# 10/08/2022 Lecture 15 ## More compactly supported cohomology :::{.remark} Last time: proper base change, which we won't prove, but we'll discuss the key geometric input. Recall the statement: for $\pi:X\to S$ proper, $F\in \Sh(X_\et)$ constructible (i.e. finite stalks and every closed subscheme admits an open subscheme where $F$ is locally constant), the derived pushforwards $\RR^i \pi_* F$ are all constructible with geometric stalks given by $H^i(X_{\bar x, \et}; \ro{F}{X_{\bar x, \et}})$. As a corollary, for proper $\pi$, forming $\RR \pi_* F$ commutes with base change. In order to go from constructible to torsion sheaves, one needs to write a torsion $F$ as a filtered colimit of constructible sheaves. Recall that properness is necessarily, evidence by $\ul{\FF_p}$ on $\AA^1\slice{\bar\FF_p}$. ::: :::{.remark} The key ideas for the proof. **Step 1**: reduce to the case where $\pi$ is a relative curve. Since $X$ is quasiprojective, after blowing up $\pi$ can be factored as a sequence of morphisms which are all relative curves. Applying the Leray spectral sequence, one you know the pushforwards through one of these maps in constructible, you've reduced to the case of 1 dimension less. **Step 2**: use dévissage to reduce to the case where $F = \mu_n$, which is a locally constant sheaf. This is not so easy! On an open, $F$ is locally constant, so passing to a fine enough cover makes it honestly constant. Reducing to this case involves push-pull arguments to get a map from $F$ to the push-pull of some constant sheaf, which is a direct sum of copies of $\mu_n$. **Step 3**: for $\pi:X\to S$ a relative curve and $F = \mu_n$, take the LES of derived pushforwards applied to the Kummer sequence to get \begin{tikzcd} 0 \\ {\pi_*\mu_n} && {\pi_* \GG_m} && {\pi_* \GG_m} \\ {\RR^1 \pi_*\mu_n} && {\RR^ 1\pi_* \GG_m} && {\RR^ 1\pi_* \GG_m} \\ {\RR^2 \pi_*\mu_n} && {0\, \cdots} \arrow[from=2-1, to=2-3] \arrow[from=2-3, to=2-5] \arrow[from=2-5, to=3-1, in=180, out=0] \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=3-5, to=4-1, in=180, out=0] \arrow[from=4-1, to=4-3] \arrow[from=1-1, to=2-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMCwxLCJcXHBpXypcXG11X24iXSxbMCwyLCJcXFJSXjEgXFxwaV8qXFxtdV9uIl0sWzAsMywiXFxSUl4yIFxccGlfKlxcbXVfbiJdLFsyLDEsIlxccGlfKiBcXEdHX20iXSxbMiwyLCJcXFJSXiAxXFxwaV8qIFxcR0dfbSJdLFsyLDMsIjBcXCwgXFxjZG90cyJdLFs0LDIsIlxcUlJeIDFcXHBpXyogXFxHR19tIl0sWzQsMSwiXFxwaV8qIFxcR0dfbSJdLFswLDAsIjAiXSxbMCwzXSxbMyw3XSxbNywxXSxbMSw0XSxbNCw2XSxbNiwyXSxbMiw1XSxbOCwwXV0=) Although it's not trivial, everything after $\RR^2 \pi_* \mu_n$ is zero. We proved that if the base was a point, so we have a curve over $k=\kbar$, then we get cohomological vanishing in degrees at least 2. This is a relative version, and to show it's true one needs to carry out a similar argument over strictly Henselian fields. Goal: show $\RR^i \pi_* \mu_n$ for $i=0,1,2$ are represented by quasifinite $\Sch\slice S$. The key geometric input: in this situation, $\RR^1 \pi_* \GG_m = \Pic(X/S)$ is representable in $\Sch\slice{S}^{\lft}$, along with some of the structure theory for this scheme, which is due to Grothendieck and highly nontrivial. If $S = \spec k$ for a field $k$, this is the Picard scheme of a curve. If the curve is smooth projective over $k=\kbar$, this is a finite disjoint union of copies of $\Jac(X)$ Note that $\Pic(X/S)$ is the functor that sends $T$ to line bundles on $X_T$, modulo the pullbacks of line bundles on $T$, sheafified. Thus is it nontrivial that it is representable. One can use this to identify \[ \RR^1\pi_* \mu_n = \ker( \Pic(X/S) \mapsvia{\cdot n} \Pic(X/S)) .\] This kernel may not be quasifinite when $X$ is just a locally finite type scheme, e.g. take the constant scheme $\bigoplus_{i\geq 0} C_2$. One thus needs structure theory of $\Pic(X/S)$, which is complicated by the fact that the functor does not commute with base change. The key input input is knowing how it *does* interact with base change, which lets one pass to individual fibers and use what we know about $\Pic$ of curves. :::{.claim} The LHS $\RR^1 \pi_* \mu_n$ is quasifinite, represented by the scheme-theoretic kernel of the RHS map, i.e. the preimage of the zero section. Moreover, \[ \RR^2 \pi_* \mu_n = \coker(\Pic(X/S) \to \Pic(X/S)) \] is quasifinite. ::: The second statement is again nontrivial: if $X/S$ is a family of smooth proper curves, this would be the cokernel of multiplication by $n$. So the claim heavily uses the structure theory, namely that the identity component of the scheme is divisible. ::: :::{.example title="How to use this"} Let $\pi:X\to S$ be a smooth proper curve, and consider computing $H^i(X_\et; C_n)$. The Leray spectral sequence gives \[ H^r(S_\et; \RR^s \pi_* C_n) \abuts H^{r+s}(X_\et; C_n) ,\] so what is the derived pushforward on the LHS? We know it is constructible, and locally constant on an open, and we know the stalks are given by \[ (\RR^s \pi_* C_n)_{\bar x} \cong H^s(X_{\bar x}; C_n) ,\] which we've computed, and the rank does not depend on the point $\bar x$. One we know a bit more, this will be enough to show that the sheaf is locally constant. Moreover, there are techniques for computing at least the ranks of a locally constant sheaf on a curve, e.g. the Grothendieck–Ogg–Shafarevich formula. ::: :::{.example title="?"} Let $X\to S$ be a proper curve, not necessarily smooth, but (say) with smooth generic fiber. The same computation as above shows over the locus in $S$ where $\pi$ is smooth, $\RR\pi_* C_n$ is locally constant. So we can understand the cohomology over a large open subset of $S$, and using techniques like cohomology with compact support and dévissage, we can get the full cohomology of $S$. ::: :::{.proposition title="?"} Let $U\in\Sch$ be separated and $F$ a constructible sheaf on $U$, then for $j: U\injects X$ a compactification into a proper $X$, \( H_c(U; F) \da H^i(X_\et; j_! F) \) does not depend on $j$ or $X$. ::: :::{.proof title="?"} Let $j_i: U\injects X_i$ for $i=1,2$ be two compactifications. Take $j_1\times j_2: U\to X_1 \times X_2$ and let $X$ be the closure of the image. This yields the following: \begin{tikzcd} && X \\ U \\ && {X_1} \arrow["\pi", from=1-3, to=3-3] \arrow["j", from=2-1, to=1-3] \arrow["{j_1}"', from=2-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwxLCJVIl0sWzIsMCwiWCJdLFsyLDIsIlhfMSJdLFsxLDIsIlxccGkiXSxbMCwxLCJqIl0sWzAsMiwial8xIiwyXV0=) Here both $X$ and $X_1$ are proper, and $\pi$ is a projection from the product. We want to compare the cohomology of $j_! F$ on $X$ with that of $(j_1)_!$ on $X_1$, so use the Leray spectral sequence to get \[ H^r(X_1; \RR^s \pi_* j_! F) \abuts H^r(X; j_! F) .\] We want to understand the RHS. We can compute the stalks $( \RR^s \pi_* j_! F)_{\bar x} = H^s(\pi\inv(\bar x); j_! F) = 0$ for $s\gt 0$; this follows because if $\bar{x} \in \im j$ then we are taking higher cohomology of a point, and if not then $F = 0$. Note that we've used that $j$ and $j_1$ are open embeddings. We get an isomorphism \[ H^r(X; \pi_* j_! F) \iso H^r(X; j_! F) .\] :::{.claim} There is a canonical isomorphism \[ \pi_* j_! F \iso (j_1)_! F .\] One you construct a map, it's enough to check on stalks, and there one can apply proper base change. ::: ::: :::{.proposition title="?"} \envlist 1. Given a SES $F\injects G \surjects H$ of constructible sheaves on $U_\et$ there is a LES in $H_c(U; \wait)$. 2. If $F$ is constructible, then $H_c(U; F)$ is finite. ::: :::{.proof title="?"} Part 1: applying the definition of $H_c$, for $j: U\injects X$ a compactification we want a LES \[ H_c^i(X_\et; j_! F) \to H_c^i(X_\et; j_! G) \to H_c^i(X_\et; j_! H)\to \cdots .\] This follows because $j_!$ is exact, so $j_! F\injects j_! G \surjects j_! H$ is exact. We proved this last time by checking exactness on stalks. Part 2: ETS that $j_! F$ is constructible on $X$, since then $H^i(X_\et; j_! F)$ is finite by proper base change. Checking constructibility: 1. The stalks are finite: they're either zero or the stalks of $F$, and $F_x$ is finite since $F$ itself is constructible. 2. For $T \subseteq X$, we need to show $\ro{j_! F}{T}$ is locally constant on an open subset of $T$. This is already true on $U$, so consider $T \intersect U$. If nonempty, this is an open subset of $T$ which is closed in $U$, and constructibility of $F$ gives an open subset of local constancy. Otherwise, then the restriction is zero. ::: :::{.remark} One might use proper base change where $\pi: X\to S$ and you stratify $S$ into loci $U$ where $\pi$ is smooth and $Z$ its complement, and reduce things about computing cohomology on $S$ to cohomology on $U$ and $Z$. We'll now approach this more systematically. ::: ## Purity, the Gysin Sequence, Cohomology of Supports :::{.remark} Fixing notation, let \( \Lambda\da \ul{C_n} \) where $n$ is invertible on the base, and let $F$ be a sheaf of \( \Lambda\dash \) modules, e.g. $\mu_n$. For such an $F$, write $F(r) \da F \tensor_{ \Lambda} \mu_n\tensorpower{}{r}$, where if $r < 0$ one first takes the dual sheaf. Note that over $k=\kbar$, after choosing an $n$th root of unity (i.e. trivializing $\mu_n$), one has $F(r) \cong F$, but these differ in general and this will manifest in different Galois actions on their cohomology. The Gysin sequences here will be similar to those that appear in topology, but these twists will appear, recording unexpected Galois actions. In topology, Gysin sequences involve the study of cohomology of sphere bundles -- here the analog will be $\AA^n\smts{\pt}$ instead of a sphere, and it will have an interesting nontrivial Galois action -- for example, we saw that $H(\GG_m; C_n)$ has a nontrivial action. Cohomology with supports will be a technique relating cohomology on $U$ to cohomology on $Z$. ::: :::{.definition title="Global sections with support"} For $Z \subseteq X$ a closed subscheme, define the functor \[ \Gamma_Z: \Sh(X_\et; \zmod) &\to \zmod \\ \Gamma_Z(X; \wait) &\da \ker(\Gamma(X; \wait) \to \ker \Gamma(U; \wait)) .\] These are global sections on $X$ which vanish when restricted to $U$. ::: :::{.exercise title="?"} Show $\Gamma_Z$ is left exact. ::: :::{.definition title="Cohomology with supports"} Define $H_Z^*(X; \wait) \da \RR \Gamma_Z(X; \wait)$ to be the right-derived functor of global sections with supports. ::: :::{.proposition title="?"} There is a LES \[ H_Z(X_\et; \wait) \to H(X_\et; \wait) \to H_Z(U_\et; \wait) \to \cdots .\] ::: :::{.proof title="?"} Let $U\injectsvia{j} X, Z\da X\sm U \injectsvia{i}$ be open/closed embeddings respectively. Take the SES \[ j_! j^* \ul\ZZ \injects \ul\ZZ \surjects i_* i^* \ul\ZZ \] and note that $\Hom(i_*i^* \ul\ZZ, \wait) \cong \Gamma_Z(X_\et; F)$. This follows by applying $\Hom(\wait, F)$ to the SES to get \begin{tikzcd} {\Hom(i_*i^* \ul\ZZ, F)} && {\Hom(\ul\ZZ, F)} && {\Hom(j_!j^* \ul \ZZ, F)} \\ && {\Gamma(X; F)} && {\Hom(j^* \ul \ZZ, F)} \\ &&&& {\Gamma(U; \ro{F}{U})} \arrow["{\text{restriction}}", from=1-3, to=1-5] \arrow[Rightarrow, no head, from=3-5, to=2-5] \arrow[Rightarrow, no head, from=2-5, to=1-5] \arrow[Rightarrow, no head, from=2-3, to=1-3] \arrow[from=1-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) and so the first term is the kernel of the restriction map, which is the definition of $\Gamma_Z(X_\et; F)$. This implies that \[ H^i_Z(X_\et; F) \cong \Ext^i_{\Sh(X_\et; \zmod)}(i_*i^* \ul\ZZ, F) ,\] since we've shown that the $i=0$ functors are isomorphic and this determines their derived functors. Thus the LES we want is precisely the LES in $\Ext$. ::: :::{.theorem title="?"} Let $Z \subseteq X\in \Sch\slice k$ be a closed subscheme over $k$ a field and suppose $Z, X$ are smooth with $Z$ of pure codimension $c$ in $X$. For $F\in \Sh(X_\et; \zmod)$ locally constant and constructible (so locally constant with finite stalks), there is a canonical isomorphism \[ H^{r-2c}(Z; F(-c)) \iso H^r_Z(X; F), \qquad \forall r \geq 0 .\] ::: :::{.example title="?"} Let $Z = \ts{\pt}\injects \AA^1$, so $c=1$. The following will be the base case in many upcoming induction arguments, including computing $H^*$ of $\PP^n$: :::{.claim} Then \[ H^{r-2}(\pt; C_n(-1)) \cong H^r_{\pt}(\AA^1_\et; C_n) = \begin{cases} C_n(-1) & r=2 \\ 0 & \text{else}. \end{cases} .\] ::: This can be proved directly without using the theorem by using the LES constructed above: \[ \cdots\to H^i_\pt(\AA^1; C_n) \to H^i(\AA^1; C_n) \to H^i(\GG_m; C_n)\to H^{i+1}_\pt(\AA^1; C_n) \to \cdots .\] One can show $H^*(\AA^1) = C_n\cdot t^0$, so $H^i = 0$ for $i\geq 0$: this follows from taking the LES for the Kummer sequence to get \begin{tikzcd} 0 \\ {\mu_n} && {k[t]\units = k\units} && k\units \\ \\ { \therefore H^1(\AA^1; \mu_n) = 0} && {\Pic(\AA^1) = 0} && {\Pic(\AA^1) = 0} \\ &&&& 0 \arrow["{\cdot n}", color={rgb,255:red,214;green,92;blue,92}, two heads, from=2-3, to=2-5] \arrow[from=1-1, to=2-1] \arrow[from=2-1, to=2-3] \arrow[from=2-5, to=4-1, out=0, in=180] \arrow[from=4-1, to=4-3] \arrow[from=4-3, to=4-5] \arrow[from=4-5, to=5-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCIwIl0sWzAsMSwiXFxtdV9uIl0sWzIsMSwia1t0XVxcdW5pdHMgPSBrXFx1bml0cyJdLFs0LDEsImtcXHVuaXRzIl0sWzAsMywiIFxcdGhlcmVmb3JlIEheMShcXEFBXjE7IFxcbXVfbikgPSAwIl0sWzIsMywiXFxQaWMoXFxBQV4xKSA9IDAiXSxbNCwzLCJcXFBpYyhcXEFBXjEpID0gMCJdLFs0LDQsIjAiXSxbMiwzLCJcXGNkb3QgbiIsMCx7ImNvbG91ciI6WzAsNjAsNjBdLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19LFswLDYwLDYwLDFdXSxbMCwxXSxbMSwyXSxbMyw0XSxbNCw1XSxbNSw2XSxbNiw3XV0=) We can now compute $H^i(\GG_m; \mu_n)$; the LES yields \[ H^i(\GG_m; \mu_n)\to H^i(\GG_m; \GG_m) \to H^i(\GG_m; \GG_m) \to H^{i+1}(\GG_m; \mu_n)\to \cdots \] which can be identified as \begin{tikzcd} 0 \\ {H^0(\GG_m; \mu_n) = \mu_n(k)} && {k[t, t\inv]\units \cong k\units \times t^\ZZ} && {k[t, t\inv]\units \cong k\units \times t^\ZZ} \\ \\ { \therefore H^1(\GG_m; \mu_n) \cong C_n} && {\Pic(\GG_m) = 0} \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, out=0, in=180] \arrow[from=4-1, to=4-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzAsMSwiSF4wKFxcR0dfbTsgXFxtdV9uKSA9IFxcbXVfbihrKSJdLFsyLDEsImtbdCwgdFxcaW52XVxcdW5pdHMgXFxjb25nIGtcXHVuaXRzIFxcdGltZXMgdF5cXFpaIl0sWzQsMSwia1t0LCB0XFxpbnZdXFx1bml0cyBcXGNvbmcga1xcdW5pdHMgXFx0aW1lcyB0XlxcWloiXSxbMCwzLCIgXFx0aGVyZWZvcmUgSF4xKFxcR0dfbTsgXFxtdV9uKSBcXGNvbmcgQ19uIl0sWzIsMywiXFxQaWMoXFxHR19tKSA9IDAiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbNCw1XV0=) where we've used that deleting a codimension 1 subvariety changes $\Pic$ by quotienting by a (not necessarily injective) copy of $\ZZ$[^hart_pic_inj] and taking $n$th power is surjective on $k\units$. Thus $H^i(\AA^1; C_n) = 0$ for $i>0$, and using the LES of cohomology with supports this yields $H^i(\GG_m; C_n)\cong H^{i+1}_{\pt}(\AA^1; C_n)$. In particular this yields an isomorphism at $i=1$, which proves the claim. [^hart_pic_inj]: See Hartshorne section 2.6. :::