2 added 13 characters in body

$\DeclareMathOperator{\Spec}{Spec}$

My question is concerned with vanishing cycles of a locally constant sheaf for a smooth morphism in the case $l = p$. In the case $l \neq p$ this is a statement in SGA7-II. See below for the precise question. First let me start with some

## Background

Let us assume that one has an affine scheme $X$ over a field $k$ of characteristic $p >0$ and a function $f: X \to \mathbb{A}^1_k$. In SGA7-II Deligne then introduces the so-called nearby and vanishing cycles of $f$ at $0$ (say).

This roughly goes as follows: Let $S$ be the strict Henselization of $k[x]_{(x)}$ and denote by k[x]_{(x)}$. Let$\bar{S}$s$ be the integral closure closed point of $\Spec S$ in and let $\bar{\eta}$ be the spectrum of a separable closure of the quotient field of $S$. Let $\bar{\eta}$ be the generic point of $\Spec \bar{S}$ and $\bar{s}$ its closed point. Consider the cartesian diagram

$\begin{array}{ccccccc} &Z&\xrightarrow{i}&X_{\bar{S}}&\xleftarrow{j}&U &X_s&\xrightarrow{i}&X_{S}&\xleftarrow{j}&X_{\bar{\eta}} \newline & \downarrow & &\downarrow && \downarrow\newline &\bar{s}&\xrightarrow{}&\bar{S}&\xleftarrow{}&\bar{\eta} &s&\xrightarrow{}&S&\xleftarrow{}&\bar{\eta} \end{array}$

where $U$ and $Z$ are the fibers over $\bar{\eta}$ and $\bar{s}$ respectively.

For a constructible sheaf $\mathcal{F}$ on $X_{\acute{e}t}$ the nearby cycles of $\mathcal{F}$ are defined as $R \Psi(\mathcal{F}) = i^\ast Rj_\ast j^\ast \mathcal{F}$ (in $D(X_s)$). There is a natural morphism $i^\ast \mathcal{F} \to R\Psi(\mathcal{F})$ whose mapping cone we denote by $R\Phi(\mathcal{F})$ -- this is called the vanishing cycles.

(Of course, Deligne's construction is much more general and there are some groups acting here which I swept under the rug. But I hope the above is sufficient to get an idea of my question).

Deligne then proves the following (SGA7-II, 2.1.5): If $\mathcal{F}$ is a locally constant sheaf of $\mathbb{Z}/l$-modules (with $l$ a prime $\neq p$) and $f: X \to S$ defines a smooth morphism, then $R\Phi(\mathcal{F}) = 0$. My question is the following: Does this statement also hold for $l = p$? I am content to make additional assumptions on $X$ (e.g. of finite type over $k$) or my base field (take a finite field if you want). In particular, I am looking for a reference where this is proved or a counterexample.

Edit: My previous definition of Nearby/Vanishing cycles contained an error. I am indebted to Brian Conrad for pointing this out to me.

1

# Vanishing cycles of a locally constant sheaf for a smooth morphism in the $l = p$-case

$\DeclareMathOperator{\Spec}{Spec}$

My question is concerned with vanishing cycles of a locally constant sheaf for a smooth morphism in the case $l = p$. In the case $l \neq p$ this is a statement in SGA7-II. See below for the precise question. First let me start with some

## Background

Let us assume that one has an affine scheme $X$ over a field $k$ of characteristic $p >0$ and a function $f: X \to \mathbb{A}^1_k$. In SGA7-II Deligne then introduces the so-called nearby and vanishing cycles of $f$ at $0$ (say).

This roughly goes as follows: Let $S$ be the Henselization of $k[x]_{(x)}$ and denote by $\bar{S}$ the integral closure of $S$ in a separable closure of the quotient field of $S$. Let $\bar{\eta}$ be the generic point of $\Spec \bar{S}$ and $\bar{s}$ its closed point. Consider the diagram

$\begin{array}{ccccccc} &Z&\xrightarrow{i}&X_{\bar{S}}&\xleftarrow{j}&U \newline & \downarrow & &\downarrow && \downarrow\newline &\bar{s}&\xrightarrow{}&\bar{S}&\xleftarrow{}&\bar{\eta} \end{array}$

where $U$ and $Z$ are the fibers over $\bar{\eta}$ and $\bar{s}$ respectively.

For a constructible sheaf $\mathcal{F}$ on $X_{\acute{e}t}$ the nearby cycles of $\mathcal{F}$ are defined as $R \Psi(\mathcal{F}) = i^\ast Rj_\ast j^\ast \mathcal{F}$ (in $D(X_s)$). There is a natural morphism $i^\ast \mathcal{F} \to R\Psi(\mathcal{F})$ whose mapping cone we denote by $R\Phi(\mathcal{F})$ -- this is called the vanishing cycles.

(Of course, Deligne's construction is much more general and there are some groups acting here which I swept under the rug. But I hope the above is sufficient to get an idea of my question).

Deligne then proves the following (SGA7-II, 2.1.5): If $\mathcal{F}$ is a locally constant sheaf of $\mathbb{Z}/l$-modules (with $l$ a prime $\neq p$) and $f: X \to S$ defines a smooth morphism, then $R\Phi(\mathcal{F}) = 0$. My question is the following: Does this statement also hold for $l = p$? I am content to make additional assumptions on $X$ (e.g. of finite type over $k$) or my base field (take a finite field if you want). In particular, I am looking for a reference where this is proved or a counterexample.