A prime-characteristic analogue of a theorem of Hartshorne-Polini
aa r X i v : . [ m a t h . A C ] M a y A PRIME-CHARACTERISTIC ANALOGUE OF A THEOREM OF HARTSHORNE-POLINI
NICHOLAS SWITALA AND WENLIANG ZHANGA
BSTRACT . Let R be an F -finite Noetherian regular ring containing an algebraically closed field k of positivecharacteristic, and let M be an F -finite F -module over R in the sense of Lyubeznik (for example, any localcohomology module of R ). We prove that the F p -dimension of the space of F -module morphisms M → E ( R/ m ) (where m is any maximal ideal of R and E ( R/ m ) is the R -injective hull of R/ m ) is equal to the k -dimension of the Frobenius stable part of Hom R ( M, E ( R/ m )) . This is a positive-characteristic analogue ofa recent result of Hartshorne and Polini for holonomic D -modules in characteristic zero. We use this result tocalculate the F -module length of certain local cohomology modules associated with projective schemes.
1. I
NTRODUCTION
In the study of finiteness properties of local cohomology modules there has been an emerging theme: theparallel between (holonomic) D -modules in characteristic zero and ( F -finite) F -modules in characteristic p > ( cf. [18],[15]). This paper continues the line of research under the same theme: we prove F -moduleanalogues of results obtained by Hartshorne and Polini in [7] and by the authors in [16] for D -modules overformal power series or polynomial rings. The theory of ( F -finite) F -modules will be reviewed in the nextsection. We begin by recalling the results for holonomic D -modules in characteristic zero. Theorem 1.1. (a) [7, Corollary 5.2] , [14, Theorem 5.1] Let R = k [[ x , . . . , x n ]] where k is a field of char-acteristic zero, let m ⊆ R be the maximal ideal, and let D = D ( R, k ) be the ring of k -linear differentialoperators on R . Denote by E the R -module H n m ( R ) , which is an R -injective hull of k . If M is a holo-nomic D -module, then dim k H n dR ( M ) = dim k H ( D ( M )) = dim k Hom D ( M, E )= max { t ∈ N | ∃ a D − module surjection M → E t } , where H i dR ( − ) denotes the de Rham cohomology of a D -module, and D ( − ) = Hom R ( − , E ) is the Matlis dual functor.(b) [16, Theorem 5.3]
Let R = k [ x , . . . , x n ] where k is a field of characteristic zero, let m ⊆ R be theirrelevant maximal ideal, and let D = D ( R, k ) be the ring of k -linear differential operators on R .Denote by E the R -module H n m ( R ) , which (with the correct choice of grading) is a graded R -injectivehull of k . If M is a finitely generated graded D -module with finite-dimensional de Rham cohomologyspaces (for example, a graded holonomic D -module), then dim k H n dR ( M ) = dim k H ( * D( M )) = dim k Hom D ( M, E )= max { t ∈ N | ∃ a D − module surjection M → E t } , where * D( − ) = * Hom R ( − , E ) is the graded Matlis dual functor. In Theorem 1.1(b), a graded D -module is a graded R -module on which the operators ∂ j = ∂∂x j ∈ D act as graded k -linear maps of degree − . If I ⊆ k [[ x , . . . , x n ]] is an ideal (resp. I ⊆ k [ x , . . . , x n ] is Mathematics Subject Classification.
Primary 13A35; secondary 13A02, 13D45.
Key words and phrases.
Matlis duality, local cohomology, Frobenius modules, F -modules.The first author gratefully acknowledges NSF support through grant DMS-1604503. The second author is partially supportedby the NSF through grant DMS-1606414 and CAREER grant DMS-1752081. a homogeneous ideal), the local cohomology modules H iI ( R ) are holonomic (resp. graded holonomic) D -modules, and so Theorem 1.1 can be applied to them. The following formulas of Hartshorne and Polinifor the D -module length of certain local cohomology modules associated with projective schemes can bededuced from this theorem (either Theorem 1.1(a) or Theorem 1.1(b) on its own suffices, though differentarguments are required in each case). Theorem 1.2. [7, Theorems 4.8, 6.4]
Let X ⊆ P nk be a smooth, irreducible projective scheme of codimen-sion c < n , where k is an algebraically closed field of characteristic zero. Let I ⊆ R = k [ x , . . . , x n ] bethe homogeneous defining ideal of X . Then we have ℓ D ( H iI ( R )) = β n − c − β n − c − + 1 i = cβ n − i − β n − i − c < i < n ≤ i < c or i ≥ n where ℓ D ( − ) denotes the length of a D -module and β j = dim k H j dR ( X ) is the k -dimension of the j thalgebraic de Rham cohomology space of X as defined in [6] . The main result of this paper, Theorem A, is an F -module analogue of Theorem 1.1. The finiteness con-dition analogous to the holonomicity of D -modules is the F -finiteness of F -modules, and local cohomologymodules satisfy this condition. De Rham cohomology for D -modules is ill-behaved in positive characteris-tic, so any analogue of Theorem 1.1 will require a replacement for H ( D ( M )) . The desired replacementturns out to be the (Frobenius) stable part of D ( M ) . (See section 2 below for the relevant definitions.)The hypotheses of Theorem A are more general than those of Theorem 1.1 and include both the cases ofpolynomial and formal power series rings; observe that there is no need to state and prove a graded versionseparately. Theorem A (Corollary 3.4) . Let R be a regular Noetherian ring containing an algebraically closed field k of characteristic p > , and let m ⊆ R be any maximal ideal. Let E = E R ( R/ m ) be the R -injective hull of R/ m , and denote by D ( − ) the exact functor Hom R ( − , E ) on the category of R -modules. Assume that theFrobenius F : R → R is a finite morphism. If M is an F -finite F -module over R , then dim F p Hom F ( M, E ) = dim k D ( M ) s = max { t ∈ N | ∃ an F − module surjection M → E t } , where ( − ) s denotes the stable part of a Frobenius module, and Hom F denotes the F p -space of F -modulemorphisms. We can use Theorem A to obtain formulas for the F -module length of certain local cohomology modulesassociated with projective schemes, which constitute a positive-characteristic analogue of Theorem 1.2.´Etale cohomology replaces algebraic de Rham cohomology in these formulas. Theorem B.
Let X ⊆ P nk be a projective scheme over an algebraically closed field k of characteristic p > , and let I ⊆ R = k [ x , . . . , x n ] be the homogeneous defining ideal of X .(a) (Theorem 4.1) Let m ⊆ R be the irrelevant maximal ideal, and let E = H n +1 m ( R ) . Then we have H j ´et ( X, F p ) ∼ = Hom F ( H n − jI ( R ) , E ) as F p -spaces, for all j > .(b) (Corollary 4.3) If X is Cohen-Macaulay, then ℓ F ( H n − jI ( R )) = dim F p ( H j ´et ( X, F p )) for all < j < dim( X ) , where ℓ F ( − ) denotes the length of a F -module. PRIME-CHARACTERISTIC ANALOGUE OF A THEOREM OF HARTSHORNE-POLINI 3
In section 2, we collect the necessary preliminary material on Frobenius modules and F -modules. Muchof this section recalls definitions and results due to Hartshorne and Speiser, Lyubeznik, and Bhatt and Lurie,but Proposition 2.13 appears to us to be new. We prove Theorem A in section 3, and finally deduce TheoremB in section 4. Acknowledgments.
The first author thanks Rankeya Datta, Linquan Ma, and Mircea Mustat¸˘a for helpfuldiscussions. 2. F
ROBENIUS MODULES AND F - MODULES
We begin with some notation and conventions. All rings considered in this paper are commutative withidentity . Except in section 1, all rings are of characteristic p > unless otherwise noted. Throughout thissection, A denotes such a ring, and we reserve the letter R for regular Noetherian rings; we will repeat theseassumptions in the hypotheses of definitions and theorems. All local rings are assumed to be Noetherian.We denote by F (or F A if the context demands) the Frobenius endomorphism F : A → A defined by F ( a ) = a p for all a ∈ A . If M is an A -module, we can consider the A -modules F ∗ M and F ∗ M . The A -module F ∗ M has the same underlying Abelian group as M , with A -action defined by a ∗ m = a p m . Onthe other hand, as an Abelian group, F ∗ M can be expressed as A ⊗ F A M , where the notation means thatwe form the tensor product by regarding A as a right A -module via the Frobenius. Explicitly, for a, b ∈ A and m ∈ M , we have a ( b ⊗ m ) = ab ⊗ m and a ⊗ bm = ab p ⊗ m .The following is just the well-known adjunction between restriction and extension of scalars; we recordit here separately so as to have a specific reference for the formulas in the sequel. Proposition 2.1.
Let A be a ring of characteristic p > , and let M be an A -module. There is a bijectivecorrespondence between A -linear maps M → F ∗ M and A -linear maps F ∗ M → M .Proof. If ϕ : M → F ∗ M is an A -linear map, the corresponding A -linear map ψ : F ∗ M = A ⊗ F A M → M is defined by ψ ( a ⊗ m ) = aϕ ( m ) . Conversely, if ψ : F ∗ M = A ⊗ F A M → M is an A -linear map, thecorresponding A -linear map ϕ : M → F ∗ M is defined by ϕ ( m ) = ψ (1 ⊗ m ) . (cid:3) The main objects of this paper are A -modules equipped with A -linear maps to or from their pushforwardsand pullbacks along the Frobenius F A . Frobenius modules over A (that is, A -modules M equipped with achoice of A -linear map M → F ∗ M ) were studied by Hartshorne and Speiser in [8] and, more recently,by Bhatt and Lurie in [1]. On the other hand, F -modules (that is, A -modules M equipped with a choice of A -linear isomorphism M → F ∗ M ), also known as unit Frobenius modules , were introduced by Lyubeznikin [12] and studied further by Emerton and Kisin in [5] and Bhatt and Lurie in [1], in the case where A isregular and Noetherian. (By a celebrated theorem of Kunz [11, Theorem 2.1], the functor F ∗ is exact underthese hypotheses, and this exactness is crucial to the theory of F -modules.)We now proceed to give the basic definitions and relationships between these objects. Definition 2.2.
Let A be a ring of characteristic p > . A Frobenius module over A is a pair ( M, ϕ M ) where M is an A -module and ϕ M : M → F ∗ M is an A -linear map. (When there is no danger of confusion, wesometimes write ϕ for ϕ M ; we also sometimes refer simply to M as a “Frobenius module”.)An A -linear map M → F ∗ M is the same thing as an additive map ϕ M : M → M such that ϕ M ( am ) = a p ϕ M ( m ) for all a ∈ A and m ∈ M . In particular, the iterates ϕ iM for i ≥ make sense. Frobenius modulesover A form a category Mod Fr A , where a morphism ( M, ϕ M ) → ( N, ϕ N ) is an A -linear map f : M → N such that ϕ N ◦ f = F ∗ f ◦ ϕ M . We denote by Hom A [ F ] ( M, N ) the F p -space of Frobenius module morphisms ( M, ϕ M ) → ( N, ϕ N ) . The reason for the notation is that a Frobenius module over A is the same thing as aleft module over the non-commutative ring A [ F ] generated over A by the symbol F , subject to the relations F a = a p F for all a ∈ A . NICHOLAS SWITALA AND WENLIANG ZHANG
Definition 2.3.
Let A be a ring of characteristic p > , and let ( M, ϕ M ) be a Frobenius module over A .(a) The (Frobenius) fixed part M ϕ =1 of M is the F p -subspace { m ∈ M | ϕ M ( m ) = m } ⊆ M .(b) Suppose that A contains a perfect field k of characteristic p > . The (Frobenius) stable part M s of M is the k -subspace ∩ i ≥ ϕ iM ( M ) ⊆ M .If k is any field of characteristic p > , the only solutions λ ∈ k to the equation λ p = λ are the elementsof the prime subfield F p ⊆ k . Therefore the fixed part M ϕ =1 can only be an F p -subspace. If k is perfect, theiterated images ϕ iM ( M ) are k -subspaces of M , and so the same is true for M s .It is clear that M ϕ =1 ⊆ M s . Under stronger hypotheses on k , we can say something more about therelationship between the fixed and stable parts: Proposition 2.4. [3, Exp. XXII, Corollaire 1.1.10]
Let A be a ring containing an algebraically closed field k of characteristic p > . Let ( M, ϕ M ) be a Frobenius module over A . If M s is a finite-dimensional k -space,then there is an isomorphism k ⊗ F p M ϕ =1 ∼ = M s of k -spaces. In particular, there exists a k -basis { m , . . . , m l } of M s such that ϕ M ( m i ) = m i for i =1 , . . . , l . Proposition 2.4 implies that if M s is a finite-dimensional k -space, then M ϕ =1 is a finite-dimensional F p -space (of the same dimension). The converse, however, is not true, as the following example shows. Example . Let R = k [ x ] where k is a perfect field of characteristic p > , let F R be the Frobeniusendomorphism on R , and consider the perfection R /p ∞ = k [ x /p ∞ ] , that is, the colimit lim −→ ( R F R −−→ R F R −−→ R F R −−→ · · · ) . The Frobenius endomorphism F R /p ∞ is bijective. Therefore, if we regard ( R /p ∞ , F R /p ∞ ) as a Frobe-nius module over R , we have ( R /p ∞ ) s = R /p ∞ , which is not a finite-dimensional k -space. However, ( R /p ∞ ) ϕ =1 is simply F p , a one-dimensional F p -space. This is true whether or not k is algebraically closed.The following result of Hartshorne and Speiser provides one useful case in which the finiteness of thestable part is known (and so Proposition 2.4 applies). Theorem 2.6. [8, Theorem 1.12]
Let A be a local ring containing a perfect field k of characteristic p > , and let ( M, ϕ M ) be a Frobenius module over A . If M is an Artinian A -module, then M s is a finite-dimensional k -space, and the induced map ϕ : M s → M s is bijective. If ( M, ϕ M ) is a Frobenius module over A , we have by Proposition 2.1 an A -linear map ψ M : F ∗ M → M . When ψ M is an isomorphism, M is called a unit Frobenius module by some authors [5, 1]. In Lyubeznik’s[12], which deals only with the case of a regular Noetherian ring R , unit Frobenius modules over R are called F -modules (or F R -modules). We will follow Lyubeznik’s notation and terminology. Definition 2.7.
Let R be a regular Noetherian ring of characteristic p > . An F -module over R (or F R -module) is a pair ( M, θ M ) where M is an R -module and θ M : M ∼ −→ F ∗ M is an R -module isomorphism,called the structure morphism . (When there is no danger of confusion, we sometimes refer simply to M asan “ F -module”.)Of course, if ( M, θ M ) is an F R -module, then ( M, ϕ M ) is a Frobenius module over R , where ϕ M : M → F ∗ M is the R -linear map that corresponds via adjunction to ψ M = θ − M .There is a category F R - Mod of F -modules over R , where a morphism ( M, θ M ) → ( N, θ N ) is an R -linear map f : M → N such that θ N ◦ f = F ∗ f ◦ θ M . We denote by Hom F ( M, N ) the F p -space of F -module morphisms ( M, θ M ) → ( N, θ N ) . In particular, we can speak of F -submodules : an F -submodule PRIME-CHARACTERISTIC ANALOGUE OF A THEOREM OF HARTSHORNE-POLINI 5 N of an F -module ( M, θ M ) is an R -submodule N ⊆ M such that the restriction θ M | N is an isomorphism N ∼ −→ F ∗ N (since R is regular, F ∗ is exact and so F ∗ N can always be identified with an R -submodule of F ∗ M ).In the fruitful analogy between D -modules in characteristic zero and F -modules in positive characteris-tic, the finiteness condition of holonomicity for D -modules corresponds to the condition of “ F -finiteness”defined below (in particular, local cohomology modules provide examples of each). Loosely speaking, an F -finite F -module is one built from a finitely generated R -module by repeatedly applying the functor F ∗ and passing to a colimit. Definition 2.8.
Let R be a regular Noetherian ring of characteristic p > , and let ( M, θ M ) be an F -module. We say that M is F -finite if there exists a finitely generated R -module M ′ and an R -linear map β : M ′ → F ∗ M ′ such that lim −→ ( M ′ β −→ F ∗ M ′ F ∗ β −−→ ( F ∗ ) M ′ → · · · ) ∼ = M, and the structure morphism θ M is induced by taking the colimit over l of ( F ∗ ) l β : ( F ∗ ) l M ′ → ( F ∗ ) l +1 M ′ .In this case we call M ′ a generator of M and β a generating morphism . Example . Let R be a regular Noetherian ring of characteristic p > . The following are the most relevantexamples of F -finite F -modules for the purposes of this paper.(a) R itself is an F -finite F -module. The corresponding Frobenius module structure is given by ϕ R = F R ,and id R is an F -module generating morphism for R . Moreover, R is a simple F -module, since F -submodules of R are ideals I ⊆ R such that the natural surjection R/I [ p ] → R/I is an isomorphism(here I [ p ] is the ideal generated by all p th powers of elements of I ), and as R is Noetherian, this canonly happen if I = (0) or I = R .(b) If I ⊆ R is an ideal and i ≥ , the local cohomology module H iI ( R ) is an F -finite F -module [12,Example 2.2(b)].(c) If m ⊆ R is a maximal ideal, then the R -injective hull E = E R ( R/ m ) of the R -module R/ m is an F -finite F -module [15, Proposition 5.4(d)]. Moreover, E is a simple F -module, since R/ m is a simple R -module as well as an F -module generator of E . Definition 2.10.
Let A be a ring of characteristic p > . We say that A is F -finite if F ∗ A is a finitelygenerated A -module.For example, if k is a perfect field of characteristic p > , the rings k [ x , . . . , x n ] and k [[ x , . . . , x n ]] are F -finite. We recall the well-known facts that if R is an F -finite regular Noetherian ring of characteristic p > , then R is F -split (meaning that the R -module homomorphism R → F ∗ R defined by the Frobeniusadmits a section) and F ∗ R is locally free as an R -module (because it is finitely generated as well as flat).In what follows, for each commutative ring A of characteristic p > and each A -module M , we willuse ( A ⊗ F A M ) | F A to denote A ⊗ F A M viewed as an A -module by restriction of scalars along F A : A → A ,so that a ∗ ( b ⊗ z ) = a p b ⊗ z = b ⊗ az for all a, b ∈ A and z ∈ M . Proposition 2.11.
Let R be a regular Noetherian ring of characteristic p > , and let J be an injective R -module. Denote by D J ( − ) the exact functor Hom R ( − , J ) on the category of R -modules. There are R -module homomorphisms δ M : F ∗ D J ( M ) → D J ( F ∗ M ) for all R -modules M , functorial in M . Furthermore, if R is F -finite and M is a finitely generated R -module,then the δ M is an isomorphism. Note the distinction between F -finiteness , a property of a ring, and F -finiteness , a property of an F -module. NICHOLAS SWITALA AND WENLIANG ZHANG
Proof.
By [9, Proposition 1.5], since R is Gorenstein, J ∼ = F ∗ J as R -modules. Fix a choice of R -moduleisomorphism θ J : J → F ∗ J . For M an R -module, we define δ M to be the composite F ∗ D J ( M ) = R ⊗ F R Hom R ( M, J ) → Hom R ( M, ( R ⊗ F R J ) | F R ) (i) → Hom R ( R ⊗ F R M, R ⊗ F R J ) (ii) → Hom R ( R ⊗ F R M, J ) = D J ( F ∗ M ) , (iii)where • the map in (i) carries r ⊗ ϕ ∈ R ⊗ F R Hom R ( M, J ) to the element of Hom R ( M, ( R ⊗ F R J ) | F R ) whose value on m ∈ M is r ⊗ ϕ ( m ) (given our construction of ( R ⊗ F R J ) | F R , the map m r ⊗ ϕ ( m ) is indeed R -linear); • the map in (ii) is the isomorphism arising from the adjunction between extension and restriction ofscalars; and • the map in (iii) is given by post-composition with θ − J .It is easy to see that all three maps are functorial in M and that the middle and bottom arrows are isomor-phisms for all R -modules M , so it remains to show that if R is F -finite, the top arrow is an isomorphismwhenever M is finitely generated. This statement can be checked locally. Let p ∈ Spec( R ) be given, and let M be a finitely generated R -module. The localization ( R ⊗ F R Hom R ( M, J )) p → Hom R ( M, ( R ⊗ F R J ) | F R ) p of the top displayed arrow at p factors naturally as ( R ⊗ F R Hom R ( M, J )) p → R p ⊗ F R p Hom R ( M, J ) p → R p ⊗ F R p Hom R p ( M p , J p ) → Hom R p ( M p , ( R p ⊗ F R p J p ) | F R p ) → Hom R ( M, ( R ⊗ F R J ) | F R ) p , where the top arrow is an isomorphism for all R -modules M and the second and fourth arrows are isomor-phisms because R is Noetherian and so M is finitely presented. Finally, the third arrow is an isomorphismbecause R is regular and F -finite, so ( F R p ) ∗ R p is a finite free R p -module, and therefore ( R p ⊗ F R p J p ) | F R p is isomorphic as an R p -module to a finite direct sum of copies of J p . This completes the proof. (cid:3) Given two F -modules over R (say M and N ) we can regard them as Frobenius modules, and considerthe F p -space of F -module (resp. Frobenius module) morphisms between them. We show in Proposition 2.13below that not only are these two sets of morphisms the same, but that this set arises as the fixed part of acertain Frobenius module structure on Hom R ( M, N ) itself, explained next. Remark . Let R be a regular Noetherian ring of characteristic p > . Let ( M, θ M ) and ( N, θ N ) be F -modules over R . Then Hom R ( M, N ) admits a natural Frobenius module structure as follows. Define ϕ : Hom R ( M, N ) → F ∗ Hom R ( M, N ) by ϕ ( f ) = θ − N ◦ (id R ⊗ f ) ◦ θ M for each f ∈ Hom R ( M, N ) . It is clear that ϕ is additive; it remains to show it is R -linear. Given any r ∈ R ,we have ϕ ( rf ) = θ − N ◦ (id R ⊗ rf ) ◦ θ M = θ − N ◦ ( µ r p ⊗ f ) ◦ θ M = µ r p ◦ θ − N ◦ (id R ⊗ f ) ◦ θ M = r p ϕ ( f ) = r ∗ ϕ ( f ) for all f ∈ Hom R ( M, N ) , where µ s (for any s ∈ R ) denotes multiplication by s . It follows that ϕ is R -linear, and hence it provides a Frobenius module structure on Hom R ( M, N ) . PRIME-CHARACTERISTIC ANALOGUE OF A THEOREM OF HARTSHORNE-POLINI 7
Proposition 2.13.
Let R be a regular Noetherian ring of characteristic p > . Let ( M, θ M ) and ( N, θ N ) be F -modules over R . Regard M (resp. N ) as a Frobenius module via the R -linear map ϕ M : M → F ∗ M (resp. ϕ N : N → F ∗ N ) corresponding via adjunction to θ − M (resp. θ − N ). Then Hom R [ F ] ( M, N ) = Hom F ( M, N ) = Hom R ( M, N ) ϕ =1 as F p -subspaces of Hom R ( M, N ) , where the Frobenius module structure ϕ on Hom R ( M, N ) is defined asin Remark 2.12.Proof. The first equality has nothing to do with the choice of ϕ . Let f ∈ Hom R ( M, N ) be given. On theone hand, the map f belongs to Hom R [ F ] ( M, N ) if and only if ϕ N ◦ f = F ∗ f ◦ ϕ M . On the other hand, f belongs to Hom F ( M, N ) if and only if θ N ◦ f = F ∗ f ◦ θ M ; equivalently, f ◦ θ − M = θ − N ◦ F ∗ f . We have f ( θ − M ( r ⊗ m )) = f ( rϕ M ( m )) = rf ( ϕ M ( m )) and θ − N ( F ∗ f ( r ⊗ m )) = θ − N ( r ⊗ f ( m )) = rϕ N ( f ( m )) for all r ∈ R and m ∈ M , and so the equality f ◦ θ − M = θ − N ◦ F ∗ f is equivalent to ϕ N ◦ f = F ∗ f ◦ ϕ M .Thus Hom R [ F ] ( M, N ) = Hom F ( M, N ) .For the second equality, observe that a map f ∈ Hom R ( M, N ) belongs to the fixed part Hom R ( M, N ) ϕ =1 if and only if f = θ − N ◦ (id R ⊗ f ) ◦ θ M , or equivalently, θ N ◦ f = F ∗ f ◦ θ M . This is exactly the criterionfor f to be an F -module morphism, completing the proof. (cid:3) Remark . Let R be a regular Noetherian ring of characteristic p > , and let ( M, θ M ) be an F -moduleover R . By Example 2.9(a), the R -module R itself is an F -module with corresponding Frobenius modulestructure given by ϕ R = F R . Under the canonical identification of M with Hom R ( R, M ) , the Frobeniusmodule structure on Hom R ( R, M ) provided by Remark 2.12 coincides with the Frobenius module structure ϕ M corresponding by adjunction to the given F -module structure on M itself. Indeed, if f : R → M isdefined by f (1) = m , then the composite R θ R −→ F ∗ R id R ⊗ f −−−−→ F ∗ M θ − M −−→ M maps ⊗ ⊗ m ϕ M ( m ) . 3. P ROOF OF THE MAIN THEOREM
Lemma 3.1.
Let R be a regular Noetherian ring of characteristic p > , and let J be an injective R -module.Denote by D J ( − ) the exact functor Hom R ( − , J ) on the category of R -modules. If M is an R -module, thenany R -module homomorphism M → F ∗ M induces a Frobenius module structure on D J ( M ) .Proof. Apply the functor D J to the given map, obtaining an R -linear map D J ( F ∗ M ) → D J ( M ) . Pre-composition with the map δ M defined in Proposition 2.11 gives an R -linear map F ∗ D J ( M ) → D J ( M ) ,which corresponds by adjunction (Proposition 2.1) to an R -linear map D J ( M ) → F ∗ D J ( M ) , the desiredFrobenius module structure. (cid:3) In particular, if M is an F -module (resp. a generator of an F -finite F -module), then for any injective R -module J , D J ( M ) has a Frobenius module structure obtained by applying Lemma 3.1 to the structuremorphism θ M (resp. the generating morphism β ). Theorem 3.2.
Let R be an F -finite regular Noetherian ring containing an algebraically closed field k ofcharacteristic p > , and let J be an injective R -module. Denote by D J ( − ) the exact functor Hom R ( − , J ) on the category of R -modules. Assume that the following conditions are satisfied:(i) J , which is an F -module by [9, Proposition 1.5] , is simple as an F -module; NICHOLAS SWITALA AND WENLIANG ZHANG (ii) for every finitely generated R -module M ′ equipped with a choice of R -module homomorphism M ′ → F ∗ M ′ , the stable part D J ( M ′ ) s (which is defined by Lemma 3.1) is a finite-dimensional k -space, andthe Frobenius structure on D J ( M ′ ) restricts to a bijection on D J ( M ′ ) s .Then, for each F -finite F -module M , the following numbers are all equal (and, in particular, are all finite):(1) the F p -dimension of Hom R [ F ] ( M, J ) = Hom F ( M, J ) ,(2) the F p -dimension of D J ( M ) ϕ =1 ,(3) the k -dimension of D J ( M ) s ,(4) the k -dimension of D J ( M ′ ) s , where M ′ is any F -module generator of M ,(5) the maximal integer t such that there exists a surjective F -module morphism (equivalently, surjectiveFrobenius module morphism) M → J t .Proof. Let M be an F -finite F -module over R , and let β : M ′ → F ∗ M ′ be an F -module generatingmorphism for M . We have already proved the equality of (1) and (2) above, in Proposition 2.13. Since M ′ is a finitely generated R -module, our condition (ii) implies that D J ( M ′ ) s is a finite-dimensional k -space,that is, that (4) is finite. If we can prove the equality of (3) and (4) and hence the finiteness of (3), then theequality of (2) and (3) will follow from Proposition 2.4. Therefore we need only prove the equality of (1)and (5) as well as the equality of (3) and (4).We begin with the equality of (1) and (5). Suppose first that there exists an F -module surjection M → J t .Post-composing it with each of the t coordinate projections J t → J produces t F p -linearly independent F -module morphisms M → J .Conversely, assume there are t such F p -linearly independent F -module morphisms ϕ , . . . , ϕ t : M → J ;we wish to construct an F -module surjection M → J t , or equivalently, an F -submodule N ⊆ M suchthat M/N is isomorphic to J t . Since J is a simple F -module by our condition (i), each ϕ i must itself besurjective, since its image is a non-zero F -submodule of J . Set M i = ker( ϕ i ) , an F -submodule of M , forall i . Since M/M i ∼ = J , we must have M i + M j = M whenever i = j . We claim that M/ ( ∩ ti =1 M i ) ∼ = J t , which will complete the proof (take N = ∩ ti =1 M i ); we do this by showing, by induction on j , that M/ ( ∩ ji =1 M i ) ∼ = J j for ≤ j ≤ t . This assertion is obvious for j = 1 . Now suppose that for some ≤ j < t we know that M/ ( ∩ ji =1 M i ) ∼ = J j as F -modules. We cannot have ∩ ji =1 M i ⊆ M j +1 , sinceotherwise ϕ j +1 would factor through M → M/ ( ∩ ji =1 M i ) and hence would lie in the F p -span of ϕ , . . . , ϕ j ,a contradiction. Therefore ∩ ji =1 M i * M j +1 , and so ∩ ji =1 M i + M j +1 = M by the simplicity. But then theshort exact sequence → M/ ( ∩ j +1 i =1 M i ) → M/ ( ∩ ji =1 M i ) ⊕ M/M j +1 → M/ ( M j +1 + ∩ ji =1 M i ) → of F -modules implies that M/ ( ∩ j +1 i =1 M i ) ∼ = M/ ( ∩ ji =1 M i ) ⊕ M/M j +1 ∼ = J j ⊕ J ∼ = J j +1 by the inductionhypothesis, as desired.Finally, we prove the equality of (3) and (4). By definition, M = lim −→ ( M ′ β −→ F ∗ M ′ F ∗ β −−→ ( F ∗ ) M ′ → · · · ) and the F -module structure on M is induced by β and its F ∗ -iterates. Applying D J ( − ) , we find D J ( M ) ∼ = lim ←− ( · · · → D J (( F ∗ ) M ′ ) D J ( F ∗ β ) −−−−−−→ D J ( F ∗ M ′ ) D ( β ) −−−→ D J ( M ′ )) . Since R is F -finite, not only M ′ but also ( F ∗ ) l M ′ for all l ≥ are finitely generated R -modules. Therefore,using Proposition 2.11 to identify D (( F ∗ ) l M ′ ) with ( F ∗ ) l D J ( M ′ ) for all l ≥ , we can rewrite the limit as D J ( M ) ∼ = lim ←− ( · · · → ( F ∗ ) D J ( M ′ ) → F ∗ D J ( M ′ ) → D J ( M ′ )) . Since ( F ∗ ) l and ( F l ) ∗ are isomorphic functors, this is exactly the leveling functor of [8, p. 47]. (In thenotation of [8], we have D J ( M ) = G ( D J ( M ′ )) .) It follows from the proof of [8, Proposition 1.2(b)] (see PRIME-CHARACTERISTIC ANALOGUE OF A THEOREM OF HARTSHORNE-POLINI 9
Remark 3.3 below) that D J ( M ) s ∼ = lim ←− ( · · · → k ⊗ F k D J ( M ′ ) s → k ⊗ F k D J ( M ′ ) s → D J ( M ′ ) s ) , where F k : k → k is the Frobenius endomorphism of k , and the maps k ⊗ F l +1 k D J ( M ′ ) s → k ⊗ F lk D J ( M ′ ) s are given by the identity on the first tensor factor and the restriction of the map D J ( M ′ ) → D J ( M ′ ) definingthe Frobenius module structure on D J ( M ′ ) in the second. But by our condition (ii), this last map restrictsto a bijection from D J ( M ′ ) s to itself. That is, the displayed limit can be identified with D J ( M ′ ) s , so that D J ( M ) s ∼ = D J ( M ′ ) s as k -spaces, completing the proof of the equality of (3) and (4) and therefore theproof of the theorem. (cid:3) Remark . In the proof of Theorem 3.2 above, we appealed to [8, Proposition 1.2(b)]. This proposition isstated in [8] only for a ring R of characteristic p > such that F ∗ R is a free R -module, a hypothesis that isstronger than ours. However, examining the proof of [8, Proposition 1.2(b)], it is clear that this hypothesisis only used in the form of the following consequence: if M is an R -module and m, m ′ ∈ M are such that ⊗ m = 1 ⊗ m ′ in F ∗ M = R ⊗ F R M , then m = m ′ . But since we assumed in Theorem 3.2 that R isregular and F -finite, it is also F -split, from which the previous statement is immediate.The following corollary of Theorem 3.2, which identifies a class of injective modules for which thehypotheses of the theorem are satisfied, is the main result of this paper. Corollary 3.4.
Let R be a regular, F -finite Noetherian ring containing an algebraically closed field k ofcharacteristic p > , and let m ⊆ R be any maximal ideal. Let E = E R ( R/ m ) be the R -injective hull of R/ m , and denote by D ( − ) the exact functor Hom R ( − , E ) on the category of R -modules. Then, for each F -finite F -module M , the following numbers are all equal (and, in particular, are all finite):(1) the F p -dimension of Hom R [ F ] ( M, E ) = Hom F ( M, E ) ,(2) the F p -dimension of D ( M ) ϕ =1 ,(3) the k -dimension of D ( M ) s ,(4) the k -dimension of D ( M ′ ) s , where M ′ is any F -module generator of M ,(5) the maximal integer t such that there exists a surjective F -module morphism (equivalently, surjectiveFrobenius module morphism) M → E t .Proof. By Example 2.9(c), E is a simple F -module, so condition (i) of Theorem 3.2 is satisfied. Now let M ′ be a finitely generated R -module equipped with a choice of R -module homomorphism M ′ → F ∗ M ′ .There exists an R -linear surjection R l → M ′ for some l ≥ ; applying the exact functor D , we obtain an R -linear injection D ( M ′ ) → D ( R l ) = E l . Since E (and hence E l ) is an Artinian R -module supportedonly at m , the same is true of D ( M ′ ) , so D ( M ′ ) has a natural structure as a module over the m -adiccompletion b R m of R . In fact, D ( M ′ ) is a Frobenius module over b R m , with the Frobenius structure given bythe same underlying additive map D ( M ′ ) → D ( M ′ ) defined by Lemma 3.1. The ring b R m is local (since m is maximal) and contains an algebraically closed (hence perfect) field k ; moreover, D ( M ′ ) is Artinian asan b R m -module. Therefore, Theorem 2.6 applies. We conclude that D ( M ′ ) s is a finite-dimensional k -spaceand the Frobenius structure on D ( M ′ ) restricts to a bijection on D ( M ′ ) s , so condition (ii) of Theorem 3.2is satisfied. The corollary now follows from Theorem 3.2 applied to J = E . (cid:3) The proof of the equality of (1) and (5) in Corollary 3.4 works in characteristic zero as well, replacing“ F -finite F -module” with “holonomic D -module”. Therefore we obtain an alternate proof of the fact thatif R = k [ x , . . . , x n ] or k [[ x , . . . , x n ]] where k is a field of characteristic zero, and M is a holonomic D ( R, k ) -module, then dim k Hom D ( M, E ) is equal to the maximal integer t for which there exists a D -linear surjection M → E t . This statement is part of [7, Corollary 5.2]. An easier “dual” statement is thefollowing [16, Lemma 2.3]: dim k Hom D ( R, M ) is equal to the maximal integer t for which there exists a D -linear injection R t → M . We can prove a version of this in the Frobenius module setting, as part ofa “dual” version of Corollary 3.4. Note, however, that Theorem 3.5 has a finite-dimensionality hypothesiswhose analogue is not needed (because it is automatically satisfied) in Corollary 3.4. Theorem 3.5.
Let R be a regular Noetherian ring containing an algebraically closed field k of characteristic p > , and let M be an F -finite F -module over R . Assume that M s is a finite-dimensional k -space. Thenthe following numbers are all equal (and, in particular, are all finite):(1) the F p -dimension of Hom R [ F ] ( R, M ) = Hom F ( R, M ) ,(2) the F p -dimension of M ϕ =1 ,(3) the k -dimension of M s ,(4) the maximal integer t such that there exists an injective F -module morphism (equivalently, injectiveFrobenius module morphism) R t → M .Proof. By Remark 2.14, we can identify M with Hom R ( R, M ) as Frobenius modules over R , and thereforewe can identify M ϕ =1 with Hom R ( R, M ) ϕ =1 as F p -spaces. Therefore the equality of (1) and (2) followsfrom Proposition 2.13. We have assumed that M s is a finite-dimensional k -space, so the equality of (2)and (3) follows from Proposition 2.4. Finally, the proof of the equality of (1) and (4) is essentially dualto the proof of the equality of (1) and (5) in Theorem 3.2, using the fact that R is a simple F -module(Example 2.9(a)). The arguments are similar enough that we omit the details, providing a sketch. An F -module injection R t → M gives rise to t F p -linearly independent F -module morphisms R → M bypre-composition with the coordinate inclusions; conversely, given t distinct isomorphic copies of R (say M , . . . , M t ) as F -submodules of M , it can be shown (since all M i are simple F -submodules) that the sum P ji =1 M i ⊆ M is a direct sum for j = 1 , . . . , t by induction on j , and the case j = t is the desiredassertion. (cid:3) Question . If R is a regular Noetherian ring containing an algebraically closed field k of characteristic p > , and M is an F -finite F -module over R , is M s a finite-dimensional k -space?A positive answer to Question 3.6 would, of course, permit us to remove the finite-dimensionality hy-pothesis in Theorem 3.5, since Proposition 2.4 would apply.4. A N APPLICATION TO LOCAL COHOMOLOGY
Our Corollary 3.4 allows us to relate certain ´etale and local cohomology groups associated with projectiveschemes.
Theorem 4.1.
Let X ⊆ P nk be a projective scheme over an algebraically closed field k of characteristic p > . Let I ⊆ R = k [ x , . . . , x n ] be the homogeneous defining ideal of X , let m ⊆ R be the irrelevantmaximal ideal, and let E = H n +1 m ( R ) . Then we have H j ´et ( X, F p ) ∼ = Hom F ( H n − jI ( R ) , E ) as F p -spaces, for all j > . In the proof of Theorem 4.1, we will need the notion of a graded
Frobenius module.
Definition 4.2.
Let R = k [ x , . . . , x n ] where k is a field of characteristic p > . View R as a gradedring ⊕ i ≥ R i with respect to the standard grading (so R = k and deg( x j ) = 1 for ≤ j ≤ n ). A graded Frobenius module over R is a pair ( M, ϕ M ) where M = ⊕ l ∈ Z M l is a graded R -module and ϕ M : M → F ∗ M is a graded R -linear homomorphism (equivalently, ϕ M is an additive map M → M such that ϕ M ( rm ) = r p ϕ M ( m ) for all r ∈ R and m ∈ M and ϕ M ( M l ) ⊆ M pl for all l ∈ Z ). PRIME-CHARACTERISTIC ANALOGUE OF A THEOREM OF HARTSHORNE-POLINI 11
Proof of Theorem 4.1.
By [8, Proposition 5.1], we can identify k ⊗ F p H j ´et ( X, F p ) with the Frobenius stablepart H j ( X, O X ) s of the Zariski cohomology (which carries a natural structure of Frobenius module over R ). By [4, Theorem A4.1], which is the well-known long exact sequence relating local cohomology andZariski cohomology, we have M l ∈ Z H j ( X, O X ( l )) ∼ = H j +1 m ( R ) as graded R -modules (in fact, as graded Frobenius modules) for j > . Since ⊕ l ∈ Z H j ( X, O X ( l )) is a graded Frobenius module, its stable part is contained in its degree-zero component, so H j ( X, O X ) s ∼ = H j +1 m ( R ) s .Graded local duality [2, Theorem 14.4.1] implies that H j +1 m ( R ) ∼ = * D(Ext ( n +1) − ( j +1) R ( R/I, R )) = * D(Ext n − jR ( R/I, R )) as graded R -modules (in fact, as graded Frobenius modules), where * D denotes the graded Matlis dualfunctor * Hom R ( − , E ) . Since Ext n − jR ( R/I, R ) is a finitely generated R -module, we have * Hom R (Ext n − jR ( R/I, R ) , E ) = Hom R (Ext n − jR ( R/I, R ) , E ) as R -modules. At this point we begin systematically to ignore the gradings. The R -linear map Ext n − jR ( R/I, R ) → Ext n − jR ( R/I [ p ] , R ) ∼ = F ∗ (Ext n − jR ( R/I, R )) induced by the surjection R/I [ p ] → R/I is a generating morphism [12, Proposition 1.11] for the F -finite F -module H n − jI ( R ) (here I [ p ] is the ideal generated by all p th powers of elements of I , and the displayedisomorphism holds because R/I is a finitely generated R -module and R is regular so that F ∗ is exact). Thering R satisfies the hypotheses of Corollary 3.4, and E = H n +1 m ( R ) is the R -injective hull of R/ m , so wehave equalities dim F p H j ´et ( X, F p ) = dim k H j ( X, O X ) s = dim k H j +1 m ( R ) s = dim k Hom R (Ext n − jR ( R/I, R ) , E ) s = dim k Hom R ( H n − jI ( R ) , E ) s = dim F p Hom F ( H n − jI ( R ) , E ) , where the last two equalities are part of Corollary 3.4 (and all these numbers are finite). This completes theproof. (cid:3) If X ⊆ P nk is a projective scheme over a field k , defined by a homogeneous ideal I ⊆ R = k [ x , . . . , x n ] ,the length of the local cohomology modules H iI ( R ) as D -modules (if char( k ) = 0 ) or as F -modules (if char( k ) = p > ) is of particular interest. Theorem 1.2 calculates these D -module lengths when char( k ) =0 and X is smooth. In positive characteristic, Katzman, Ma, Smirnov, and the second author’s [10, Theorem4.3] gives an explicit formula for the F -module length of H cI ( R ) (where c is the codimension of X in P nk )under the assumption that R/I has an isolated non- F -rational point at the origin. Our Theorem 4.1 givessome new information about the F -module length of H iI ( R ) in the case i = c . Corollary 4.3.
Keep the notation and hypotheses of Theorem 4.1.(a) Let j > be given, and suppose that the local cohomology module H n − jI ( R ) is supported only at m .Then H n − jI ( R ) ∼ = E λ j , where λ j = dim F p ( H j ´et ( X, F p )) and hence the F -module length of H n − jI ( R ) is precisely dim F p ( H j ´et ( X, F p )) . (b) In particular, if X is Cohen-Macaulay ( i.e. , each local ring O X,x of X is a Cohen-Macaulay local ring),then H n − jI ( R ) ∼ = E λ j , where λ j = dim F p ( H j ´et ( X, F p )) , for all 0 < j < dim( X ) . Proof.
Under the assumption on H n − jI ( R ) , we have H n − jI ( R ) ∼ = E a j (as F -modules) for some non-negativeinteger a j by [12, Theorem 1.4]. It is straightforward to check that Hom F ( E a j , E ) ∼ = F a j p . Therefore, a j = λ j = dim F p ( H j ´et ( X, F p )) by Theorem 4.1, proving part (a).When X is Cohen-Macaulay, it follows from the Peskine-Szpiro vanishing theorem [13, PropositionIII.4.1] that Supp R ( H n − jI ( R )) ⊆ { m } for all j such that n − j = codim( X, P nk ) = n − dim( X ) . Part (b),and hence the corollary, follows. (cid:3) Remark . Let X be a projective scheme over any field k of characteristic p > . Let I be the homogeneousdefining ideal in R = k [ x , . . . , x n ] of X for a choice of embedding X ֒ → P nk . The numbers λ i,j = dim k (Ext iR ( k, H n +1 − jI ( R ))) are called the Lyubeznik numbers of X and were shown in [17, Theorem 1.1] to be independent of the choiceof the embedding of X . As a consequence of Corollary 4.3, one can calculate explicitly each λ i,j under theassumption that X is Cohen-Macaulay; we will leave the details to the interested reader. (The λ i,j do notchange under a field extension, so the hypothesis in Corollary 4.3 that k is algebraically closed causes nodifficulty.) Remark . Let
X, R, I be the same as in Remark 4.4 and, additionally, assume that X is irreducible andeach local ring O X,x is F -rational for each point x ∈ X . Our Theorem 4.1 implies that H n − dim( X ) I ( R ) admits a F -module quotient that is isomorphic to E λ d where λ d = dim F p ( H dim( X )´et ( X, F p )) . This does notfully recover the prime-characteristic analogue of Theorem 1.2 in the case when i = c = n − dim( X ) ; thelength differs by one from the desired analogous result. The reason is that the simple F -submodule of H cI ( R ) does not admit any non-zero F -module morphism to E . However, it follows directly from [10, Theorem 4.3]that H cI ( R ) admits a simple F -submodule H such that H cI ( R ) /H ∼ = E λ d . This completes our analogue ofthe theorem of Hartshorne-Polini. R EFERENCES [1] B. Bhatt and J. Lurie. A Riemann-Hilbert correspondence in positive characteristic. arXiv:1711.04148 , 2017.[2] M. Brodmann and R. Sharp.
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