User axel st&#228;bler - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:39:18Z http://mathoverflow.net/feeds/user/31051 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120840/vanishing-cycles-of-a-locally-constant-sheaf-for-a-smooth-morphism-in-the-l-p/124960#124960 Answer by Axel Stäbler for Vanishing cycles of a locally constant sheaf for a smooth morphism in the $l = p$-case Axel Stäbler 2013-03-19T13:20:04Z 2013-03-19T13:20:04Z <p>$\DeclareMathOperator{\ord}{ord}$ $\DeclareMathOperator{\Spec}{Spec}$</p> <p>This negative answer to my question is a slight expansion of an email sent to me by Brian Conrad. Since I merely added some details for my own benefit this post is community wiki.</p> <p>As Damian Rössler pointed out in his comment the essential ingredient is the failure of the smooth base change theorem in the $l = p$ case. This is can be done via Artin-Schreier examples (cf. Milne's book on étale cohomology VI, Remark 4.4).</p> <p>Let $S$ be the spectrum of $R = k[[t]]$ where $k$ is an algebraically closed field of characteristic $p > 0$ and write $K$ for the quotient field of $R$. Let now $X = \mathbb{A}^1_{R} = \Spec k[[t]][y]$. Let us consider the constant sheaf $\mathcal{F} = \mathbb{Z}/p\mathbb{Z}$ on $X$ and its vanishing cycles complex at the origin $0$ of the special fiber. Denote by $A$ the strict henselization of $R[y]_{(t,y)}$.</p> <p>Using SGA7-I, Exposé I, Proposition 2.3 we have <code>$$(1) \quad (i^\ast R^n j_\ast j^\ast \mathcal{F})_0 = H^n(A_{K_s}, \mathcal{F}) = \varinjlim H^n(A_F, \mathcal{F}) ,$$</code> where $F$ varies over the finite extensions of $K$ in $K_s$. Since strict henselization is compatible with finite base change each $A_F$ is the analogue of $A_K$ with $R$ replaced by its normalization in $F$ (which is again local since $R$ is henselian).</p> <p>We claim that $(1)$ is non-zero (in fact infinite). Admitting this for the moment we obtain a counterexample to my question. Indeed, part of the long exact sequence associated to the exact triangle $i^\ast \mathcal{F} \to R\Psi \mathcal{F} \to R\Phi \mathcal{F}$ is $$i^\ast R^1 j_\ast j^\ast \mathcal{F} \to R^1\Phi \mathcal{F} \to H^1(i^\ast\mathcal{F}),$$ but $i^\ast$ is exact so that $H^1(i^\ast \mathcal{F})$ is zero.</p> <p>In order to show that $(1)$ is nonzero we consider the Artin-Schreier sequence <code>$$0 \to \mathcal{F} \to \mathcal{O}_{A_{K_s}} \to \mathcal{O}_{A_{K_s}} \to 0.$$</code> Since $\Spec A_{K_s}$ is affine the higher étale cohomology groups of <code>$\mathcal{O}_{A_{K_s}}$</code> vanish. It is therefore sufficient to show that the cokernel of $f \mapsto f^p - f$ on $A_{K_s}$ is infinite.</p> <blockquote> <p>Claim: For any non-zero elements $c_i$ in the maximal ideal of the valuation ring of $K_s$ with pairwise distinct valuation the elements $y/c_i$ represent $\mathbb{F}_p$-linearly independent Artin-Schreier classes.</p> </blockquote> <p>It suffices to proves this in each $H^1(A_F, \mathcal{F})$ separately (assuming all $c_i$ are contained in $A_F$). Hence, by replacing $R$ with with its normalization in $F$ we may assume that $F = K$ and all $c_i$ contained in $K$.</p> <p>A non-trivial $\mathbb{F}_p$-linear combination of such $1/c_i$'s is an element of $K^\times$ with negative valuation. So it suffices to show that for nonzero $c \in R$ with $\ord(c) > 0$ the element $y/c \in A_K$ is not of the form $f^p - f$ for any $f \in A_K$. Here the valuation function $\ord$ is induced by $t^n \mapsto n$. In particular, $\ord(y/c) = -\ord(c) &lt; 0$. If such $f$ exists then we must have $$0 > \ord(f^p - f) \geq \min(p \ord(f), \ord(f)),$$ so $\ord(f) &lt; 0$. We therefore have $f = h/t^e$ with $e > 0$ and $h \in A$ not divisible by $t$.</p> <p>Thus we get $$-\ord(c) = \ord(y/c) = \ord(f^p - f) = p \ord(f) = -pe.$$ This means that $c$ is contained in $t^{ep}R^\times$ and by changing $y$ by a unit we may assume that $c = t^{ep}$. Hence, $$\frac{y}{t^{ep}} = \frac{y}{c} = f^p - f = \frac{(h^p - t^{(p-1)e} h)}{t^{ep}}.$$ Equivalently we obtain the following equation in $A$ $$y = h^p - t^{(p-1)e}h = h(h^{p-1} - t^{(p-1)e}).$$</p> <p>But $A$ is a regular local ring, so in particular a UFD, in which $y$ is irreducible. It follows that precisely one of the factors on the right hand side is a unit. Since $h^p = y + t^{(p-1)e}h$ lies in the maximal ideal of $A$ so does $h$. But then, since $e > 0$, $h^{p-1} - t^{(p-1)e}$ is also contained in the maximal ideal. This is a contradiction.</p> http://mathoverflow.net/questions/120840/vanishing-cycles-of-a-locally-constant-sheaf-for-a-smooth-morphism-in-the-l-p Vanishing cycles of a locally constant sheaf for a smooth morphism in the $l = p$-case Axel Stäbler 2013-02-05T07:07:23Z 2013-03-19T13:20:04Z <p>$\DeclareMathOperator{\Spec}{Spec}$</p> <p>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</p> <h2>Background</h2> <p>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).</p> <p>This roughly goes as follows: Let $S$ be the strict Henselization of $k[x]_{(x)}$. Let $s$ be the closed point of $\Spec S$ and let $\bar{\eta}$ be the spectrum of a separable closure of the quotient field of $S$. Consider the cartesian diagram</p> <p>$\begin{array}{ccccccc} &amp;X_s&amp;\xrightarrow{i}&amp;X_{S}&amp;\xleftarrow{j}&amp;X_{\bar{\eta}} \newline &amp; \downarrow &amp; &amp;\downarrow &amp;&amp; \downarrow\newline &amp;s&amp;\xrightarrow{}&amp;S&amp;\xleftarrow{}&amp;\bar{\eta} \end{array}$</p> <p>For a constructible sheaf $\mathcal{F}$ on $X_{\acute{e}t}$ the <em>nearby cycles</em> 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 <em>vanishing cycles</em>.</p> <p>(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).</p> <blockquote> <p>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.</p> </blockquote> <hr> <p>Edit: My previous definition of Nearby/Vanishing cycles contained an error. I am indebted to Brian Conrad for pointing this out to me.</p>