# 10/13/2022 Lecture 16 ## The Gysin sequence and purity :::{.remark} Last time: purity and the Gysin sequence. We introduced cohomology with supports, the derived functor of global sections supported on a closed subscheme $Z$, and tried to understand it when $Z$ was smooth in an ambient smooth variety. For today, sheaves $F$ will have *order prime to the characteristic*, i.e. the stalks have order prime to the characteristic. ::: :::{.theorem title="Purity"} If $Z \injects X$ is a closed immersion with $X, Z$ smooth and $\codim_X Z = c$, for $F$ locally constant and constructible, \[ H^{r-2c}(X;F(-c)) =H^r_Z(X; F), \qquad \forall r\geq 0 .\] ::: :::{.remark} We regard this as a computation of the RHS, which can be viewed as the cohomology of a deleted neighborhood of $Z$. ::: :::{.corollary title="Existence of a Gysin sequence"} Suppose $X, Z$ as above with $U \da X\sm Z$ the open complement of $Z$. For $0 \leq r \leq 2c-1$ (small $r$), the restriction map is an isomorphism \[ H^r(X_\et; F)\iso H^r(U_\et; F) .\] For large $r$, there is a Gysin LES: \begin{tikzcd} 0 \\ {H^{2c-1}(X; F)} && {H^{2c-1}(U; F)} && {H^{0}(Z; F(-c))} \\ \\ {H^{2c}(X; F)} && {H^{2c}(U; F)} && {H^{1}(Z; F(-c))} \\ \\ \cdots \arrow[from=1-1, to=2-1] \arrow[from=2-1, to=2-3] \arrow[from=2-3, to=2-5] \arrow[from=2-5, to=4-1, in=180, out=0] \arrow[from=4-1, to=4-3] \arrow[from=4-3, to=4-5] \arrow[from=4-5, to=6-1, in=180, out=0] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCIwIl0sWzAsMSwiSF57MmMtMX0oWDsgRikiXSxbMiwxLCJIXnsyYy0xfShVOyBGKSJdLFs0LDEsIkheezB9KFo7IEYoLWMpKSJdLFswLDMsIkheezJjfShYOyBGKSJdLFsyLDMsIkheezJjfShVOyBGKSJdLFs0LDMsIkheezF9KFo7IEYoLWMpKSJdLFswLDUsIlxcY2RvdHMiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbNCw1XSxbNSw2XSxbNiw3XV0=) ::: :::{.remark} How to interpret: deleting $Z$ of high codimension in $X$, then $H^*(X) = H^*(Z)$ in low degrees and there is a LES relating $H^*(X), H^*(Z),$ and $H^*(X\sm Z)$ in high degrees. These results are called *purity* because the cohomology of high codimension subvarieties doesn't matter -- the cohomology is supported purely in low codimension. See Zariski-Nagata purity, i.e. purity of the branch locus, which says that an étale cover of a nice scheme has a branch locus concentrated in codimension 1. This is the analog of the above statement for $H^1$, and says that $G\dash$covers only depend on codimension 1, i.e. deleting codimension 2 subvarieties doesn't change covers. This is part of a very long story, and the main question is what happens when $X$ or $Z$ is not smooth. One can define a notion of flat cohomology, replacing $F$ with a finite flat group scheme, and people are currently working on this -- there is a recent result of Scholze et al proved a strong purity theorem in this situation using perfectoid techniques. ::: :::{.remark} The topological situation: consider the maps \[ H^{2c-1+i}(U; \ro F U) \to H^i(Z; F(-c)) .\] Let $\tilde Z$ be a delete neighborhood of $Z$, i.e. an $\varepsilon\dash$neighborhood of $Z$ with $Z$ removed. This yields a sphere bundle over $Z$, since it's the normal bundle over $Z$ with the zero section deleted. There is a map $\pi: \tilde Z\to Z$ homotopic to a sphere bundle, and taking the Leray spectral sequence yields exactly two nonzero columns and so there is only one page where the differentials can be nontrivial. For $F$ defined on all of $X$, not just $U$, this produces a LES \[ H^{2c-1+i}(Z; F) \to H^{2c-1+i}(\tilde Z; F) \mapsvia{\del} H^i(Z; F) \to H^{2c+i}(Z; F) \] where $\del$ is a differential in the spectral sequence. There is a restriction map $H^{2c-1+i}(U; F)\to H^{2c-1+i}(\tilde Z; F)$. This is called the **Thom-Gysin exact sequence**. Morally speaking, regarding this as de Rham cohomology, the map $H^{2c-1+i} \to H^i$ is integration along the fibers. ::: :::{.proof title="the purity theorem implies the corollary"} In the LES for cohomology with supports, replace $H_Z^*$ with $H^*(Z; F(-c))$. Why does this work? For the isomorphisms in low degrees, if $r < 2c$ then $H^{r-2c} = 0$ and one gets a LES comparing two things that are identically zero. In higher degrees, one just makes the direct replacement. ::: :::{.example title="Cohomology of $\PP^N$"} Suppose $\characteristic k \nmid n$. First we compute $H^*(\AA^1; \mu_n) = \mu_n t^0$ -- this follows from the LES in the Kummer sequence, since $H^*(\AA^1; \GG_m)$ vanishes for $i \geq 2$ and is $\Pic(\AA^1) = 0$ in degree $i=1$. By a Kunneth theorem, $H^i(\AA^n; \mu_n) = \mu_n t^0$. Now use the Gysin sequence for the pair $(\PP^n, \PP^{n-1})$ to get \[ H^r(\PP^n; C_n) = H^r(\AA^n; C_n),\qquad 0\leq r < 1 ,\] and for higher $r$ we get \begin{tikzcd} 0 \\ {H^1(\PP^n; \mu_n)} && {H^1(\AA^n; \mu_n)} && {H^0(\PP^{n-1}; C_n)} \\ \\ {H^2(\PP^n; \mu_n)} && {H^2(\AA^n; \mu_n)} && {H^1(\PP^{n-1}; C_n)} \\ \\ \cdots \arrow[from=1-1, to=2-1] \arrow[from=2-1, to=2-3] \arrow[from=2-3, to=2-5] \arrow[from=2-5, to=4-1, in=180, out=0] \arrow[from=4-1, to=4-3] \arrow[from=4-3, to=4-5] \arrow[from=4-5, to=6-1, in=180, out=0] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCIwIl0sWzAsMSwiSF4xKFxcUFBebjsgXFxtdV9uKSJdLFsyLDEsIkheMShcXEFBXm47IFxcbXVfbikiXSxbNCwxLCJIXjAoXFxQUF57bi0xfTsgQ19uKSJdLFswLDMsIkheMihcXFBQXm47IFxcbXVfbikiXSxbMiwzLCJIXjIoXFxQUF5uOyBcXG11X24pIl0sWzQsMywiSF4xKFxcUFBee24tMX07IENfbikiXSxbMCw1LCJcXGNkb3RzIl0sWzAsMV0sWzEsMl0sWzIsM10sWzMsNF0sWzQsNV0sWzUsNl0sWzYsN11d) using that $\mu_n(-1) = C_n$. Thus $H^1(\PP^n; \mu_n) = 0$ and $H^i(\PP^n; \mu_n) \cong H^{i-2}(\PP^{n-1}; C_n)$ for $i\geq 2$. Inducting on $n$ yields \[ H^*(\PP^n; C_n) = \bigoplus_{r\leq 2n \in 2\ZZ_{\geq 0}} C_n\qty{-{r\over 2}} t^{r} .\] Note that one could also compute this by computing $H^*$ of $\AA^n$ over a strictly Henselian local ring, one could use the Leray spectral sequence for $\AA^n\to \AA^{n-1}$. ::: ## Sketch proof of purity :::{.proof title="of purity, sketch"} Let $j: U \injects X$ be an open immersion with $i: Z\da X\sm U\injects X$ closed. The first step is to reduce to a local statement. ::: :::{.definition title="Upper shriek, sections supported on $Z$"} Define the *sections of $F$ supported on $Z$*: \[ i^! \da i^* \ker\qty{F \to j_* j^* F} \in \Sh(Z_\et) ,\] where the inner map takes a section and restricts it to its value on $U$. This is a local version of $\Gamma_Z$, and is a special case of Verdier's upper shriek. ::: :::{.proposition title="?"} $i_*$ is left adjoint to $i^!$. ::: :::{.exercise title="Essential"} Prove this. ::: :::{.remark} This shouldn't be surprising if you've seen Verdier duality. We also knew that $i_*$ has a left adjoint $i^*$, and having one might expect a right adjoint since $i_*$ is exact. ::: :::{.corollary title="?"} $i^!$ is left exact and preserves injectives, which follows from having an exact left adjoint. ::: :::{.proposition title="Local version of purity"} Let $Z,X,F$ be as in the theorem, then \[ \RR^{2c}i^! F = i^* F(-c), \qquad \RR^r i^! F =0 \, \text{ for }r\neq 0, 2c .\] The $r=0$ case is clear since $F$ is lcc and thus has no sections supported on $Z$. ::: :::{.proof title="The proposition implies the theorem"} The claim is that for any lcc sheaf $F$, we have \[ \Gamma(Z; i^! F) = \Gamma_Z(X; F) .\] This follows from the fact that $i^!$ is defined by taking a global section of $F$ on $X$ which vanishes on $U$, which is precisely the RHS here. Now the Grothendieck spectral sequence for $\RR \Gamma \circ \RR i^! = \RR \Gamma_Z$ yields \[ H^r(Z; \RR^s i^! F)\abuts H_Z^{r+s}(X; F) ,\] using that $i^!$ preserves injectives. Note that the spectral sequence has one column, since the $r=0$ column vanishes since $F$ lcc $\implies i^! F =0$. By the claim, $\RR^2 i^! F = 0$ for $s\neq 2c$, so $H^r(Z; F(-c)) = H^{r+2c}_Z(X; F)$. ::: :::{.proof title="of the proposition, sketch"} We claim we can reduce to the special case of the pair $(\AA^m, \AA^{m-c})$ where the 2nd factor is embedded as coordinate hyperplanes. This is a form of the structure theorem for smooth morphisms: anything smooth is étale-locally affine space, and smooth pair is étale-locally affine space and a coordinate hyperplane. The proof for pairs is the same as the proof for smooth morphisms. One can then induct on $m$ and $c$, and we did the base case $m=1,c=1$ last time (cohomology supported at the origin of $\AA^1$). Doing it in general can be done with Čech cohomology, covering by $\AA^m\sm U_i$ for $U_i$ various coordinate hyperplanes containing $\AA^{m-c}$. ::: :::{.remark} Where the sphere bundle appeared: the pair $(\AA^m, \AA^{m-c})$ yields a quotient that looks something like $S^n \times \CC^{n'}$ for some $n, n'$. For $c=0$, this is literally a sphere. ::: ## Comparison theorem :::{.remark} Continuing in the quest for computational tools, we'll prove (modulo a hard geometric statement) Artin's comparison theorem, which compares étale cohomology to singular cohomology. The main geometric tool: elementary fibrations, which we'll introduce over the next two classes and use it to prove this theorem along with some finiteness theorems for $H^*_\et$. ::: :::{.definition title="Elementary fibrations"} An **elementary fibration** is a diagram of the following form: \begin{tikzcd} U && Y && Z \\ \\ && S \arrow["j", hook, from=1-1, to=1-3] \arrow["i"', hook', from=1-5, to=1-3] \arrow["h", from=1-3, to=3-3] \arrow["f"', from=1-1, to=3-3] \arrow["g", from=1-5, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJVIl0sWzIsMCwiWSJdLFs0LDAsIloiXSxbMiwyLCJTIl0sWzAsMSwiaiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzIsMSwiaSIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoiYm90dG9tIn19fV0sWzEsMywiaCJdLFswLDMsImYiLDJdLFsyLDMsImciXV0=) where - $j$ is a Zariski-open embedding, so $U \subseteq Y$ is an open subset, - $j(U)$ is fiberwise dense in $Y$, i.e. $j(U)$ is dense in each fiber $Y_s$ for $s\in S$, - $Z = Y\sm U$, - $h$ is smooth projective with geometrically irreducible fibers, and is relative dimension 1, and - $g$ is finite étale. In words: we have an object of interest $U$, and we write it as an open subset of a smooth proper curve bundle over $S$ whose complement is finite étale, i.e. a disjoint union of étale multisections. The key property: the topology of the fibers of $f$ is constant, so e.g. over $\CC$ the fibers are all Riemann surfaces with the same genus and number of punctures. ::: :::{.proposition title="due to Artin"} This may seem restrictive, but it's not: for $X$ smooth over $k=\kbar$, for each $x\in X$ there exists a Zariski open $U\ni x$ fitting into an elementary fibration. This in a neighborhood of every point, a smooth variety can be written as a fiber bundle with punctured curve fibers. ::: :::{.proof title="?"} Pick any $U$, embed in $\PP^N$, and project away from a line until the base $S$ has dimension $\dim U - 1$. Doing this generically and deleting a bad locus yields the desired result. ::: :::{.theorem title="?"} Let $X\in\Var\slice \CC$ and let $F$ be a constructible abelian sheaf on $X_\et$. There are three sites: $X^\an, X_\et^\an, X_\et$, and there are morphisms of sites, \begin{tikzcd} && {X^\an_\et} \\ \\ {X^\an} &&&& {X_\et} \arrow["\pi"', from=1-3, to=3-1] \arrow["\an", from=1-3, to=3-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMiwwLCJYXlxcYW5fXFxldCJdLFswLDIsIlheXFxhbiJdLFs0LDIsIlhfXFxldCJdLFswLDEsIlxccGkiLDJdLFswLDIsIlxcYW4iXV0=) 1. $\pi$ induces isomorphisms on the cohomology of all abelian sheaves, and in fact an equivalence of categories $\pi^*: \Sh(X^\an)\iso \Sh(X^\an_\et)$. 2. For $F$ as in the theorem, there is an induced isomorphism \[ \an^*: H^i(X_\et; F)\iso H^i(X^{\an}_\et; \an^* F) .\] ::: :::{.remark} The corollary will be that cohomology on the associated analytic site will be the same as the étale cohomology. We need to define what these sites are: - $X^\an$ is the site associated to the Euclidean topology, i.e. the complex analytic space associated to $X$. Locally choose an embedding $X\injects \AA^N\slice \CC$, which is given by polynomials that cut out a topological space subspace of $\CC^n$. One checks that this topology is independent of the embedding, which was worked out in the 40s/50s. - $X^\an_\et$ is the category of complex analytic spaces mapping to $X^\an$ via local analytic isomorphisms, with covers the usual covers. There are objects in $X^{\an}_\et\sm X^\an$, such as covering spaces, since these are not literally open sets in $X$. For example, if $X$ is a Riemann surface, $\HH\in X^{\an}_\et$ despite not being an algebraic variety. The morphism $\pi$ corresponds to an inclusion of categories, regarding an open subset $U \injects X$, it can be regarded as a local analytic isomorphism. The morphism $\an$ is taking the associated analytic space, and one shows that analytification of an étale morphism is a local analytic isomorphism. ::: :::{.corollary title="?"} For $F$ as in the theorem, there is a canonical isomorphism \[ H^i(X_\et; F) = H^i(X^\an; F^\an) \] where $F^\an \da \pi_* \an^* F$, noting that $\pi$ induces an equivalence of categories on sheaves. ::: :::{.remark} We'll prove this for $X$ smooth and $F$ lcc using elementary fibrations. ::: :::{.exercise title="?"} Show that $\pi$ induces an equivalence of categories of sheaves of sets. Use that an analytic étale cover can be refined by objects in $X^\an$, namely open discs. :::