free, the existence of the bijection described in

Theorem 3.5(a) implies that for any non-unitL∈I_{0(R,u), annHLA[T;f]6= annHA[T;f] =}

0. Theorem 3.2(b) implies N_(L) ∼₌ H_{R,A annHLA[T;f] so} H_{R,A annHLA[T;f]= 0.}

F

Assume now thatH is notT-torsion free, i.e.,Hn6= 0. The short exact sequence

0→Hn→H →H/Hn→0

yields the short exact sequence

0→ H/Hn

∨

→ R

uR →H

∨

n →0.

Notice that as the functor Hom(−, E) is faithful,H∨

n 6= 0, and soHn∨∼=R/Ifor some ideal uR⊆I R. NowH_{R,A Hnis theF-finite quotient ofH with generating morphism}

R I

up−1

−−−→ R

I[p]

and this vanishes because of Theorem 4.2(ii) in [L], i.e.,I∈I_0(R,u).

We now describe the parameter test ideal of A. Henceforth we shall always denote

Hdim(m_A A)(A) withH.

Theorem 5.3. Assume that Ris complete. The parameter test ideal of Ais given by

\

I∈I_{(R,u)|htIA >0 .}

Proof. Writeτfor the parameter test ideal ofAand letτbe its pre-image inR. Recall that

τ is anF-ideal (Proposition 4.5 in [Sm1],) i.e., annHτ is anA[T;f]-submodule ofH, and

H_{R,A annHτhas generating morphism}

(annHτ)∨ up−1

−−−→FR (annHτ)∨

.

But

(annHτ)

∨_∼

= (A/τ)∨∨∼=R/(τ+uR) so the generating morphism ofH_{R,A annHτis}

R/(τ+uR) up −1

−−−→R/(τ[p]_+up_R)

and so we must haveτ∈I_(R,u).

AsAis Cohen-Macaulay,τ = (0 :A0∗H) (cf. Proposition 4.4 in [Sm1].)

By Theorem 3.2(b), for each I ∈ I_{(R,u), the ideal IA is an F-ideal and, if htI > 0,}

annHIA= annHIA[T;f]⊆0∗H and so

τ= (0 :A0∗H)⊆

\

(0 :AannHIA)|IA∈I(R,u),htIA >0 .

ButH is an injective hull ofA/mAso

and

τ ⊆\ IA|IA∈I_{(R,u),htIA >0 .}

But asτ is one of the ideals in this intersection, we obtainτ=\ IA∈I_{(R,u)|htIA >}

0 .

6. The Gorenstein case

In this section we generalise the results so far to the case whereAis Gorenstein.

Write δ = dimR−dimA and E = EA(A/mA). Local duality implies Extδ

R(A, R) = HdimA

m (A)

∨_∼

= HomHdimA

m_A (A), E

and sinceAis Gorenstein this is justA=R/uR.

Now Extδ

R(R/uR, A)∼=R/uR, ExtδR(R/upR, A)∼=R/upRandHR,A Hdimm_AA

= Hδ

m(R)

has generating morphismR/u→R/up_{Rgiven by multiplication by some element ofRwhich}

we denoteε(u) (this is unique up to multiplication by a unit.) Unlike the complete

intersec-tion case, the map R/u−−−→ε(u) R/upR may not be injective, i.e., this generating morphism

of Hδm(R) is not aroot. However, if define

Ku:=

[

e≥0

upe+1R:Rε(u)1+p+···+p

e

we obtain a rootR/Ku

ε(u)

−−−→R/Ku[p](cf. Proposition 2.3 in [L].)

We now extend naturally our definition ofI_{(R,u) whenAis Gorenstein as follows.}

Definition 6.1. IfA=R/uR is Gorenstein we define I_(R,u) to be the set of all idealsI

ofRcontainingKufor whichε(u)I⊆I[p].

Now a routine modification of the proofs of the previous sections gives the following two

theorems.

Theorem 6.2. AssumeAis Gorenstein and thatHdimA

mA (A) isT-torsion-free.

(a) The mapI7→IAis a bijection betweenI_{(R,u)and the A-specialH}dimA

mA (A)-ideals.

(b) There exists a unique minimal elementτ in{I|I∈I_{(R,u), htIA >0}and that τ}

is a parameter-test-ideal forA.

(c) A isF-rational if and only if I_{(R,u) ={0, R}.}

Theorem 6.3. Assume that R is complete and that A is Gorenstein. The parameter test

ideal of Ais given by

\

F

Acknowledgement

My interest in the study of local cohomology modules as modules over skew polynomial

rings was aroused by many interesting conversations with Rodney Sharp, during one of

which I learnt about Example 4.1.

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Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH,

United Kingdom,Fax number: +44-(0)114-222-3769