promoting access to White Rose research papers
White Rose Research Online
eprints@whiterose.ac.uk
Universities of Leeds, Sheffield and York
http://eprints.whiterose.ac.uk/
White Rose Research Online URL for this paper:
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 polynomialsPni=0siTi, wheren≥0 and
s0, . . . , sn∈S; however, its multiplication is subject to the rule
T s=f(s)T =spT 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 HdimmAA(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 functorHR,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 ofHR,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 ofHR,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
HR,A
HdimmAA(A)
∼
= HdimuR R−dimA(R);
this has generating root
R
uR
up−1
−−−→ R
upR (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]+upR.
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]+upR upR ∼=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]+upR ∼=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]+upR upR
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 haveup−1I⊆I[p]+upR, i.e.,I∈I(R,u). (b) For anyI∈I(R,u), the map
R I+uR
up−1
−−−→ R
I[p]+upR ∼=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 defineN(I) to be theF-module quotient of Hnu
R(R) with generating morphism
R I+uR
up−1
−−−→ R
I[p]+upR ∼=FR
R I+uR
.
Lemma 2.4. Assume thatR is complete. Let H be an ArtinianA[T;f]-module and write
M =HR,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 functorHR,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 ∼=HR,A(N∨
F
Notation 2.5. LetM be a leftA[T, f]-module. We shall write ATαM for the A-module
generated byTαM. Note thatATαM 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. BothATαann
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-annATαM = 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 annHdim(A)
mA (A)I is
a leftA[T;f]-module, i.e., if annHdim(A)
mA (A)
I= annHdim(A)
mA (A)
IA[T;f].
Theorem 3.2. Assume thatRis complete. Consider theFR-finiteF-moduleM = HnuR(R)
with generating root
R
uR
up−1
−−−→ R
upR
and consider the ArtinianA[T;f]moduleH = Hmdim(A A)(A). LetN be a homomorphic image
(a) M =HR,A(−)(H)and has generating root H∨∼=R/uR−−−→up−1 R/upR∼=F R(H∨). (b) If N has generating morphism
R I+uR
up−1
−−−→ R
I[p]+upR
thenIA is an F-ideal, N ∼=HR,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 moduleHR,A annHJA[T;f]has generating morphism
R I+uR
up−1
−−−→ R
I[p]+upR
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 ∼=HR,A(L) whereL=
R I+uR
∨
is aA[T;f]-submodule ofH =E. But
R I+uR
∨
= annE(I+uR)
= ann(annuRE)I
= annEI.
ButLis aA[T;f]-submodule ofE and so IAis anF-ideal andL= annEIA[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 annEIA[T;f] and Lemma 1.9 in [S] implies thatI is radical.
To prove part (c) we recall Lemma 2.2 which states that HR,A annHJA[T;f] has
generating morphism
R I+uR
up−1
−−−→ R
I[p]+upR
F
Part (b) implies thatHR,A annHJA[T;f]=HR,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=x2a−y2bandA=R/uR. Its
easy to verify that (x, y, a2)R∈I(R, x2a−y2b) 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 ATiL
has the formATαM 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) =
HR,A(L). Now use part (b) of Theorem 3.2 and deduce thatN(I)=∼HR,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) =nHR,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(mA 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 IAb. This
also shows that all ideals inI(R,b u) are expanded fromR, and now sinceRb 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 H1xyR(R) = HR,A H1
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)2A
x−y
−−−→ A
(x−y)3A
x−y
F
First notice that in H, for all n≥1 and 0< α ≤n,xα+ (x−y)nA=x+ (x−y)n−α+1
andyα+ (x−y)nA=y+ (x−y)n−α+1 soH is theK-span of{x+ (x−y)A} ∪X∪Y ∪U
where
X=x+ (x−y)nA|n≥2 ,
Y =y+ (x−y)nA|n≥2 ,
U =1 + (x−y)nA|n≥1
and notice also that the action of the Frobenius mapfonHis such thatT(xα+ (x−y)nA) =
xαp+ (x−y)npAandT(yα+ (x−y)nA) =yαp+ (x−y)npAfor allα≥0.
Next notice that any A[T, f]-submodule M of H which contains an element 1 + (x−
y)nA ∈ U must coincide with H: for 1 ≤m < n we have (x−y)n−m(1 + (x−y)nA) = (x−y)n−m+ (x−y)nA = 1 + (x−y)mA, whereas for m > n, pick an e ≥ 0 such that
npe> m, write
Te(1 + (x−y)nA) = 1 + (x−y)npeA∈M
and use the previous case (m < n) to deduce that 1 + (x−y)mA∈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
SpanKX and SpanKY, and SpanK{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)nA ∈ M for some n ≥ 2, then
X ⊂M. If 1≤m < n,
xn−m(x+ (x−y)nA) =xn−m+1+ (x−y)nA=x+ (x−y)n−(n−m)A=x+ (x−y)mA
whereas, ifm > n≥2, pick ane≥0 such thatnpe−pe+ 1> mand write
Te(x+ (x−y)nA) =xpe
+ (x−y)npe
A=x+ (x−y)npe−pe+1 A∈M
and using the previous case (m < n) we deduce thatx+ (x−y)mA∈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,(x2+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
I0(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∈I0(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]+upR ∼=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]+upR 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]+upeRfor 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∈I0(R,u), annHLA[T;f]6= annHA[T;f] =
0. Theorem 3.2(b) implies N(L) ∼= HR,A annHLA[T;f] so HR,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. NowHR,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∈I0(R,u).
We now describe the parameter test ideal of A. Henceforth we shall always denote
Hdim(mA 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
HR,A annHτhas generating morphism
(annHτ)∨ up−1
−−−→FR (annHτ)∨
.
But
(annHτ)
∨∼
= (A/τ)∨∨∼=R/(τ+uR) so the generating morphism ofHR,A annHτis
R/(τ+uR) up −1
−−−→R/(τ[p]+upR)
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
mA (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 HdimmAA
= Hδ
m(R)
has generating morphismR/u→R/upRgiven 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-specialHdimA
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.
References
[BS] M. P. Brodmann and R. Y. Sharp.Local cohomology: an algebraic introduction with geometric
appli-cations.Cambridge Studies in Advanced Mathematics,60. Cambridge University Press, Cambridge,
1998.
[F] R. Fedder.F-purity and rational singularity.Transactions of the AMS,278(1983), no. 2, pp. 461–
480.
[HH1] M. Hochster and C. Huneke.Tight closure, invariant theory, and the Brianon-Skoda theorem.Journal
of the AMS3(1990), no. 1, 31–116.
[HH2] M. Hochster and C. Huneke.F-regularity, test elements, and smooth base change.Transactions of the AMS,346(1994), no. 1, pp. 1–62.
[K] E. Kunz.Characterizations of regular local rings for characteristicp.American Journal of Mathe-matics91(1969), pp. 772–784.
[L] G. Lyubeznik.F-modules: applications to local cohomology andD-modules in characteristicp >0.
J. Reine Angew. Math.491(1997), pp. 65–130.
[S] R. Y. Sharp.Graded annihilators of modules over the Frobenius skew polynomial ring, and tight
closure.Preprint.
[Sm1] K. E. Smith.Test ideals in local rings.Transactions of the AMS347(1995), no. 9, pp. 3453–3472.
[Sm2] K. E. Smith. F-rational rings have rational singularities.American Journal of Mathematics 119
(1997), no. 1, pp. 159–180.
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH,
United Kingdom,Fax number: +44-(0)114-222-3769