aa r X i v : . [ m a t h . A C ] J un A THEOREM ABOUT MAXIMAL COHEN-MACAULAY MODULES
THOMAS POLSTRA
Abstract.
It is shown that for any local strongly F -regular ring there exists natural num-ber e so that if M is any finitely generated maximal Cohen-Macaulay module, then thepushforward of M under the e th iterate of the Frobenius endomorphism contains a freesummand. Consequently, the torsion subgroup of the divisor class group of a local strongly F -regular ring is finite. Introduction
Let R be a Noetherian commutative ring of prime characteristic p > e ∈ N let F e : R → R be the e th iterate of the Frobenius endomorphism of R . Throughout thisarticle we assume R is F -finite, i.e. the Frobenius endomorphism is a finite map. If M is an R -module then F e ∗ M is the R -module obtained from M via restriction of scalars under F e .The ring R is said to be strongly F -regular if for each nonzero r ∈ R there exists an e ∈ N and ϕ ∈ Hom R ( F e ∗ R, R ) such that ϕ ( F e ∗ r ) = 1. Every local strongly F -regular ring is a Cohen-Macaulay normal domain. In particular, studying the class of Cohen-Macaulay modules ina strongly F -regular ring is a warranted venture. The main contribution of this article isthe following uniform property concerning the class of finitely generated Cohen-Macaulaymodules in strongly F -regular rings. Theorem A.
Let ( R, m , k ) be a local F -finite and strongly F -regular ring of prime char-acteristic p > . There exists an e ∈ N so that if M is any finitely generated maximalCohen-Macaulay R -module then there exists an onto R -linear map F e ∗ M → R , i.e. R canbe realized as direct summand of F e ∗ M . More precisely, we show any minimal generator of a maximal Cohen-Macaulay module M can be sent to 1 in R under an R -linear map F e ∗ M → R in Theorem 3.1. A newapplication of Theorem A is that the torsion subgroup of the divisor class group of any localstrongly F -regular ring is finite, see Corollary 3.3. When coupled with results originatingfrom [CRST18, Car17], it is now known that the torsion group of the divisor class groupof a strongly F -regular ring is finite and every torsion divisor has order bounded by thereciprocal of the F -signature of R .We utilize Theorem A to rederive two other important properties of local strongly F -regular rings. First and foremost we reprove that the divisor class group of a 2-dimensionalstrongly F -regular ring is finite in Corollary 3.2. It is important to note that our proof ofthis property does not require an understanding of resolution of singularities of excellentsurfaces with at worst rational singularities by quadratic transforms. We also show thatthe F -signature of a local strongly F -regular ring is positive, recapturing the main result of[AL03] in an elementary, novel, and streamlined fashion. Polstra was supported in part by NSF Postdoctoral Research Fellowship DMS . Preliminary results and notation
Suppose ( R, m , k ) is a local F -finite strongly F -regular ring. Every strongly F -regular ringis a normal domain and therefore has a well-defined divisor class group on X = Spec( R ).For the sake of convenience, we recall several elementary properties concerning divisors innormal rings of prime characteristic. For each Weil divisor D on X we write R ( D ) forthe global sections of O X ( D ). We refer to R ( D ) as a divisorial ideal. Every divisorialideal is a rank 1 module satisfying Serre’s condition ( S ). Conversely, every rank 1 modulesatisfying Serre’s condition ( S ) is isomorphic to a divisorial ideal. Suppose D , D are Weildivisors on X . Since X is affine, the divisors D , D are linearly equivalent if and only if R ( D ) ∼ = R ( D ). The module Hom R ( R ( D ) , R ( D )) is isomorphic to the divisorial ideal R ( D − D ). Moreover, the reflexification of R ( D ) ⊗ R R ( D ) is isomorphic to R ( D + D ).In particular, since reflexification commutes with restriction of scalars under Frobenius, thereflexification of ( F e ∗ R ( D )) ⊗ R R ( D ) is isomorphic to F e ∗ R ( D + p e D ).Now suppose that ( R, m , k ) is a Cohen-Macaulay normal F -finite domain, e.g. strongly F -regular, of prime characteristic p >
0. Every F -finite ring is the homomorphic image of aregular ring by [Gab04, Rem. 13.6] and therefore has a canonical module ω R . Because thecanonical module of a normal domain is ( S ) and rank 1 we have that ω R ∼ = R ( K X ) for someWeil divisor K X . We refer to such a divisor as a canonical divisor.The following lemma is characteristic-free and is an observation that ( S )-modules satisfya Krull-Schmidt type condition for rank 1-summands. Lemma 2.1.
Let ( R, m , k ) be a normal local domain. Let C be a finitely generated ( S ) -module, M a rank module, and suppose that C ∼ = M ⊕ a ⊕ N ∼ = M ⊕ a ⊕ N are choices ofdirect sum decompositions of C so that M cannot be realized as a direct summand of either N or N . Then a = a .Proof. Without loss of generality we may assume a > a >
0. Because C satisfiesSerre’s condition ( S ), the rank 1 module M is ( S ), and therefore M ∼ = R ( D ) for someWeil divisor D . The divisorial ideal R ( D ) is a direct summand of a finitely generated ( S )module N if and only if R is a direct summand Hom R ( N, R ( D )). Indeed, this follows fromthe fact that the natural map N → Hom R (Hom R ( N, R ( D )) , R ( D )) is an isomorphism onthe regular locus and therefore an isomorphism globally by [Har94, Proposition 1.11]. Itfollows that Hom R ( C, R ( D )) ∼ = R ⊕ a ⊕ Hom R ( N , R ( D )) ∼ = R ⊕ a ⊕ Hom R ( N , R ( D )) arechoices of direct sum decompositions of Hom R ( C, R ( D )) where neither Hom R ( N , R ( D )) norHom R ( N , R ( D )) have a free summand. In other words, a and a are the maximal numberof free summands appearing in choices of direct sum decompositions of Hom R ( C, R ( D )).But such numbers can be computed after completion and hence a = a since complete localrings satisfy the Krull-Schmidt condition. (cid:3) Corollary 2.2.
Let ( R, m , k ) be a local normal domain and C a finitely generated ( S ) -module. If D , D , . . . , D t are divisors representing distinct elements of the divisor classgroup of R and R ( D i ) is a direct summand of C for each ≤ i ≤ t , then R ( D ) ⊕ · · ·⊕ R ( D t ) is a direct summand of C . The reflexification of an R -module M is the module Hom R (Hom R ( M, R ) , R ). If R is normal and M isa finitely generated ( S ) module then the natural map M → Hom R (Hom R ( M, R ) , R ) is an isomorphism,[Har94, Theorem 1.9]. roof. Let 1 ≤ i ≤ t − C ∼ = R ( D ) ⊕ · · · ⊕ R ( D i ) ⊕ N. We claim that R ( D i +1 ) is a direct summand of N . By Lemma 2.1 it is enough to show that R ( D i +1 ) is not a summand of R ( D ) ⊕ · · · ⊕ R ( D i ). By Hom-ing into R ( D i +1 ) it is enoughto show R is not a direct summand of R ( D − D i +1 ) ⊕ · · · ⊕ R ( D i − D i +1 ). Suppose by wayof contradiction that R ( D − D i +1 ) ⊕ · · · ⊕ R ( D i − D i +1 ) had R as a summand. Then thereexists an onto R -linear map R ( D − D i +1 ) ⊕ · · · ⊕ R ( D i − D i +1 ) → R . Since R is local thereexists 1 ≤ j ≤ i so that the image of R ( D j − D i +1 ) in R contains a unit. Hence R is a directsummand of R ( D j − D i +1 ). By rank considerations R ( D j − D i +1 ) ∼ = R , i.e. D j and D i +1 arelinearly equivalent divisors, a contradiction to our assumption that D j and D i +1 representdistinct elements of the divisor class group. (cid:3) We return to the assumption that ( R, m , k ) is a local ring of prime characteristic p > M is a finitely generated R -module and e ∈ N then we let I e ( M ) = { η ∈ M | ϕ ( F e ∗ η ) ∈ m , ∀ ϕ ∈ Hom R ( F e ∗ M, R ) } . Observe that an element η ∈ M avoids I e ( M ) if and only if there exists ϕ ∈ Hom R ( F e ∗ M, R )so that ϕ ( F e ∗ η ) = 1. If M = R then we let I e = I e ( R ). The ring R is strongly F -regular ifand only if T e ∈ N I e = 0. We record some basic properties of these subsets of M . Lemma 2.3.
Let ( R, m , k ) be a local F -finite ring of prime characteristic p > and M afinitely generated R -module.(1) The sets I e ( M ) form submodules of M .(2) For each e ∈ N there is an inclusion m [ p e ] M ⊆ I e ( M ) .(3) If e ≥ e ′ then I e ( M ) ⊆ I e ′ ( M ) .(4) If R is strongly F -regular and M is torsion-free then T e ∈ N I e ( M ) = 0 .Proof. Properties (1)-(3) are straight-forward and left to the reader to verify. For (4) let η ∈ M be a nonzero element and consider an R -linear map ψ : M → R such that ψ ( η ) = r = 0. We are assuming R is strongly F -regular, i.e. T e ∈ N I e = 0. Hence r I e forsome e and therefore there exists ϕ ∈ Hom R ( F e ∗ R, R ) so that ϕ ( F e ∗ r ) = 1. In particular, ϕ ◦ F e ∗ ψ ∈ Hom R ( F e ∗ M, R ) and ϕ ◦ F e ∗ ψ ( η ) = 1. (cid:3) Main results
Theorem 3.1.
Let ( R, m , k ) be an F -finite and strongly F -regular ring of prime character-istic p > . There exists an e ∈ N , depending only on R , so that if M is a finitely generatedmaximal Cohen-Macaulay R -module and η ∈ M \ m M , then there exists ϕ ∈ Hom R ( F e ∗ M, R ) such that ϕ ( F e ∗ η ) = 1 .Proof. Begin by surjecting a finitely generated free module R ⊕ n onto Hom R ( M, R ( K X )) andthen consider a short exact of the following form:0 → N → R ⊕ n → Hom R ( M, R ( K X )) → . The modules Hom R ( M, R ( K X )) and N are maximal Cohen-Macaulay. Hence there is a shortexact sequence 0 → M → R ( K X ) ⊕ n → Hom R ( N, R ( K X )) → btained by applying Hom R ( − , R ( K X )), see [BH93, Theorem 3.3.10]. Let C denote the max-imal Cohen-Macaulay R -module Hom R ( N, R ( K X )) and let x = x , . . . , x d be a full systemof parameters of R . Then Tor R ( R/ ( x ) , C )) = 0 since C is maximal Cohen-Macaulay andTor R ( R/ ( x ) , C )) agrees with the first Koszul homology on the C -regular sequence x , . . . , x d .Hence there is short exact sequence0 → M ( x ) M → R ( K X ) ⊕ n ( x ) R ( K X ) ⊕ n → C ( x ) C → . In particular, under the inclusion M → R ( K X ) ⊕ n we find that( x ) M = M ∩ ( x ) R ( K X ) ⊕ n . (1)By (3) and (4) of Lemma 2.3, utilizing faithfully flat descent to the completion of R ,and Chevalley’s Lemma, [Che43, Section 2, Lemma 7], there exists an e ∈ N so that I e ( R ( K X )) ⊆ ( x ) R ( K X ). Suppose that η ∈ M \ m M . Then under the inclusion M → R ( K X ) ⊕ n we find that η ( x ) R ( K X ) ⊕ n by (1). In particular, η avoids I e ( R ( K X )) ⊕ n andhence there is a commutative diagram F e ∗ M F e ∗ R ( K X ) R ϕ ψ so that ϕ ( F e ∗ η ) = ψ ( F e ∗ η ) = 1. (cid:3) The first application given of Theorem 3.1 is that the divisor class group of a 2-dimensionalstrongly F -regular ring is finite. Before this article, the proof that 2-dimensional strongly F -regular rings would have gone as follows: every strongly F -regular ring is weakly F -regular, in particular is F -rational. Hence R has pseudorational singularities by [Smi97,Theorem 3.1]. But every F -finite ring is excellent by [Kun76, Theorem 1] and thereforehas rational singularities since resolution of singularities of excellent surfaces is known by[Lip78]. Every 2-dimensional rational surface singularity has finite divisor class group by[Lip69, Proposition 17.1], a result that requires classifying minimal resolution of singularitiesof rational surfaces by quadratic transforms.For the following corollaries we remind the reader that divisorial ideals in a normal domain R are torsion-free and therefore have the same dimension as R . Corollary 3.2.
Let ( R, m , k ) be an F -finite and strongly F -regular ring of prime character-istic p > and Krull dimension . Then the divisor class group of R is finite.Proof. Every divisorial ideal in a 2-dimensional normal domain is Cohen-Macaulay. ByTheorem 3.1 there is an e ∈ N so that if D is any Weil divisor then F e ∗ R ( − p e D ) has afree summand. Reflexifying F e ∗ R ( − p e D ) ⊗ R R ( D ) then shows R ( D ) is a direct summandof F e ∗ R . Therefore every divisorial ideal can be realized as direct summand of F e ∗ R . ByCorollary 2.2, if D , . . . , D t are divisors, no two of which are linearly equivalent, then R ( D ) ⊕· · · ⊕ R ( D t ) is a direct summand of F e ∗ R . By rank considerations on F e ∗ R there can onlybe finitely many many Weil divisors up to linear equivalence. (cid:3) Corollary 3.3.
Let ( R, m , k ) be an F -finite and strongly F -regular ring of prime character-istic p > . Then the torsion subgroup of the divisor class group of R is finite. roof. If D is torsion divisor in a strongly F -regular ring R then the fractional ideal R ( D )is Cohen-Macaulay, see [PS14, Corollary 3.3] and [DS16, Corollary 3.12], cf [Wat91, Corol-lary 2.9]. By Theorem 3.1 there exists an e ∈ N so that if D is a torsion divisor then F e ∗ R ( − p e D ) has a free summand. The proof now follows as in the proof of Corollary 3.2. (cid:3) Finiteness of the prime-to- p subgroup of the torsion group of an F -finite local ring withalgebraically closed residue field is something observed in [Car17, Corollary 5.1]. However,the techniques of Corollary 3.3 are more elementary, do not require the residue field to bealgebraically closed, and show finiteness of the entire torsion subgroup.The last result we recapture is Aberbach’s and Leuschke’s theorem that the F -signature ofa local strongly F -regular ring is positive. The F -signature of a ring of prime characteristicis the asymptotic measurement of the number of free summands of F e ∗ R as compared tothe generic rank of F e ∗ R . The study of F -signature originates in [SVdB97, HL02]. When( R, m , k ) is local of dimension d and ℓ ( M ) denotes the length of finite length R -module M ,then the F -signature of R is realized as the limit s ( R ) = lim e →∞ ℓ ( R/I e ) p ed , a limit which always exists by [Tuc12], cf [PT18].Before reestablishing the positivity of the F -signature of strongly F -regular rings we brieflydiscuss the related prime characteristic invariant Hilbert-Kunz multiplicity. If ( R, m , k ) islocal ring of prime characteristic p > d then the Hilbert-Kunz multi-plicity of R is the limit e HK ( R ) = lim e →∞ ℓ ( R/ m [ p e ] ) p ed , a limit which always exists by [Mon83]. The Hilbert-Kunz multiplicity of a local ring isbounded below by 1. With this simple fact in mind we are prepared to recapture [AL03,Main Result]. Corollary 3.4.
Let ( R, m , k ) be an F -finite and strongly F -regular ring of prime character-istic p > and Krull dimension d . Then s ( R ) > .Proof. Let e be as in Theorem 3.1. Then I e + e ⊆ m [ p e ] for all e ∈ N . Indeed, if r ∈ R \ m [ p e ] then F e ∗ r ∈ F e ∗ R \ m F e ∗ R . Hence by Theorem 3.1 there exists an R -linear map F e + e ∗ R → R mapping F e + e ∗ r to 1, i.e., r I e + e .It now follows that the F -signature of R is positive since s ( R ) = lim e →∞ ℓ ( R/I e + e ) p ( e + e ) d ≥ lim e →∞ ℓ ( R/ m [ p e ] ) p ( e + e ) d = e HK ( R ) p e d > . (cid:3) Let ( R, m , k ) be a local normal domain with divisor class group Cl( R ). If R is essentially offinite type over C and has rational singularities then Cl( R ) ⊗ Z Q is finitely generated over Q ,see [Kaw88, Lemma 1]. Since F -rational rings have pseudorational singularities one mighthope an analogous statement holds for local F -finite rings with F -rational singularities. Inparticular, one might also expect local strongly F -regular rings to have finitely generateddivisor class groups by Corollary 3.3. uestion 1. Let ( R, m , k ) be a local F -finite and F -rational ring of prime characteristic p > with divisor class group Cl( R ) . Is Cl( R ) ⊗ Z Q a finitely generated Q -vector space?Moreover, do local strongly F -regular rings have finitely generated divisor class group? Utilizing results of Boutot, [Bou78], along with local uniformization of threefolds, [Abh66,CP19], Javier Carvajal-Rojas and Axel St¨abler have recently answered Question 1 for 3-dimensional F -finite rings of prime characteristic with perfect residue field, see [CS20, Sec-tion 4]. However, even if local uniformization is shown to hold in higher dimensions, themethods of [CS20] will not suffice to answer Question 1 for rings of dimension greater than3. References [Abh66] Shreeram Shankar Abhyankar.
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Department of Mathematics, University of Utah, Salt Lake City, UT 84102 USA
E-mail address : [email protected]@math.utah.edu