We show that the F-signature of a local ring of characteristic p, defined by
Huneke and Leuschke, is positive if and only if the ring is strongly F-regular.Comment: revised version, incorporating referee's comments. 6 page

Aberbach, Ian M.

Leuschke, Graham J.

English

arXiv

This is a revised version, incorporating referee's comments, of an article published in Mathematical Research Letters, 10 (2003), no. 1, 51-56.We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular

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arXiv:math/0208067v2  [math.AC]  25 Nov 2002THE F -SIGNATURE AND STRONG F -REGULARITYIAN M. ABERBACH AND GRAHAM J. LEUSCHKEAbstract. We show that the F -signature of a local ring of characteristic p, defined byHuneke and Leuschke, is positive if and only if the ring is strongly F -regular.In [7], Huneke and Leuschke define the F -signature of an F -finite local ring of primecharacteristic with perfect residue field. The F -signature, denoted s(R), is an asymptoticmeasure of the proportion of R-free direct summands in a direct-sum decomposition of R1/pe,the ring of peth roots of R. This proportion seems to give subtle information on the natureof the singularity defining R. For example, the F -signature of any of the two-dimensionalquotient singularities (An), (Dn), (E6), (E7), (E8) is the reciprocal of the order of the groupG defining the singularity [7, Example 18]. The main theorem of [7] on F -signatures is asfollows.Theorem 0.1. [7, Theorem 11] Let (R,m, k) be a reduced complete F -finite Cohen–Macaulaylocal ring containing a field of prime characteristic p. Assume that k is perfect. Then(1) If s(R) > 0, then R is weakly F -regular.(2) If in addition R is Gorenstein, then s(R) exists, and is positive if and only if R isweakly F -regular.(See below for definitions of the F -signature and weak F -regularity.) In this note, weextend this theorem in two directions: we remove the assumption in (2) that R be Gorenstein,and we replace “weakly F -regular” by “strongly F -regular” throughout. Our main theoremis thus as follows.Theorem 0.2. Let (R,m, k) be a reduced excellent F -finite local ring containing a field ofcharacteristic p, and let d = dimR. Then the following are equivalent:(1) lim inf aqqd+α(R)> 0.(2) lim sup aqqd+α(R)> 0.(3) R is strongly F -regular.Date: February 1, 2008.Aberbach was partially supported by the National Science Foundation and the National Security Agency.Leuschke was partially supported by an NSF Postdoctoral Fellowship.12 IAN M. ABERBACH AND GRAHAM J. LEUSCHKEIn particular, if the F -signature s(R) is known to exist, then s(R) is positive if and only ifR is strongly F -regular.We also extend the definition of the F -signature to the case of an imperfect residue field.This allows us to prove that s(R) behaves well with respect to localization (Proposition 1.3).Our results do not address the existence of the limit defining s(R). Yao has shown thats(R) exists whenever R is Gorenstein on the punctured spectrum [10].1. The Main ResultThroughout what follows, (R,m, k) is a reduced Noetherian local ring of dimension d,containing a field of positive characteristic p. We use q to denote a varying power of p. Setd = dim(R) and α(R) = logp[k : kp]. We assume throughout that R is F -finite, that is, theFrobenius endomorphism F : R−→R defined by F (r) = rp is a module-finite ring homomor-phism. Equivalently, for each q = pe, R1/q = {r1/q | r ∈ R} is a finitely generated R-module.In particular, this implies that α(R) < ∞, and that R is excellent [8, Propositions 1.1 and2.5]. Also, when computing length over R, we have λ(R/I [q]R) = λ(R1/q/IR1/q)/qα(R).We first define the F -signature of R.Definition 1.1. Let (R,m, k) be as above. For each q = pe, decompose R1/q as a direct sumof finitely generated R-modules Raq ⊕Mq, where Mq has no nonzero free direct summands.The F -signature of R iss(R) = limq−→∞aqqd+α(R),provided the limit exists.Our formulation differs slightly from the original definition in [7], where it is assumed thatk is perfect, or equivalently that α(R) = 0. This reformulation allows us to show that s(R)cannot decrease upon localization. We use a lemma due to Kunz ([8]).Lemma 1.2. Let R be an F -finite Noetherian ring of characteristic p. Then for any primeideals P ⊆ Q of R, [k(P ) : k(P )p] = [k(Q) : k(Q)p]pdimRQ/PRQ. In other words, α(RP ) =α(RQ) + htQ/P .Proposition 1.3. Let (R,m) be an F -finite local ring and P a prime ideal. For q = pe, letaq be the number of nonzero R-free direct summands in R1/q, and let bq be the correspondingquantity for RP . Thenbqqdim(RP )+α(RP )≥aqqdim(R)+α(R).In particular, if both s(R) and s(RP ) exist, then s(RP ) ≥ s(R).THE F -SIGNATURE AND STRONG F -REGULARITY 3Proof. We have (RP )1/q ∼= (R1/q)P , so the number of RP -free direct summands in (RP )1/qis at least the number of R-free summands in R1/q. A straightforward computation usingLemma 1.2 now gives the result. We now begin to work toward showing that s(R) is positive if and only if R is stronglyF -regular. We refer the reader to [6] for basic notions concerning the theory of tight closure,including finitistic tight closure, but review briefly the ideas used in the proof.A Noetherian ring R of characteristic p is said to be weakly F -regular provided everyideal of R is tightly closed. Equivalently, the zero module is finitistically tightly closed inE = ER(k), the injective hull of the residue field of R. In other symbols, 0∗fgE = 0. We saythat R is strongly F -regular if for every c ∈ R not in any minimal prime of R, the inclusionRc1/q ⊂ R1/q splits for q  0. Equivalently, the zero module is tightly closed in E, that is,0∗E = 0. Weak and strong F -regularity are conjecturally equivalent, but this is known onlyin low dimension and in some special cases.A test element for R is an element c, not in any minimal prime of R, such that cI∗ ⊆ Ifor every ideal I of R, and the test ideal, denoted τ(R), is the ideal generated by all testelements. For a reduced local ring R, τ(R) = AnnR 0∗fgE by [5, Theorem 8.23]. Thus R isweakly F -regular if and only if τ(R) = R. On the other hand, the CS test ideal, cf. [9]and [2], is the ideal τ˜(R) = AnnR 0∗E. By work of [9] and [2], the CS test ideal behaves wellunder localization, so defines the non-strongly F -regular locus of Spec(R). In particular, Ris strongly F -regular if and only if τ˜(R) = R.It is known that a weakly F -regular ring is F -pure, that is, the Frobenius morphism is apure homomorphism, and that for an F -pure ring both τ(R) and τ˜(R) are radical ideals.A local ring (R,m, k) is said to be approximately Gorenstein provided there is a sequence{It} of m-primary irreducible ideals cofinal with the powers of m. When R is Cohen–Macaulay and has a canonical ideal J (so is Gorenstein at all associated primes), such afamily can be obtained as follows: Let x1, . . . , xd be a system of parameters such that x1 ∈ Jand x2, . . . , xd form a system of parameters for R/J . Then It := (xt−11 J, xt2, . . . , xtd)R, fort ≥ 1, gives the required family. Furthermore, the direct limit lim−→tR/It, where the maps inthe direct system are R/Itx1···xd−→ R/It+1, is isomorphic to ER(k). If u1 ∈ R is a representativefor the socle generator of R/I1, then ut := (x1 · · ·xd)t−1u1 generates the socle of R/It, andeach ut maps in the limit to u, the socle element of ER(k).More generally ([4, Thm. 1.7]), if R is any locally excellent Noetherian ring that is locallyGorenstein at associated primes, then R is approximately Gorenstein.4 IAN M. ABERBACH AND GRAHAM J. LEUSCHKEThe following result of Hochster, together with its corollary below, explains our interestin approximately Gorenstein rings. It can be thought of as a generalization of [7, Lemma12].Proposition 1.4. [4, Theorem 2.6] Let (R,m) be an approximately Gorenstein local ringand let {It} be a sequence of irreducible ideals cofinal with the powers of m. Let f : R−→Mbe a homomorphism of finitely generated R-modules. Then f is a split injection if and onlyif f ⊗R R/It is injective for every t.Proposition 1.5. Let (R,m) be an approximately Gorenstein local ring with a family ofirreducible ideals {It} as above, and let ut ∈ R represent a socle generator for R/It. Letf : R−→M be a homomorphism of finitely generated R-modules. IfM has no free summands,then there exists t0 > 0 such that utM ⊆ ItM for all t ≥ t0.Proof. By Proposition 1.4, f ⊗ R/It fails to be injective for some t. Since ut is the uniquesocle element of R/It, we have f(ut) ∈ ItM , that is, utM ⊆ ItM . This continues to hold forall t′ ≥ t, since there is an injection R/It−→R/It′ with ut 7→ ut′. We also use a result of Aberbach, which says that, in some sense, elements not in tightclosures are very far from being in Frobenius powers.Theorem 1.6. [1, Prop. 2.4] Let (R,m) be an excellent local domain such that the completionis also a domain. Let N = lim−→tR/Jt be a direct limit system of cyclic modules. Fix u 6∈ 0∗N .Then there exists q0 such that ⋃t(J[q]t : uqt ) ⊆ m[q/q0]for all q  0 (where the sequence {ut} represents u ∈ N and ut 7→ ut+1).Proof of Theorem 0.2. The Cohen-Macaulayness of R is forced by the assumptions ([7, The-orem 11] and [5]), so we may assume throughout that R is Cohen-Macaulay.That (1) implies (2) is trivial. So assume that (2) holds. We proceed by induction on thedimension d, the case d = 0 being trivial. If d > 0, then Proposition 1.3 shows that we mayassume by induction on d that R is strongly F -regular on the punctured spectrum. We willshow that 0∗E = 0, where as above E = ER(k) is the injective hull of the residue field of R.Since τ˜(R) = AnnR 0∗E is a radical ideal and is known to define the non-strongly F -regularlocus of R (see [2]), and R is strongly F -regular on the punctured spectrum, AnnR 0∗E containsthe maximal ideal m. If τ˜(R) = R, then we are done, so we assume τ˜ (R) = m. Then0∗E = soc(E).THE F -SIGNATURE AND STRONG F -REGULARITY 5As in the discussion above, E = ER(k) ∼= lim−→tR/It for a family of irreducible ideals It.Let u be a socle generator for E and {ut} ⊆ R a sequence of representatives for the soclegenerators of R/It, converging to u.Fix a power q of the characteristic, and decompose R1/q ∼= Raq ⊕Mq, where Mq has nononzero free summands. Then for each t, we haveλ(R/I[q]t)− λ(R/(It, ut)[q])=λ(R1/q/ItR1/q)qα(R)−λ(R1/q/(It, ut)R1/q)qα(R)=aqλ(R/It) + λ(Mq/ItMq)qα(R)−aqλ (R/(It, ut))− λ (Mq/(It, ut)Mq)qα(R)=aqλ(R/It)− aqλ(R/(It, ut))qα(R)+λ(Mq/ItMq)− λ(Mq/(It, ut)Mq)qα(R)=aq + ct,qqα(R),for some ct,q ≥ 0. By Proposition 1.5, there exists t0 > 0 such that utMq ⊆ ItMq for t ≥ t0,that is, ct,q = 0 for t ≥ t0. On the other hand, λ(R/I[q]t )− λ(R/(It, ut)[q]) = λ(R/(I[q]t : uqt ))is equal to 1 for large t since (I[q]t : uqt ) = m for large t. Thus, for large t,limq−→∞aq + ct,qqd+α(R)= limq−→∞1qd+α(R)= 0,a contradiction.Lastly, assume that R is strongly F -regular and keep the same notation. We then have0∗E = 0, so u 6∈ 0∗E. By Theorem 1.6, then, there exists q0 such that(I[q]t :R uqt ) ⊆ m[q/q0]for all q ≥ q0. Fix q ≥ q0. Then there exists t0 such that for all t ≥ t0 we haveaqqα(R)= λ(R/I[q]t)− λ(R/(I[q]t , uqt ))= λ(I[q]t : uqt)≥ λ(R/m[q/q0]).Divide by qd and pass to the limit; we see that lim inf aqqd+α(R)≥ eHK(m, R)/qd0 > 0. Thus (1)holds.6 IAN M. ABERBACH AND GRAHAM J. LEUSCHKEThe last statement is immediate if there is a limit. The F -signature suggests a form of dimension that we may attach to an F -finite reducedlocal ring. Let sj = limq−→∞aqqj+α(R)for 0 ≤ j ≤ d = dim(R) and set s−1 = 1. Then wecan define the s-dimension of R as sdim(R) = max{j ≥ −1|sj > 0}. A ring which is F -purethen has non-negative s-dimension, and Theorem 0.2 says that R is strongly F -regular ifand only if sdim(R) = dim(R).References1. I. Aberbach, Extension of weakly and strongly F -regular rings by flat maps, J. Algebra 241 (2001), 799–807.2. I. Aberbach and F. Enescu, Test ideals and base change problems in tight closure theory, Trans. Amer.Math. Soc. (to appear).3. M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973),25–43.4. , Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977),no. 2, 463–488.5. M. Hochster and C. Huneke, Tight closure, invariant theory, and the Brianc¸on-Skoda theorem, J. Amer.Math. Soc. 3 (1990), 31–116.6. C. Huneke, Tight Closure and its Applications, CBMS Lecture notes in Mathematics, vol. 88, AmericanMath. Soc., 1996.7. C. Huneke and G. Leuschke, Two theorems about maximal Cohen–Macaulay modules, Math. Annalen(2002), 1–2, to appear.8. E. Kunz, On Noetherian rings of characteristic p, Amer. J. Math. 98 (1976), no. 4, 999–1013.9. G. Lyubeznik and K. E. Smith, On the commutation of the test ideal with localization and completion,Trans. Amer. Math. Soc. 353 (2001), no. 8, 3149–3180 (electronic).10. Y. Yao, Observations on the F -signature of local rings of characteristic p, preprint.Mathematics Department, University of Missouri, Columbia, MO 65211 USAE-mail address: aberbach@math.missouri.eduURL: http://www.math.missouri.edu/people/iaberbach.htmlDepartment of Mathematics, University of Kansas, Lawrence, KS 66045 USAE-mail address: gleuschke@math.ukans.eduURL: http://www.math.ukans.edu/~gleuschke

The F-signature and strong F-regularity

We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular

The F-signature and strong F-regularity

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