Length of local cohomology in positive characteristic and ordinarity

TL;DR

This paper computes the length of a local cohomology module as a D-module in positive characteristic, linking it to Frobenius actions and singularity resolutions, and explores its relation to characteristic zero cases.

Contribution

It provides an explicit formula for the D-module length of local cohomology in positive characteristic and connects it to Frobenius actions and singularity resolutions.

Findings

Length depends on Frobenius action on top cohomology

Length differs from characteristic zero case

Relation to ordinarity in characteristic zero

Abstract

Let $D$ be the ring of Grothendieck differential operators of the ring $R$ of polynomials in $d\geq3$ variables with coefficients in a perfect field of positive characteristic $p.$ We compute the $D$-module length of the first local cohomology module $H^1_f(R)$ of $R$ with respect to an irreducible polynomial $f$ with an isolated singularity, for $p$ large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the structure sheaf of the exceptional divisor of a resolution of the singularity. Our proof rests on a tight closure computation due to Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero $\mathcal{D}$-module whose length is expected to equal that above, for ordinary primes.

Equations

$$1+\mathrm{ql}(\overline{k(u)}\otimes H^{n-1}(X_u,{\mathcal{O}}_{X_u}))=1+{\dim }_{\overline{k(u)}}((\overline{k(u)}\otimes H^{n-1}(Y_u,{\mathcal{O}}_{Y_u}){)}^{\ast }),$$

$$0\to \mathcal{L}\to H_{g_u}^1(R_{k(u)})\to \mathcal{K}\to 0$$

$$0\to {\hat{R}}_0{\otimes }_{R_{k(u)}}\mathcal{L}\to {\hat{R}}_0{\otimes }_{R_{k(u)}}{H}_{g_u}^1(R_{k(u)})\to {\hat{R}}_0{\otimes }_{R_{k(u)}}\mathcal{K}\to 0$$

$$0\to \overline{k(u)}\otimes \mathcal{L}\to H_{g_u}^1(R_{\overline{k(u)}})\to \overline{k(u)}\otimes \mathcal{K}\to 0$$

$$0\to \overline{k(u)}\otimes {\mathcal{L}}^{\prime }\to H_{g_u}^1(\widehat{R_{\overline{0}}})\to \overline{k(u)}\otimes {\mathcal{K}}^{\prime }\to 0,$$

Full text

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK; tbitoun@gmail.com

1. Introduction

In this note, we compute the positive characteristic $D$-module length of the first local cohomology module of the structure sheaf with support in a hypersurface, in a large class of examples. Our main result can also be seen as part of our study of the $b$-function in positive characteristic, see [2]. On the one hand, in [2] using $D$-module (or unit $F$-module) techniques, for $D$ the ring of Grothendieck differential operators, we associate to a non-constant polynomial $f$ with coefficients in a perfect field of positive characteristic a set of $p$-adic integers, called the roots of the $b$-function of $f.$ On the other, one may consider the $F$-jumping exponents of the generalised test ideals of $f,$ see [12]. These are positive real numbers which are characterised by their intersection with the unit interval $(0,1]$ and have been shown to be rational numbers in [7]. In [2] we prove that the roots of the $b$-function of $f$ are exactly the opposites of the $F$-jumping exponents of $f$ which are in $\mathbb{Q}\cap\mathbb{Z}_{p}.$ It would thus seem that the information provided by the $F$-jumping exponents of $f$ which are not in $\mathbb{Q}\cap\mathbb{Z}_{p},$ i.e. whose denominator is divisible by $p,$ let us call them irregular, is lost in the theory. A consequence of the results presented here is that not all the information is lost. Namely the absence of irregular $F$-jumping exponents is well-known to be closely related to phenomena of ordinarity, see [19]. We claim that at the very least the $D$-module (or unit $F$-module) length of the module $N_{f}$ used to define the $b$-function in [2] distinguishes ordinary primes from supersingular ones, for large enough primes. More precisely, using the terminology of [2], one can see that the joint eigenspace of the action of the higher Euler operators on $N_{f}$ corresponding to the root -1 of the $b$-function of $f$ is isomorphic to the first local cohomology module $H^{1}_{f}(R)=\frac{R[\frac{1}{f}]}{R},$ where $R$ is the ring of polynomials. Since this assertion does not appear in the literature, let us give a proof. By [2, Proposition 1], this joint eigenspace is the limit of the direct system: $\frac{D^{(0)}f^{p-1}}{D^{(0)}f^{p}}\xrightarrow{.f^{p^{2}-p}}\frac{D^{(1)}f^{p^{2}-1}}{D^{(1)}f^{p^{2}}}\xrightarrow{.f^{p^{3}-p^{2}}}\cdots\xrightarrow{.f^{p^{e+1}-p^{e}}}\frac{D^{(e)}f^{p^{e+1}-1}}{D^{(e)}f^{p^{e+1}}}\xrightarrow{.f^{p^{e+2}-p^{e+1}}}\frac{D^{(e+1)}f^{p^{e+2}-1}}{D^{(e+1)}f^{p^{e+2}}}\xrightarrow{.f^{p^{e+3}-p^{e+2}}}\cdots,$ where $D^{(e)}$ is the ring of Grothendieck differential operators of level $e$ on $\operatorname{Spec}(R).$ Since $D^{(e)}f^{p^{e+1}-1}=R$ for every large enough natural number $e$ by [18, Proof of Lemma 6.8], and the $D$-submodule $D\frac{1}{f}$ of $R[\frac{1}{f}]$ generated by $\frac{1}{f}$ is $R[\frac{1}{f}]$ itself by [1, Theorem 1.1], the assertion immediately follows from the description of $D\frac{1}{f}$ given in [7, Remark 2.7].

For a $d$-dimensional proper variety $Z$ over a field of characteristic $p>0,$ we let the $p$-genus $g_{p}(Z)$ of $Z$ be the dimension of the stable part [14] of $\overline{k}\otimes H^{d}(Z,\mathcal{O}_{Z}),$ that is $\dim_{\overline{k}}(\cap_{l\geq 0}F^{l}(\overline{k}\otimes H^{d}(Z,\mathcal{O}_{Z}))),$ where $F$ is the Frobenius action on coherent cohomology and $\overline{k}$ 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 $n\geq 3$ variables with an isolated singularity at the origin and let $Y\xrightarrow{\pi}X$ be a resolution of the singularity. Then for almost all $p,$ the $D$-module length of $H^{1}_{f_{p}}(R)$ is $1+g_{p}(Z_{p}),$ where $Z_{p}$ is the reduction modulo $p$ of the exceptional fibre of $\pi.$

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 [math] 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 $E$ in $\mathbb{P}^{2}_{\mathbb{Q}},$ of genus $g=1.$ Let $R_{\mathbb{C}}$ be the ring of complex polynomials in $3$ variables and for all primes $p,$ let $R_{p}$ be the ring of $\mathbb{F}_{p}$-polynomials in $3$ variables. Then the $D_{R_{\mathbb{C}}}$-module length of the local cohomology module $H^{1}_{f}(R_{\mathbb{C}})$ 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 $D_{p}$-module length of $H^{1}_{f_{p}}(R_{p})$ is $2=1+g$ if $E_{p}$ is ordinary and $1$ if $E_{p}$ is supersingular, where $E_{p}$ (resp. $f_{p}$) is the reduction of $E$ (resp. $f$) modulo $p$ and $D_{p}=D_{R_{\mathbb{F}_{p}}}$ is the ring of Grothendieck differential operators on $\mathbb{A}^{n}_{\mathbb{F}_{p}}.$ 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 generality, the $D_{R_{\mathbb{C}}}$-submodule $D_{R_{\mathbb{C}}}\frac{1}{f}$ of the local cohomology module $H^{1}_{f}(R_{\mathbb{C}})$ generated by the class of $\frac{1}{f}$ (which need not be equal to $H^{1}_{f}(R_{\mathbb{C}})$) is a better behaved characteristic zero analogue of $H^{1}_{f_{p}}(R_{p})$ than the whole local cohomology module $H^{1}_{f}(R_{\mathbb{C}}).$ (Recall that the left $D_{p}$-module $H^{1}_{f_{p}}(R_{p})$ is generated by the class of $\frac{1}{f_{p}},$ by [1, Theorem 1.1].) For example, in the case of the elliptic curve above, we have that the $D_{R_{\mathbb{C}}}$-module length of $D_{R_{\mathbb{C}}}\frac{1}{f}$ is equal to the $D_{p}$-module length of $H^{1}_{f_{p}}(R_{p}),$ for almost all ordinary primes $p$ of $E.$ The $D_{R_{\mathbb{C}}}$-module $D_{R_{\mathbb{C}}}\frac{1}{f}$ 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 $n\geq 2$ and all fields $K,$ we let $R_{K}$ be the ring of polynomials in $n+1$ variables $\{x_{0},\dots,x_{n}\}$ with coefficients in $K$ and $D_{K}=D_{R_{K}}$ be the ring of Grothendieck differential operators on $\mathbb{A}^{n+1}_{K}.$

Let $k$ be a perfect field of positive characteristic $p,$ we set $D=D_{R_{k}}.$ If $A$ is a $k$-algebra, we denote by $A[F]$ the twisted polynomial ring over $A$ whose multiplication is defined by $Fa=a^{p}F,$ for all $a$ in $A.$ If $M$ is a left $A[F]$-module, we denote by $F^{*}M\xrightarrow{\beta_{M}}M$ 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 $f\in R_{k}$ be a non-constant polynomial in $n+1$ variables. The first local cohomology module $H_{f}^{1}(R_{k})$ of $R_{k}$ with respect to $f$ is the left $D$-module cokernel of the natural inclusion $R_{k}\subset R_{k}[\frac{1}{f}].$

Remark 1**.**

The Frobenius endomorphism of $R_{k}$ induces a finitely generated unit $F$-module structure on $H_{f}^{1}(R_{k}).$ The associated action of $D$ is the natural one. Hence it follows immediately from [17, Theorem 3.2] that $H_{f}^{1}(R_{k})$ 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].

The purpose of this note is to give an expression for the length of $H_{f}^{1}(R_{k}),$ 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 $A[F]$-module.

• •

$M^{\ast}:=\cap_{n\geq 0}F^{n}M,$* where $F^{n}M$ is the $A$-submodule of $M$ generated by the image $F^{n}(M)$*

• •

$M_{nil}:=\cup_{n\geq 0}\ker\{M\xrightarrow{F^{n}}M\}$**

Definition 3**.**

Let $A$ be a local Noetherian $k$-algebra of Frobenius endomorphism $F$ and let $M$ be a left $A[F]$-module. Suppose that $M$ is Artinian as an $A$-module.

• •

A finite chain of length $s$ of $A[F]$-submodules $0=M_{0}\subset\dots\subset M_{s}=M$ is quasimaximal if $(\frac{M_{i}/M_{i-1}}{(M_{i}/M_{i-1})_{nil}})^{\ast}$ is a simple left $A[F]$-module, for all $i\in\{1,\dots,s\}.$

• •

If $M_{nil}\subsetneq M,$ then $M$ has a quasimaximal chain of submodules and all such chains are of the same length, called the quasilength $\mathrm{ql}(M)$ of $M.$ If $M=M_{nil},$ then we set $\mathrm{ql}(M)=0.$ See **[17, Theorems 4.5 and 4.6]**.

Finally, we recall the definition of the Lyubeznik-Matlis duality.

Definition 4**.**

Let $\widehat{R}_{0}$ be a complete local regular Noetherian $k$-algebra and let $M$ be an Artinian $\widehat{R}_{0}$-module.

• •

The Matlis duality functor is the contravariant functor $(-)^{\vee}:=Hom_{\widehat{R}_{0}}(-,E)$ from the category of $\widehat{R}_{0}$-modules to itself, where $E$ is an injective hull of the residue field of $\widehat{R}_{0}$ in the category of $\widehat{R}_{0}$-modules. Moreover, the Matlis dual $M^{\vee}$ of an Artinian module $M$ is a finitely generated $\widehat{R}_{0}$-module.

• •

Suppose further that $M$ is a left $\widehat{R}_{0}[F]$-module. The Lyubeznik-Matlis dual $\mathcal{D}(M)$ of $M$ is the finitely generated unit $\widehat{R}_{0}[F]$-module given by the direct limit of the following direct system in the category of $\widehat{R}_{0}$-modules: $M^{\vee}\xrightarrow{{\beta}^{\vee}_{M}}F^{*}(M^{\vee})\xrightarrow{F^{*}{\beta}^{\vee}_{M}}{F^{*}}^{2}(M^{\vee})\to\dots,$ where we have used the canonical isomorphism $(F^{*}M)^{\vee}\cong F^{*}(M^{\vee})$ of **[17, Lemma 4.1]**. Lyubeznik-Matlis duality $\mathcal{D}$ is a contravariant functor from the category of left $\widehat{R}_{0}[F]$-modules which are Artinian as $\widehat{R}_{0}$-modules, to the category of finitely generated unit $\widehat{R}_{0}[F]$-modules.

To state our main result, we need to introduce the following notation:

Let $L$ be a field of characteristic [math] and let $g$ be a non-constant polynomial in $n+1$ variables $\{x_{0},\dots,x_{n}\},$ with coefficients in $L.$ Let $(A,\mathfrak{m})$ be the local ring of the zero-locus of $g$ at a singular point $z.$ Let us fix a resolution $X\xrightarrow{\pi}Z=\text{Spec}(A)$ of the singularity $z$ and let $Y$ be the fibre of $\pi$ at $z.$

Definition 5**.**

Let $B\subset L$ be a finitely generated subring, containing $1.$ We say that $B$ is a ring of definition of $\pi$ if the coefficients of $g$ are contained in $B$ and there is a resolution of singularities of $B$-schemes $X_{B}\xrightarrow{\pi_{B}}Z_{B}$ whose base-change $L\otimes_{Frac(B)}\pi_{B}$ is isomorphic to $\pi.$

For each closed point $u$ of $\text{Spec}(B),$ we let $X_{u}\xrightarrow{\pi_{u}}Z_{u}$ (resp. $g_{u},$ resp. $Y_{u}$) be the fibre of $\pi_{B}$ (resp. $g,$ resp. $Y$) over $k(u).$ Finally, we consider the coherent cohomology groups $H^{l}(X_{u},\mathcal{O}_{X_{u}})$ (resp. $H^{l}(Y_{u},\mathcal{O}_{Y_{u}})$ ) as left $A[F]$-modules (resp. $k(u)[F]$-modules) for the action of the Frobenius endomorphism on the cohomology. Here is our main result:

Theorem 2**.**

Suppose that $n\geq 2$ and that $g$ is absolutely irreducible with an isolated singularity at the origin. Then there is a ring of definition $B\subset L$ of $\pi$ such that, for all closed points $u$ of $\text{Spec}(B):$

• (i)

The unit $F$-module length of the first local cohomology group $H_{g_{u}}^{1}(R_{k(u)})$ is

$1+\mathrm{ql}(H^{n-1}(X_{u},\mathcal{O}_{X_{u}}))=1+\mathrm{ql}(H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}})).$**

• (ii)

The $D_{k(u)}$-module length of $H_{g_{u}}^{1}(R_{k(u)})$ is

[TABLE]

where $\overline{k(u)}$ is any algebraic closure of $k(u)$ and $(-)^{\ast}$ is the operation on $\overline{k(u)}[F]$-modules from Definition 2.

Proof.

By Ostrowski's Theorem, see [9, Lemma 11] for a quick proof, there is a definition ring $B^{\prime}$ of $\pi$ such that, for all closed points $u$ of $\text{Spec}(B^{\prime}),$ $g_{u}$ is absolutely irreducible.

For every closed point $u$ of $\text{Spec}(B^{\prime}),$ we will use the following notation: $(A_{u},\mathfrak{m}_{u})$ is the local ring of the singularity, $(R_{0},\mathfrak{m}):=((R_{k(u)})_{(x_{0},\dots,x_{n})},(x_{0},\dots,x_{n})(R_{k(u)})_{(x_{0},\dots,x_{n})})$ and $(R_{\overline{0}},\overline{\mathfrak{m}}):=((R_{\overline{k(u)}})_{(x_{0},\dots,x_{n})},(x_{0},\dots,x_{n})(R_{\overline{k(u)}})_{(x_{0},\dots,x_{n})}).$ We denote their completion with respect to their maximal ideal by $(\widehat{A}_{0},\widehat{\mathfrak{m}_{0}}),(\widehat{R}_{0},\widehat{\mathfrak{m}})$ and $(\widehat{R_{\overline{0}}},\widehat{\overline{\mathfrak{m}}}),$ respectively.

We have a short exact sequence of both $D_{k(u)}$- and unit $F$-modules:

[TABLE]

where $\mathcal{L}$ is the intersection homology module $\mathcal{L}(\mathbb{A}^{n+1}_{k(u)},\{g_{u}=0\})$ of [6] and $\mathcal{K}$ is supported at the origin. Tensoring with the completion $\widehat{R}_{0}$ of the local ring at the origin, we get a short exact sequence:

[TABLE]

which we can rewrite as $0\to\mathcal{L}^{\prime}\to H^{1}_{g_{u}}(\widehat{R}_{0})\to\mathcal{K}^{\prime}\to 0,$ where $\mathcal{L}^{\prime}=\mathcal{L}(\frac{\widehat{R}_{0}}{g_{u}\widehat{R}_{0}},\widehat{R}_{0})$ and $\mathcal{K}^{\prime}=\widehat{R}_{0}\otimes_{R_{k(u)}}\mathcal{K}.$ Indeed $\mathcal{L}^{\prime}\cong\widehat{R}_{0}\otimes_{R_{k(u)}}\mathcal{L}$ by [6, Theorem 4.6] and it is well-known that local cohomology commutes with base-change by the completion. Clearly the length of $\mathcal{K}^{\prime}$ as a $D_{\widehat{R}_{0}}$-module (resp. unit $F$-module) equals the length of $\mathcal{K}$ as a $D_{k(u)}$-module (resp. unit $F$-module). Hence so is the case for $H_{g_{u}}^{1}(R_{k(u)})$ and $H^{1}_{g_{u}}(\widehat{R}_{0}),$ since $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are irreducible.

Let $D^{\prime}:=D_{\widehat{R}_{0}}.$ We also let $\lg_{uF}(-)$ be the unit $F$-module length and $\lg_{D^{\prime}}(-)$ to be the $D^{\prime}$-module length. The proof of (i) thus reduces to: There exists a ring of definition $B\supset B^{\prime}$ of $\pi$ such that $\operatorname{Spec}(B)\subset\operatorname{Spec}(B^{\prime})$ is a dense open subset and, for all closed points $u$ of $\operatorname{Spec}(B),$ $\lg_{uF}(\mathcal{K}^{\prime})=\mathrm{ql}(H^{n-1}(X_{u},\mathcal{O}_{X_{u}})).$ Let us prove this assertion.

We will use the notation of [6]. Using Lyubeznik-Matlis duality $\mathcal{D},$ see Definition 4, for all closed points $u$ of $\text{Spec}(B^{\prime}),$ we have $H^{1}_{g_{u}}(\widehat{R}_{0})\cong\mathcal{D}(H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0}))$ by [6, Proposition 2.16]. By [6, Theorem 4.4], we also have $\mathcal{L}^{\prime}\cong\mathcal{D}(\frac{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})}{0^{\ast}_{H^{n}_{\widehat{\mathfrak{m}}}}(\widehat{A}_{0})}),$ where $0^{\ast}_{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})}\subset H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})$ is the tight closure of zero. Since Lyubeznik-Matlis duality exchanges unit $F$-module length with quasilength by the proof of [17, Theorems 4.5], we have $\lg_{uF}(H^{1}_{g_{u}}(\widehat{R}_{0}))=1+\mathrm{ql}(0^{\ast}_{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})}).$ Moreover by Lemma 1 applied to $R=R_{0}$ and $M=A_{u},$ $\mathrm{ql}(0^{\ast}_{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})})$ is equal to $\mathrm{ql}(0^{\ast}_{H^{n}_{\mathfrak{m}}(A_{u})}).$

Finally, since $A$ is an isolated singularity and $n\geq 2,$ it is normal. Hence by [11, Theorem 4.7], there exists a ring of definition $B\supset B^{\prime}$ of $\pi$ such that $\operatorname{Spec}(B)\subset\operatorname{Spec}(B^{\prime})$ is a dense open subset and, for all closed points $u$ of $\operatorname{Spec}(B),$ $0^{\ast}_{H^{n}_{\mathfrak{m}}(A_{u})}\cong H^{n-1}(X_{u},\mathcal{O}_{X_{u}}),$ as $A_{u}[F]$-modules. But by Lemma 3, $\mathrm{ql}(H^{n-1}(X_{u},\mathcal{O}_{X_{u}}))=\mathrm{ql}(H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}}))$ as $H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}})\cong\frac{H^{n-1}(X_{u},\mathcal{O}_{X_{u}})}{\mathfrak{m}_{u}H^{n-1}(X_{u},\mathcal{O}_{X_{u}})}$ by Lemma 4. This concludes the proof of (i).

We now prove (ii). From (1), we deduce the short exact sequence

[TABLE]

Therefore, tensoring with the completion $\widehat{R_{\overline{0}}}$ of $R_{\overline{0}},$ we also have the short exact sequence

[TABLE]

with $\mathcal{L}^{\prime}$ and $\mathcal{K}^{\prime}$ as above. Note that $\overline{k(u)}\otimes\mathcal{L}^{\prime}=\mathcal{L}(\frac{\widehat{R_{\overline{0}}}}{g_{u}\widehat{R_{\overline{0}}}},\widehat{R_{\overline{0}}})$ by [4, Lemma 5.16]. Moreover, since the injective hull $H^{n+1}_{\widehat{\mathfrak{m}}\otimes\overline{k(u)}}(\widehat{R_{\overline{0}}})$ of $\overline{k(u)}=\frac{\widehat{R_{\overline{0}}}}{\widehat{\mathfrak{m}}\otimes\overline{k(u)}}$ is isomorphic to $H^{n+1}_{\widehat{\mathfrak{m}}}(\widehat{R}_{0})\otimes\overline{k(u)},$ it is easy to see that Matlis duality commutes with the field extension $-\otimes_{k(u)}\overline{k(u)}.$ Hence Lyubeznik-Matlis duality commutes with $-\otimes_{k(u)}\overline{k(u)}.$ Thus we have that $\overline{k(u)}\otimes\mathcal{L}^{\prime}\cong\overline{k(u)}\otimes\mathcal{D}(\frac{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})}{0^{\ast}_{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})}})\cong\mathcal{D}(\frac{H^{n}_{\overline{k(u)}\otimes\mathfrak{m}}(\overline{k(u)}\otimes\widehat{A}_{0})}{\overline{k(u)}\otimes 0^{\ast}_{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})}}).$ Therefore the length of the unit $F$-module $H_{g_{u}}^{1}(\widehat{R_{\overline{0}}})$ is equal to $1+\mathrm{ql}(\overline{k(u)}\otimes 0^{\ast}_{H^{n}_{\widehat{\mathfrak{m}}}(\widehat{A}_{0})})=1+\mathrm{ql}(\overline{k(u)}\otimes H^{n-1}(X_{u},\mathcal{O}_{X_{u}})).$ Also, similarly as above, the unit $F$-module length of $H_{g_{u}}^{1}(R_{\overline{k(u)}})$ is equal to the length of $H_{g_{u}}^{1}(\widehat{R_{\overline{0}}}),$ as a unit $F$-module. We thus deduce that the length of the unit $F$-module $H_{g_{u}}^{1}(R_{\overline{k(u)}})$ is $1+\mathrm{ql}(\overline{k(u)}\otimes H^{n-1}(X_{u},\mathcal{O}_{X_{u}})).$ Moreover $\mathrm{ql}(\overline{k(u)}\otimes H^{n-1}(X_{u},\mathcal{O}_{X_{u}}))=\mathrm{ql}(\overline{k(u)}\otimes H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}}))$ by Lemmas 3 and 4, and $\mathrm{ql}(\overline{k(u)}\otimes H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}}))=\dim_{\overline{k(u)}}((\overline{k(u)}\otimes H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}}))^{\ast})$ by Lemma 2.

But by [5, Theorem 1.1], the length of $H_{g_{u}}^{1}(R_{\overline{k(u)}})$ as a unit $F$-module is equal to its length as a $D_{\overline{k(u)}}$-module. Finally, we claim that the $D_{\overline{k(u)}}$-module length of $H_{g_{u}}^{1}(R_{\overline{k(u)}})$ is equal to the $D_{k(u)}$-module length of $H_{g_{u}}^{1}(R_{k(u)}).$ This implies part (ii) of the theorem.

Let us prove this last claim. We let $\lg_{D_{\overline{k(u)}}}(-)$ (resp. $\lg_{D}(-)$) denote the $D_{\overline{k(u)}}$-module (resp. $D_{k(u)}$-module) length. Localising at the origin, one sees by [4, Lemma 5.16] that $\overline{k(u)}\otimes\mathcal{L}$ is the intersection homology module. Thus $\lg_{D_{\overline{k(u)}}}(\overline{k(u)}\otimes\mathcal{L})=1=\lg_{D}(\mathcal{L}).$ Hence the claim reduces to the equality $\lg_{D_{\overline{k(u)}}}(\overline{k(u)}\otimes\mathcal{K})=\lg_{D}(\mathcal{K}).$ But this follows 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 $\lg_{D_{\overline{k(u)}}}(\overline{k(u)}\otimes\mathcal{K})=\dim_{\overline{k(u)}}(\overline{k(u)}\otimes V)=\dim_{k(u)}(V)=\lg_{D}(\mathcal{K}),$ for a certain finite dimensional $k(u)$-vector space $V.$ ∎

Recall that a ring $R$ is $F$-finite if it is of positive characteristic, Noetherian and if the Frobenius map $\operatorname{Spec}(R)\xrightarrow{F}\operatorname{Spec}(R)$ is finite.

Lemma 1**.**

Let $(R,\mathfrak{m})$ be an $F$-finite regular local ring and let $M$ be a finitely generated $R$-module. Then for all $i\geq 0,$ the canonical isomorphism $H^{i}_{\mathfrak{m}}(M)\widetilde{\to}H^{i}_{\widehat{\mathfrak{m}}}(\widehat{M})$ induces an isomorphism of tight closures $0^{\ast}_{H^{i}_{\mathfrak{m}}(M)}\cong 0^{\ast}_{H^{i}_{\widehat{\mathfrak{m}}}(\widehat{M})},$ where $(\widehat{R},\widehat{\mathfrak{m}})$ (resp. $\widehat{M}$) is the $\mathfrak{m}$-adic completion of $(R,\mathfrak{m})$ (resp. $M$).

Proof.

The isomorphism $H^{i}_{\mathfrak{m}}(M)\widetilde{\to}H^{i}_{\widehat{\mathfrak{m}}}(\widehat{M})$ is well-known, see [16, Proposition 2.15]. Furthermore the existence of completely stable big test elements for $R$ ([15, Theorem p.77]) immediately implies the equality of tight closures. ∎

Lemma 2**.**

Let $\overline{k}$ be an algebraically closed field of positive characteristic $p$ and let $V$ be a $\overline{k}$-finite dimensional left $\overline{k}[F]$-module. Then the quasilength of $V$ is $\dim_{\overline{k}}(V^{\ast}),$ where $(-)^{\ast}$ is the operation on $\overline{k}[F]$-modules from Definition 2.

Proof.

By definition of quasilength, we have $\mathrm{ql}(V)=\mathrm{ql}(V^{\ast}).$ Moreover, $F$ acts surjectively and thus injectively on $V^{\ast}.$ Hence, by [5, Proposition 4.6] for example, $V^{\ast}$ has a $\overline{k}$-basis of vectors fixed by $F.$ The lemma easily follows. ∎

Lemma 3**.**

Let $(A,\mathfrak{m})$ be a local Noetherian $k$-algebra with Frobenius endomorphism $F$ and let $M$ be a left $A[F]$-module. Suppose that $M$ is Artinian and Noetherian as an $A$-module. If $M$ is supported at the maximal ideal $\mathfrak{m},$ then $M$ and $\frac{M}{\mathfrak{m}M}$ have the same quasilength.

Proof.

Let us show that $\mathfrak{m}M\subset M_{nil}.$ This implies the lemma since $\mathrm{ql}(M_{nil})=0.$

Since $M$ is supported at $\mathfrak{m}$ and is Noetherian, $\mathfrak{m}^{l}M=0$ for some $l\geq 0.$ Thus $F^{r}(\mathfrak{m}M)\subset\mathfrak{m}^{p^{r}}M=0$ for some $r\geq 0.$ Hence $\mathfrak{m}M\subset M_{nil},$ as claimed. ∎

Lemma 4**.**

Let $(A,\mathfrak{m})$ be a Noetherian local ring and let $X\xrightarrow{\pi}Z=\operatorname{Spec}(A)$ be a projective morphism of special fibre $Y.$ Suppose that the fibres of $\pi$ are of dimension at most $d.$ Then for all quasi-coherent sheaves $\mathcal{F}$ on $X,$ the canonical morphism $H^{d}(X,\mathcal{F})\to H^{d}(Y,\frac{\mathcal{F}}{\mathfrak{m}\mathcal{F}})$ induces an isomorphism $\frac{H^{d}(X,\mathcal{F})}{\mathfrak{m}H^{d}(X,\mathcal{F})}\to H^{d}(Y,\frac{\mathcal{F}}{\mathfrak{m}\mathcal{F}}).$

Proof.

We first claim that for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and for all integers $i\geq d+1,H^{i}(X,\mathcal{F})=0.$ This is well-known. Indeed it follows from the theorem on formal functions that $R^{i}\pi_{*}(\mathcal{G})=0$ for all $i\geq d+1,$ and for all coherent sheaves $\mathcal{G}$ on $X$ ([13, Corollary III 11.2]). Thus, since $Z$ is affine, we have that $H^{i}(X,\mathcal{G})=0,$ for all $i\geq d+1,$ and for all coherent sheaves $\mathcal{G}$ 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]).

Let $\{r_{1},\dots,r_{N}\}$ be a set of generators of the maximal ideal $\mathfrak{m}.$ We have the following short exact sequences of quasi-coherent sheaves on $X:$ $0\to\mathfrak{m}\mathcal{F}\to\mathcal{F}\to\frac{\mathcal{F}}{\mathfrak{m}\mathcal{F}}\to 0$ and $0\to J\to\oplus^{l=N}_{l=1}\mathcal{F}\xrightarrow{\phi}\mathfrak{m}\mathcal{F}\to 0,$ where $\phi(f_{1},\dots,f_{N})=r_{1}f_{1}+\dots+r_{N}f_{N}$ and $J$ is the kernel of $\phi.$ Moreover by the claim, we have that $H^{d+1}(X,\mathfrak{m}\mathcal{F})=H^{d+1}(X,J)=0.$ Thus in the associated long exact sequences in Zariski cohomology, the morphisms $H^{d}(X,\mathcal{F})\to H^{d}(X,\frac{\mathcal{F}}{\mathfrak{m}\mathcal{F}})$ and $\oplus^{l=N}_{l=1}H^{d}(X,\mathcal{F})\to H^{d}(X,\mathfrak{m}\mathcal{F})$ are surjective. The latter implies that the image of the morphism $H^{d}(X,\mathfrak{m}\mathcal{F})\to H^{d}(X,\mathcal{F})$ from the long exact sequence is $\mathfrak{m}H^{d}(X,\mathcal{F}).$ 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 $\mathbb{P}^{n}.$ We first fix the notation.

Definition 6**.**

Let $B\subset L$ 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 projective hypersurface $Y_{B}$ of $\mathbb{P}^{n}_{B}$ whose base-change $L\otimes_{Frac(B)}Y_{B}$ is isomorphic to $Y.$

Given such an hypersurface $Y_{B},$ for each closed point $u$ of $\text{Spec}(B),$ we let $Y_{u}$ be the fibre of $Y_{B}$ over $k(u).$ 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 $\mathbb{P}^{n}.$ Then there is a ring of definition $B\subset L$ of $Y$ such that, for all closed points $u$ of $\text{Spec}(B):$

• (i)

The unit $F$-module length of $H_{g_{u}}^{1}(R_{k(u)})$ is $1+\mathrm{ql}(H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}})).$

• (ii)

The $D_{k(u)}$-module length of $H_{g_{u}}^{1}(R_{k(u)})$ is $1+\dim_{\overline{k(u)}}(\overline{k(u)}\otimes_{k(u)}H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}}))^{\ast},$ where $\overline{k(u)}$ is any algebraic closure of $k(u)$ and $(-)^{\ast}$ is the operation on $\overline{k(u)}[F]$-modules from Definition 2.

Proof.

It is well-known that in this case the blow-up of the origin is a resolution $\pi^{\prime}$ of the singularity and that the fibre at the origin is isomorphic to $Y.$ The result then immediately follows from Theorem 2 applied to $\pi^{\prime}.$ ∎

Thus the $D_{k}$-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 curve $E$ in $\mathbb{P}^{2}_{\mathbb{Q}}.$ Then, for almost all primes $p,$ the $D_{R_{\mathbb{F}_{p}}}$-module length of $H^{1}_{g_{p}}(R_{\mathbb{F}_{p}})$ is $2$ if $E_{p}$ is ordinary and $1$ if $E_{p}$ is supersingular, where $E_{p}$ (resp. $g_{p}$) is the reduction of $E$ (resp. $g$) modulo $p.$

3. Comparison with Characteristic Zero

Here, given a complex polynomial $g,$ we consider a holonomic $D_{\mathbb{C}}$-module $N_{g}$ whose length compares well to the $D_{k(u)}$-module length of $H_{g_{u}}^{1}(R_{k(u)}).$

Definition 7**.**

Let $g\in R_{\mathbb{C}}$ be a complex polynomial. Then $N_{g}$ is the left $D_{\mathbb{C}}$-submodule of the first local cohomology $D_{\mathbb{C}}$-module $H^{1}_{g}(R_{\mathbb{C}})$ generated by the class of $\frac{1}{g}.$

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 $D_{\mathbb{C}}$-module length of $N_{g}$ is $1+\dim_{\mathbb{C}}H^{n-1}(Y,\mathcal{O}_{Y}).$

Remark 2**.**

There is a ring of definition $B\subset\mathbb{C}$ of $Y$ such that, for all closed points $u$ of $\text{Spec}(B),$ $\dim_{\mathbb{C}}H^{n-1}(Y,\mathcal{O}_{Y})=\dim_{\overline{k(u)}}(\overline{k(u)}\otimes_{k(u)}H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}})).$ Hence by Corollary 1 and Theorem 3, there is a ring of definition $B^{\prime}\supset B$ of $Y$ such that for all closed points $u$ of $\text{Spec}(B^{\prime}),$ if the Frobenius $F$ acts bijectively on $\overline{k(u)}\otimes_{k(u)}H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}}),$ then the length of $N_{g}$ is equal to the $D_{k(u)}$-module length of $H_{g_{u}}^{1}(R_{k(u)}).$ Indeed in that case, $\dim_{\mathbb{C}}H^{n-1}(Y,\mathcal{O}_{Y})=\dim_{\overline{k(u)}}(\overline{k(u)}\otimes_{k(u)}H^{n-1}(Y_{u},\mathcal{O}_{Y_{u}}))^{\ast}.$ This property of the Frobenius is called weak ordinarity and is expected to hold for a dense set of closed points of $\operatorname{Spec}(B^{\prime}),$ 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 $B\subset\mathbb{C}$ containing the coefficients of $g$ such that:

1. (1)

For all closed points $u\in\operatorname{Spec}(B),$ $\lg_{D_{k(u)}}(H_{g_{u}}^{1}(R_{k(u)}))\leq\lg_{D_{\mathbb{C}}}(N_{g})$? 2. (2)

There is a dense set of closed points of $\operatorname{Spec}(B)$ for which $\lg_{D_{k(u)}}(H_{g_{u}}^{1}(R_{k(u)}))=\lg_{D_{\mathbb{C}}}(N_{g})$?

As explained in Remark 2, for $g$ homogeneous with an isolated singularity and $n\geq 2,$ 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 $n\geq 2.$

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.