Length of Local Cohomology in Positive Characteristic and Ordinarity 3

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1 of level e on SpecpRq. Since Dpeqf pe` ´1 “ R for every large enough natural number e by 1 1 1 1 [18, Proof of Lemma 6.8], and the D-submodule D f of Rr f s generated by f is Rr f s itself 1 by [1, Theorem 1.1], the assertion immediately follows from the description of D f given in [7, Remark 2.7]. For a d-dimensional proper variety Z over a field of characteristic p ą 0, we let the d p-genus gppZq of Z be the dimension of the stable part [14] of k b H pZ, OZq, that is l d dimkpXlě0F pk b H pZ, OZqqq, where F is the Frobenius action on coherent cohomology and k is an algebraic closure of the base field. Our main result is (see Theorem 2 for the precise general formulation): Theorem 1. Suppose that f is an irreducible complex polynomial in n ě 3 variables with an isolated singularity at the origin and let Y ÝÑπ X be a resolution of the singularity. 1 Then for almost all p, the D-module length of Hfp pRq is 1 ` gppZpq, where Zp is the reduction modulo p of the exceptional fibre of π. The proof, which mostly belongs to the theory of unit F -modules, uses Blickle’s intersection homology D-module [6] and Lyubeznik’s enhancement of Matlis duality [17] to reduce the main unit F -module length computation to a geometric description of the tight closure of 0 in the local cohomology of the singularity, due to Hara [11]. One then deduces the D-module length from Blickle’s length comparison result [5] and an application of Haastert’s positive characteristic Kashiwara’s equivalence [10]. We note that in characteristic zero, the D-module length of the first local cohomology module is of a quite different nature. It actually is a topological invariant. For example, let f be a rational cubic in three variables which is the equation of an elliptic curve E 2 in PQ, of genus g “ 1. Let RC be the ring of complex polynomials in 3 variables and

for all primes p, let Rp be the ring of Fp-polynomials in 3 variables. Then the DRC - 1 module length of the local cohomology module Hf pRCq is 3 “ 1 ` 2g, see e.g. [3, Remark 1.2]. But as will be seen in Example 1, for almost all primes p, the Dp-module length 1 of Hfp pRpq is 2 “ 1 ` g if Ep is ordinary and 1 if Ep is supersingular, where Ep (resp. fp) is the reduction of E (resp. f) modulo p and Dp “ DRFp is the ring of Grothendieck An differential operators on Fp . Thus for (almost) all primes p, the lengths of the first local cohomology modules in characteristic zero and in characteristic p are different. We end this note with a section on comparison with characteristic zero, arguing that in great 1 1 generality, the DRC -submodule DRC f of the local cohomology module Hf pRCq generated 1 1 by the class of f (which need not be equal to Hf pRCq) is a better behaved characteristic 1 1 zero analogue of Hfp pRpq than the whole local cohomology module Hf pRCq. (Recall that the left D -module H1 pR q is generated by the class of 1 , by [1, Theorem 1.1].) For p fp p fp example, in the case of the elliptic curve above, we have that the DRC -module length of D 1 D H1 R , p E. RC f is equal to the p-module length of fp p pq for almost all ordinary primes of 1 The DRC -module DRC f is studied in detail in [3]. (See also [20] for a different approach.)

1.1. Notation. Throughout the note we will use the following notation: For an integer n ě 2 and all fields K, we let RK be the ring of polynomials in n`1 variables tx0,...,xnu with coefficients in K and DK “ DRK be the ring of Grothendieck differential operators n`1 on AK .

Let k be a perfect ﬁeld of positive characteristic p, we set D “ DRk . If A is a k-algebra, we denote by ArF s the twisted polynomial ring over A whose multiplication is deﬁned LENGTH OF LOCAL COHOMOLOGY IN POSITIVE CHARACTERISTIC AND ORDINARITY 3

β by F a “ apF, for all a in A. If M is a left ArF s-module, we denote by F ˚M ÝÝÑM M the A-linear morphism induced by the action of F on M, where F ˚ is the functor from the category of A-modules to itself, given by the extension of scalars by the Frobenius endomorphism of A.

2. Length of the First Local Cohomology in Positive Characteristic We ﬁrst recall some deﬁnitions.

Deﬁnition 1. Let f P Rk be a non-constant polynomial in n ` 1 variables. The ﬁrst local 1 cohomology module Hf pRkq of Rk with respect to f is the left D-module cokernel of the 1 natural inclusion Rk Ă Rkr f s.

Remark 1. The Frobenius endomorphism of Rk induces a finitely generated unit F - 1 module structure on Hf pRkq. The associated action of D is the natural one. Hence it 1 follows immediately from [17, Theorem 3.2] that Hf pRkq is of finite length as a unit F -module. It is thus of finite length as a left D-module by [17, Theorem 5.7]. 1 The purpose of this note is to give an expression for the length of Hf pRkq, when f has an isolated singularity. It will be in terms of the quasilength of a certain F -module. We now recall the definitions from [17, Section 4]. Definition 2. Let A be a local Noetherian k-algebra with Frobenius endomorphism F and let M be a left ArF s-module. ˚ n n ‚ M :“ Xně0F M, where F M is the A-submodule of M generated by the image F npMq F n ‚ Mnil :“Yně0 kertM ÝÝÑ Mu Definition 3. Let A be a local Noetherian k-algebra of Frobenius endomorphism F and let M be a left ArF s-module. Suppose that M is Artinian as an A-module.

‚ A finite chain of length s of ArF s-submodules 0 “ M0 Ă ¨¨¨ Ă Ms “ M is quasimaximal if p Mi{Mi´1 q˚ is a simple left ArF s-module, for all i P t1,...,su. pMi{Mi´1qnil ‚ If Mnil Ĺ M, then M has a quasimaximal chain of submodules and all such chains are of the same length, called the quasilength qlpMq of M. If M “ Mnil, then we set qlpMq“ 0. See [17, Theorems 4.5 and 4.6]. Finally, we recall the definition of the Lyubeznik-Matlis duality. p Definition 4. Let R0 be a complete local regular Noetherian k-algebra and let M be an p Artinian R0-module.

_ p ‚ The Matlis duality functor is the contravariant functor p´q :“ HomR0 p´, Eq p from the category of R0-modules to itself, where E is an injective hull of the p p _ residue field of R0 in the category of R0-modules. Moreover, the Matlis dual M p of an Artinian module M is a finitely generated R0-module. p ‚ Suppose further that M is a left R0rF s-module. The Lyubeznik-Matlis dual DpMq p of M is the finitely generated unit R0rF s-module given by the direct limit of the β_ F ˚β_ p _ M ˚ _ M following direct system in the category of R0-modules: M ÝÝÑ F pM q ÝÝÝÝÑ F ˚2pM _q Ñ ..., where we have used the canonical isomorphism pF ˚Mq_ – 4 THOMAS BITOUN

F ˚pM _q of [17, Lemma 4.1]. Lyubeznik-Matlis duality D is a contravariant func- p p tor from the category of left R0rF s-modules which are Artinian as R0-modules, to p the category of finitely generated unit R0rF s-modules. To state our main result, we need to introduce the following notation: Let L be a field of characteristic 0 and let g be a non-constant polynomial in n ` 1 variables tx0,...,xnu, with coefficients in L. Let pA, mq be the local ring of the zero-locus π of g at a singular point z. Let us fix a resolution X ÝÑ Z “ SpecpAq of the singularity z and let Y be the fibre of π at z. Definition 5. Let B Ă L be a finitely generated subring, containing 1. We say that B is a ring of definition of π if the coefficients of g are contained in B and there is a resolution πB of singularities of B-schemes XB ÝÑ ZB whose base-change L bFracpBq πB is isomorphic to π.

πu For each closed point u of SpecpBq, we let Xu ÝÑ Zu (resp. gu, resp. Yu) be the fibre of πB (resp. g, resp. Y ) over kpuq. Finally, we consider the coherent cohomology groups l l H pXu, OXu q (resp. H pYu, OYu q ) as left ArF s-modules (resp. kpuqrF s-modules) for the action of the Frobenius endomorphism on the cohomology. Here is our main result: Theorem 2. Suppose that n ě 2 and that g is absolutely irreducible with an isolated singularity at the origin. Then there is a ring of definition B Ă L of π such that, for all closed points u of SpecpBq : 1 (i) The unit F -module length of the first local cohomology group Hgu pRkpuqq is n´1 n´1 1 ` qlpH pXu, OXu qq “ 1 ` qlpH pYu, OYu qq. 1 (ii) The Dkpuq-module length of Hgu pRkpuqq is n´1 O n´1 O ˚ 1 ` qlpkpuqb H pXu, Xu qq “ 1 ` dimkpuqppkpuqb H pYu, Yu qq q, where kpuq is any algebraic closure of kpuq and p´q˚ is the operation on kpuqrF s- modules from Definition 2. Proof. By Ostrowski’s Theorem, see [9, Lemma 11] for a quick proof, there is a definition 1 1 ring B of π such that, for all closed points u of SpecpB q, gu is absolutely irreducible. 1 For every closed point u of SpecpB q, we will use the following notation: pAu, muq is the m local ring of the singularity, pR0, q :“ ppRkpuqqpx0,...,xnq, px0,...,xnqpRkpuqqpx0,...,xnqq and m pR0, q :“ ppRkpuqqpx0,...,xnq, px0,...,xnqpRkpuqqpx0,...,xnqq. We denote their completion with p p x p respect to their maximal ideal by pA0, mx0q, pR0, mpq and pR0, mq, respectively. We have a short exact sequence of both Dkpuq- and unit F -modules: L 1 K (1) 0 Ñ Ñ Hgu pRkpuqqÑ Ñ 0 L L n`1 K where is the intersection homology module pAkpuq, tgu “ 0uq of [6] and is supported p at the origin. Tensoring with the completion R0 of the local ring at the origin, we get a short exact sequence: p p 1 p 0 L 0 0 K 0 Ñ R bRkpuq Ñ R bRkpuq Hgu pRkpuqqÑ R bRkpuq Ñ 0 p 1 1 p 1 1 R0 p which we can rewrite as 0 Ñ L Ñ Hg pR0q Ñ K Ñ 0, where L “ Lp p , R0q and u guR0 1 p 1 p K 0 K L 0 L “ R bRkpuq . Indeed – R bRkpuq by [6, Theorem 4.6] and it is well-known that local cohomology commutes with base-change by the completion. Clearly the length of LENGTH OF LOCAL COHOMOLOGY IN POSITIVE CHARACTERISTIC AND ORDINARITY 5

K1 p K as a DR0 -module (resp. unit F -module) equals the length of as a Dkpuq-module 1 1 p 1 0 L L (resp. unit F -module). Hence so is the case for Hgu pRkpuqq and Hgu pR q, since and are irreducible. 1 p Let D :“ DR0 . We also let lguF p´q be the unit F -module length and lgD1 p´q to be the D1-module length. The proof of (i) thus reduces to: There exists a ring of definition B Ą B1 of π such that SpecpBq Ă SpecpB1q is a dense open subset and, for all closed 1 n´1 points u of SpecpBq, lguF pK q“ qlpH pXu, OXu qq. Let us prove this assertion. We will use the notation of [6]. Using Lyubeznik-Matlis duality D, see Definition 4, for 1 1 p n p 0 D 0 all closed points u of SpecpB q, we have Hgu pR q– pHmp pA qq by [6, Proposition 2.16]. n p L1 D Hmp pA0q ˚ n p By [6, Theorem 4.4], we also have – p 0˚ p q, where 0 n p Ă Hmp pA0q is the tight n pA0q Hmp pA0q Hmp closure of zero. Since Lyubeznik-Matlis duality exchanges unit F -module length with 1 p ˚ 0 quasilength by the proof of [17, Theorems 4.5], we have lguF pHgu pR qq “ 1 ` qlp0 n p q. Hmp pA0q ˚ ˚ R R0 M A , n . Moreover by Lemma 1 applied to “ and “ u qlp0 n p q is equal to qlp0HmpAuqq Hmp pA0q Finally, since A is an isolated singularity and n ě 2, it is normal. Hence by [11, Theorem 4.7], there exists a ring of definition B Ą B1 of π such that SpecpBqĂ SpecpB1q is a dense ˚ n´1 u B , n H X , O , A F open subset and, for all closed points of Specp q 0HmpAuq – p u Xu q as ur s- n´1 n´1 n´1 modules. But by Lemma 3, qlpH pXu, OXu qq “ qlpH pYu, OYu qq as H pYu, OYu q– n´1 O H pXu, Xu q m n´1 O by Lemma 4. This concludes the proof of (i). uH pXu, Xu q We now prove (ii). From (1), we deduce the short exact sequence L 1 K 0 Ñ kpuqb Ñ Hgu pRkpuqqÑ kpuqb Ñ 0 x Therefore, tensoring with the completion R0 of R0, we also have the short exact sequence L1 1 x K1 0 Ñ kpuqb Ñ Hgu pR0qÑ kpuqb Ñ 0, x 1 1 1 R0 x with L and K as above. Note that kpuqb L “ Lp x , R0q by [4, Lemma 5.16]. More- guR0 x n`1 x R0 n`1 p over, since the injective hull H pR0q of kpuq “ is isomorphic to Hp pR0qb mpbkpuq mpbkpuq m

kpuq, it is easy to see that Matlis duality commutes with the ﬁeld extension ´bkpuq

kpuq. Hence Lyubeznik-Matlis duality commutes with ´bkpuq kpuq. Thus we have that p n p Hn pkpuqbA0q 1 Hmp pA0q kpuqbm kpuqb L – kpuqb Dp ˚ q – Dp ˚ q. Therefore the length of the unit 0 p kpuqb0 HnpA0q n p mp Hmp pA0q 1 x ˚ n´1 O F -module Hgu pR0q is equal to 1 ` qlpkpuqb 0 n p q “ 1 ` qlpkpuqb H pXu, Xu qq. Hmp pA0q F H1 R Also, similarly as above, the unit -module length of gu p kpuqq is equal to the length of 1 x Hgu pR0q, as a unit F -module. We thus deduce that the length of the unit F -module H1 R k u Hn´1 X , O . k u Hn´1 X , O gu p kpuqq is 1 ` qlp p q b p u Xu qq Moreover qlp p q b p u Xu qq “ n´1 O n´1 O qlpkpuqbH pYu, Yu qq by Lemmas 3 and 4, and qlpkpuqbH pYu, Yu qq “ dimkpuqppkpuqb n´1 ˚ H pYu, OYu qq q by Lemma 2. H1 R F But by [5, Theorem 1.1], the length of gu p kpuqq as a unit -module is equal to its D D H1 R length as a kpuq-module. Finally, we claim that the kpuq-module length of gu p kpuqq 1 is equal to the Dkpuq-module length of Hgu pRkpuqq. This implies part (ii) of the theorem. Let us prove this last claim. We let lgD p´q (resp. lgDp´q) denote the D -module kpuq kpuq (resp. Dkpuq-module) length. Localising at the origin, one sees by [4, Lemma 5.16] that 6 THOMAS BITOUN

kpuqb L is the intersection homology module. Thus lgD pkpuqb Lq “ 1 “ lgDpLq. kpuq

Hence the claim reduces to the equality lgD pkpuqb Kq “ lgDpKq. But this follows kpuq immediately from the compatibility with base-ﬁeld extension of Kashiwara’s equivalence, see [10, Corollary 8.13] (the proof of which is well-known to be valid over an arbitrary ﬁeld

of positive characteristic). Indeed by Kashiwara’s equivalence, we have lgD pkpuqbKq“ kpuq K dimkpuqpkpuqb V q “ dimkpuqpV q “ lgDp q, for a certain finite dimensional kpuq-vector space V. Recall that a ring R is F -finite if it is of positive characteristic, Noetherian and if the Frobenius map SpecpRq ÝÑF SpecpRq is finite. Lemma 1. Let pR, mq be an F -finite regular local ring and let M be a finitely generated i i x R-module. Then for all i ě 0, the canonical isomorphism HmpMqÑr Hmp pMq induces an ˚ ˚ p p x isomorphism of tight closures 0 i – 0 , where pR, mq (resp. M) is the m-adic Hm M i x p q Hmp pMq completion of pR, mq (resp. M).

i i x Proof. The isomorphism HmpMqÑr Hmp pMq is well-known, see [16, Proposition 2.15]. Fur- thermore the existence of completely stable big test elements for R ([15, Theorem p.77]) immediately implies the equality of tight closures. Lemma 2. Let k be an algebraically closed field of positive characteristic p and let V be ˚ a k-finite dimensional left krF s-module. Then the quasilength of V is dimkpV q, where p´q˚ is the operation on krF s-modules from Definition 2. Proof. By definition of quasilength, we have qlpV q“ qlpV ˚q. Moreover, F acts surjectively and thus injectively on V ˚. Hence, by [5, Proposition 4.6] for example, V ˚ has a k-basis of vectors fixed by F. The lemma easily follows. Lemma 3. Let pA, mq be a local Noetherian k-algebra with Frobenius endomorphism F and let M be a left ArF s-module. Suppose that M is Artinian and Noetherian as an m M A-module. If M is supported at the maximal ideal , then M and mM have the same quasilength.

Proof. Let us show that mM Ă Mnil. This implies the lemma since qlpMnilq“ 0. Since M is supported at m and is Noetherian, mlM “ 0 for some l ě 0. Thus F rpmMqĂ pr m M “ 0 for some r ě 0. Hence mM Ă Mnil, as claimed. Lemma 4. Let pA, mq be a Noetherian local ring and let X ÝÑπ Z “ SpecpAq be a projective morphism of special fibre Y. Suppose that the fibres of π are of dimension at most d. Then d d F for all quasi-coherent sheaves F on X, the canonical morphism H pX, Fq Ñ H pY, mF q HdpX,Fq d F induces an isomorphism mHdpX,Fq Ñ H pY, mF q. Proof. We first claim that for all quasi-coherent sheaves F on X and for all integers i ě d ` 1,HipX, Fq “ 0. This is well-known. Indeed it follows from the theorem on i formal functions that R π˚pGq “ 0 for all i ě d ` 1, and for all coherent sheaves G on X ([13, Corollary III 11.2]). Thus, since Z is affine, we have that HipX, Gq “ 0, for all i ě d ` 1, and for all coherent sheaves G on X. Since a quasi-coherent sheaf is the union of its coherent subsheaves, the claim then follows from the commutation of cohomology with direct limits ([13, Proposition III 2.9]). LENGTH OF LOCAL COHOMOLOGY IN POSITIVE CHARACTERISTIC AND ORDINARITY 7

Let tr1,...,rN u be a set of generators of the maximal ideal m. We have the following m F short exact sequences of quasi-coherent sheaves on X : 0 Ñ F Ñ F Ñ mF Ñ 0 l“N φ m and 0 Ñ J Ñ ‘l“1 F ÝÑ F Ñ 0, where φpf1,...,fN q “ r1f1 `¨¨¨` rN fN and J is the kernel of φ. Moreover by the claim, we have that Hd`1pX, mFq “ Hd`1pX,Jq “ 0. Thus in the associated long exact sequences in Zariski cohomology, the morphisms d d F l“N d d m H pX, Fq Ñ H pX, mF q and ‘l“1 H pX, Fq Ñ H pX, Fq are surjective. The latter implies that the image of the morphism HdpX, mFq Ñ HdpX, Fq from the long exact sequence is mHdpX, Fq. This concludes the proof of the lemma. If g is homogeneous, Theorem 2 may be rephrased without mentioning a resolution of the singularity. Let Y be the hypersurface defined by g in Pn. We first fix the notation. Definition 6. Let B Ă L be a finitely generated subring, containing 1. We say that B is a ring of definition of Y if the coefficients of g are contained in B and there is a smooth n projective hypersurface YB of PB whose base-change L bFracpBq YB is isomorphic to Y.

Given such an hypersurface YB, for each closed point u of SpecpBq, we let Yu be the fibre of YB over kpuq. Here is the result: Corollary 1. Under the same hypotheses as in Theorem 2, assume that g is homogeneous and let Y be the hypersurface defined by g in Pn. Then there is a ring of definition B Ă L of Y such that, for all closed points u of SpecpBq : 1 n´1 O (i) The unit F -module length of Hgu pRkpuqq is 1 ` qlpH pYu, Yu qq. 1 n´1 O ˚ (ii) The Dkpuq-module length of Hgu pRkpuqq is 1 ` dimkpuqpkpuqbkpuq H pYu, Yu qq , where kpuq is any algebraic closure of kpuq and p´q˚ is the operation on kpuqrF s- modules from Definition 2. Proof. It is well-known that in this case the blow-up of the origin is a resolution π1 of the singularity and that the fibre at the origin is isomorphic to Y. The result then immediately follows from Theorem 2 applied to π1.

Thus the Dk-module length of the ﬁrst local cohomology is closely related to ordinarity. Here is a simple example: Example 1. Let g be a rational cubic in three variables which is the equation of an elliptic P2 1 curve E in Q. Then, for almost all primes p, the DRFp -module length of Hgp pRFp q is 2 if Ep is ordinary and 1 if Ep is supersingular, where Ep (resp. gp) is the reduction of E (resp. g) modulo p.

3. Comparison with Characteristic Zero

Here, given a complex polynomial g, we consider a holonomic DC-module Ng whose 1 length compares well to the Dkpuq-module length of Hgu pRkpuqq.

Deﬁnition 7. Let g P RC be a complex polynomial. Then Ng is the left DC-submodule of 1 1 the ﬁrst local cohomology DC-module Hg pRCq generated by the class of g . The following is proved in [3, Theorem 1.1]. Theorem 3. Let g be a non-constant homogeneous complex polynomial in n`1 variables with an isolated singularity at the origin. Then, using the notation of Corollary 1, the n´1 DC-module length of Ng is 1 ` dimC H pY, OY q. 8 THOMAS BITOUN

Remark 2. There is a ring of definition B Ă C of Y such that, for all closed points u n´1 O n´1 O of SpecpBq, dimC H pY, Y q “ dimkpuqpkpuqbkpuq H pYu, Yu qq. Hence by Corollary 1 and Theorem 3, there is a ring of definition B1 Ą B of Y such that for all closed 1 n´1 points u of SpecpB q, if the Frobenius F acts bijectively on kpuqbkpuq H pYu, OYu q, then 1 the length of Ng is equal to the Dkpuq-module length of Hgu pRkpuqq. Indeed in that case, n´1 O n´1 O ˚ dimC H pY, Y q“ dimkpuqpkpuqbkpuq H pYu, Yu qq . This property of the Frobenius is called weak ordinarity and is expected to hold for a dense set of closed points of SpecpB1q, see [19, Conjecture 1.1]. We would like to put forward the following questions: Question 1. Let g be a non-constant complex polynomial in n ` 1 variables. Is there a unitary finitely generated subring B Ă C containing the coefficients of g such that: u B , H1 R N (1) For all closed points P Specp q lgDkpuq p gu p kpuqqq ď lgDC p gq? B H1 R (2) There is a dense set of closed points of Specp q for which lgDkpuq p gu p kpuqqq “

lgDC pNgq? As explained in Remark 2, for g homogeneous with an isolated singularity and n ě 2, the first part of Question 1 has a positive answer. In the same case, the second part has a positive answer as well, if the weak ordinarity conjecture of [19, Conjecture 1.1] is satisfied by Y. We finally note that by Theorem 2, [3, Conjecture 1.4] (which is equivalent to [8, Conjecture 3.8]) implies a positive answer to the first part of Question 1, for g (not necessarily homogeneous) with an isolated singularity at the origin and n ě 2. 4. Funding This work was supported by the Engineering and Physical Sciences Research Council [EP/L005190/1]. It is my pleasure to thank Manuel Blickle and Gennady Lyubeznik for interesting correspondence regarding the length of the positive characteristic first local cohomology module in the homogeneous case. I am also very grateful to Karl Schwede for explaining to me a proof of Lemma 1. Finally, many thanks go to Johannes Nicaise and Travis Schedler for communicating to me proofs of Lemma 4, as well as to the referee for pointing out that a proof was needed in the first place and for suggesting a simplification in the final argument. References

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