# 10/06/2020 Lecture 14 ## Cohomology with compact support :::{.remark} Last time: étale cohomology of a curve. Today: compactly supported cohomology and Gysin sequences. Before moving on to proofs of the Weil conjectures, we'll focus on how to actually compute étale cohomology, which will use some big theorems. For example, this will include the proper base change theorem, which we likely won't prove since this would take around two weeks to set up! Recall that the étale cohomology of a smooth curve looks like the singular cohomology of a complex surface: forgetting Galois equivariance, we have \[ H(X_\et; ?) = C_{\ell^n} + C_{\ell^n}^{2g}t + C_{\ell^n}t^2 .\] ::: :::{.definition title="Extension by zero"} Let $j: U \embeds X$ be a Zariski-open embedding, define $j_!$ by sheafifying the following presheaf: \[ j_!: \Sh(U; \zmod) &\to \Sh(X; \zmod) \\ F & \mapsto V \mapsto \begin{cases} F(V\fiberprod{X} U) & j(V) \subseteq U \\ 0 & \text{else}. \end{cases} .\] ::: :::{.remark} Note that $j_! F(U) = \globsec{F; U}$. There is a $j_!$ defined for more general morphisms, but this requires the notion of a Nagata compactification. This is our first step toward understanding a notion of *compactly supported sections*. In topology, this can be defined in two ways: - The right-derived functors of taking compactly supported sections, or - Compactify the space, extend the constant sheaf by zero, and take the usual cohomology. Only the second works in algebraic geometry! ::: :::{.exercise title="?"} Show that $j_!$ is left adjoint to $j^*$. > Hint: check on presheaves and use the adjoint property of sheafification to reduce to checking on presheaves. ::: :::{.remark} The stalks at geometric points are given by \[ (j_! F)_{\bar x} = \begin{cases} F_{\bar x} & \bar{x} \in \im j \\ 0 & \bar{x}\not\in \im j. \end{cases} .\] ::: :::{.corollary title="?"} The functor $j_!$ is exact. This follows by checking on stalks. ::: :::{.proposition title="?"} Let $F\in \Sh(X_\et)$ and let $U \injects^{j}_{\text{open}} X \injectsfrom^{i}_{\text{closed}} Z = X\sm U$. Then there is a SES \[ j_! j^* F \injects F \surjects i_* i^* F .\] ::: :::{.proof title="?"} The first map is given by mapping sections and using the left-adjointness of sheafification, and the second is from restricting sections. Exactness can be checked on stalks, so choose $\bar{x} \in X$ a geometric point. There are two cases: - If $\bar{x}\in U$, we get $0\to F_{\bar x} \mapsvia{\id} F_{\bar x}\to 0\to 0$ since the $i_* i^* F$ is supported on $Z$. - If $\bar{x}\in U$, we get $0\to 0 \to F_{\bar x} \mapsvia{\id} F_{\bar x} \to 0$ which is again exact. ::: :::{.definition title="Cohomology with compact support"} Let $F\in \Sh(X_\et)$ and $j: U\to X$ with $X$ proper, and define the **compactly supported cohomology** as \[ H_c^i(U_\et; F) \da H^i(X_\et; j_! F) .\] ::: :::{.question} \envlist - Why does such an $X$ exist? One can not always embed $U$ into a proper variety. - Is this definition independent of $j: U\to X$? Answers to both require proper base change. ::: :::{.remark} This is supposed to be analogous to the following: for $X$ a manifold, take the complex of compactly supported smooth differential forms. These are locally of the form $f \dx_1\wedge \cdots \wedge \dx_n$ where $f\in C^\infty$ is smooth and compactly supported. This forms a chain complex, and its cohomology is compactly supported de Rham cohomology. More generally, if $X$ is a compact manifold, every form is compactly supported. So if $U$ is an open manifold, one can embed $U\injects X$ into a compact manifold and extend by zero to get compactly supported forms on $X$; this is what the above definition is trying to capture for an arbitrary sheaf. Note that proper schemes are always separated, so if $U$ isn't separated then it can't be embedded into a proper $X$. The following theorem says that this is the only thing that can go wrong: ::: :::{.theorem title="Nagata"} If $U \to S$ is a separated morphism, then there is a factorization \begin{tikzcd} & X \\ \\ U && S \arrow["{\exists\text{proper}}", dashed, from=1-2, to=3-3] \arrow["{\text{separated}}"', from=3-1, to=3-3] \arrow["{\exists\text{open}}", dashed, from=3-1, to=1-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJVIl0sWzIsMiwiUyJdLFsxLDAsIlgiXSxbMiwxLCJcXGV4aXN0c1xcdGV4dHtwcm9wZXJ9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzAsMSwiXFx0ZXh0e3NlcGFyYXRlZH0iLDJdLFswLDIsIlxcZXhpc3RzXFx0ZXh0e29wZW59IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d) Thus is $U$ is a point, $S$ is separated and always admits a compactification. Note that if $S$ is quasiprojective, one can always take its closure in $\PP^N$. ::: :::{.remark} Independence for torsion sheaves will follow from proper base change. ::: :::{.proposition title="?"} Let $U$ be a connected regular curve over $k=\kbar$ with $\characteristic(k) \nmid n$. There is a canonical isomorphism \[ H^2_c(U_\et; \mu_n) \iso C_n .\] This presumes independence of compactification, but note that for curves there is a canonical compactification. ::: :::{.proof title="?"} Let $j: U\injects X$ be the canonical regular compactification with $i: Z\da X\sm U\to X$ a disjoint union of finitely many points. One can then compute $H^i(X; j_! \mu_n)$ by using the SES \[ j_! j^* \mu_n \injects \mu_n\surjects i_*i^* \mu_n .\] Note $j^* \mu_n = \mu_n$ by checking its definition as a factor as maps into $\mu_n$, and $i_* i^* \mu_n \bigoplus I_i$ is a sum of skyscraper sheaves. This yields a \begin{tikzcd} H^i_c(U; \mu_n) \ar[r] & H^i(X; \mu_n) \ar[r] & H^i(X; i_* i^* \mu_n) \ar[dll, out=0, in=180] \\ H^{i+1}_c(U; \mu_n) \ar[r] & \cdots \ar[r] & \cdots \end{tikzcd} How does one compute the cohomology of skyscraper sheaves? Note \[ H^i(X; i_* i^* \mu_n) = \begin{cases} \bigoplus_{x\in Z} \mu_n(k) & i = 0 \\ 0 & i > 0. \end{cases} .\] One can see this by noting that the Leray spectral sequence gives $H^i(X; \RR^j i_* i^* \mu_n)\abuts H^{i+j}(Z; \mu_n)$, and $\RR^{j} i_* i^* \mu_n = 0$ for $j > 0$ since $i_*$ is exact as a pushforward along a finite map. Writing out the LES, one can compute - $H^0_c(U; \mu_n) \da H^0(X; j_! \mu_n) = 0$ - $H^0(X; \mu_n) = \mu_n(k)$ - $H^0(X; i_* i^* \mu_n) = \bigoplus _{x\in Z} \mu_n(k)$, and the map into this is the diagonal embedding - $H^1_c(U; \mu_n) = ?$ - $H^1_c(X; \mu_n) = \Pic(X)[n] \cong_{\zmod} C_n^{2g}$ - $H^1_c(X; i_* i^* \mu_n) = 0$ as a skyscraper sheaf - $H^2_c(U; \mu_n) = ?$ - $H^2_c(X; \mu_n) = C_n$ from last time - $H^2_c(X; i_* i^* \mu_n) = 0$, and the rest are zeros. Thus $H^2_c(U; \mu_n) = C_n$ and $H^1_c(U; \mu_n)$ is an extension of $\Pic(X)[n] \cong C_n^{2g}$ by \( \bigoplus _{x\in X} \mu_n(k) / \Delta \cong \mu_n \sumpower{m-1} \) where $m = \size Z$ is the number of deleted points, which is what one would expect for a punctured Riemann surface. ::: ## Proper base change and finiteness :::{.remark} So far, we only really know the cohomology of curves, but importantly we already have finiteness results: all of the computations above yielded finitely generated $C_n\dash$modules. Proper base change says that this will hold in greater generality, in the sense that the cohomology of proper varieties has good finiteness properties (including derived pushforwards). We'll get an étale analog of the coherence of proper pushforwards in coherent cohomology, which says that proper pushforwards of coherent sheaves through a proper morphisms are again coherent sheaves. ::: :::{.definition title="Constructible sheaves"} A sheaf $F\in \Sh(X_\et)$ is **constructible** iff a. For every closed embedding $i: Z\injects X$ there exists a nonempty open $U \subset Z$ such that $\ro{i^* F}{U}$ is a locally constant sheaf, i.e. there is a cover $V\to U$ such that $\ro{i^* F}{V}$ is the sheafification of a constant presheaf. b. The stalks $F_x$ are all finite. ::: :::{.example title="?"} For $j: U\injects X$ open, $j_!\mu_n$ is constructible. The second condition follows immediately since the stalks are either $\mu_n$ or zero. For the first condition, if $Z\injects X$ and $Z \intersect U$ is nonempty, use this intersection -- note that $\mu_n$ is not a constant sheaf since there is a Galois action, and thus it is not the sheafification of an abelian group. However, we can take the cover where we adjoin $n$th roots of unity to $Z \intersect U$ to get a constant sheaf. If $Z \intersect U$ is empty, one gets the zero sheaf, which is constant. ::: :::{.example title="?"} Any $F$ represented by a quasi-finite scheme is constructible. The sheaf $j_! \mu_n$ looks like the following: ![](figures/2023-02-10_17-06-21.png) Over $U$, one gets a cope of $\mu_n$. Note that if any open $V$ is not contained in $U$, any sections over $X\sm U$ have to land in $1$. If $V$ is connected, this forces the entire image to be contained in $1$, forcing it to be zero. What sheafification does for $j_!$ is that for any non-connected open, one evaluated the sheaf on each component separately. ::: :::{.theorem title="Characterization of constructible sheaves (hard!)"} Let $F: X_\Et\to X_\et$ be the map corresponding to the obvious inclusion of sites that regards an étale $X\dash$scheme as an $X\dash$scheme. Let $F\in \Sh(X_\Et)$, then 1. There is a natural map $f^* f_* F\iso F$ (the unit of the adjunction, using that $f^*$ is left adjoint) which is an isomorphism. 2. $f_* F$ is constructible. Then $F$ is represented by a quasifinite $X\dash$scheme. ::: :::{.remark} What this means: $f_* F$ is the restriction of functors, and $f^*(\wait)$ is pullback, so condition 1 says that $F$ is determined by its values on the small site. This says that if you have a constructible sheaf on the small site, it is representable, and moreover the corresponding pullback sheaf on the big site is also representable. This is stronger version of "any constructible sheaf is representable", and we write it this way because a quasifinite $X\dash$scheme may not in general be in the small site. So this is a formal way of capturing what it means to be representable on the small site, since it is not literally representable in the subcategory $X_{\et}$ since the representing object may be in $X_{\Et}$. For example, take any quasifinite $X\dash$scheme which is not étale and take its functor of points. For an example of a sheaf failing the first condition, consider the sheaf of relative Kähler differentials on $T$. This is quasicoherent, and for smooth $X\dash$schemes (where $X$ is also smooth), this is a vector bundle of constant rank $\dim T$. This can't be pulled back from $X$, since the dimension jumps. ::: :::{.theorem title="Proper base change"} Let $\pi:X\to S$ be a proper morphism and $F\in \Sh(X_\et, \zmod)$. Then $R^i \pi_*$ are constructible for $i \geq 0$ and the stalks can be computed as $(R_i \pi_* F)_{\bar x} \cong H(X_{\bar x}; \ro{F}{X_{\bar x}})$ for $X_{\bar x}$ is the fiber over $x$, and this holds for all geometric points $\bar x$. ::: :::{.corollary title="?"} Suppose $X$ is proper over $k = k^\sep$ and $F\in \Sh(X_\et)$ is constructible. Then 1. $H^i(X_\et; F)$ is finite. 2. Independent of base change along separably closed fields: if $k \injects L$ is an extension of separably closed fields, then there is an isomorphism \[ H^i(X_\et; F) \iso H^i(X_{L, \et}; \ro{F}{X_{L, \et}}) .\] ::: :::{.slogan} The stalks of the cohomology are the cohomologies of the fibers. This is very special to proper morphisms! ::: :::{.remark} As an example, taking a family of smooth proper curves over a base, the cohomology $\RR \pi_* F$ is not easy to compute on the base since it's the sheafification of presheaves of taking cohomology, and it's not clear how to work with this directly. However, the stalk of that sheaf is just the cohomology of a fiber, which is a single smooth proper curve. ::: :::{.proof title="?"} Part 1: a constructible sheaf on a point is just a group, and finiteness follows from constructibility. Part 2: regard this as a stalk $(\RR^i \pi_* F)_{\spec L\to \spec k}$, and the stalk of a constant sheaf is a constant group ::: :::{.example title="?"} Properness is necessary -- can you produce a scheme with a constructible sheaf whose cohomology is not finite? One needs a non-proper scheme, so take $\AA_1/\bar\FF_p$ and $F = \ul{C_p}$; then by the Artin-Schreier exact sequence, $H^i(X_\et; C_p)$ is infinite. ::: :::{.theorem title="Proper base change"} Consider the following pullback: \begin{tikzcd} {X'} && X \\ \\ T && S \arrow["f", from=3-1, to=3-3] \arrow["\pi", from=1-3, to=3-3] \arrow["{\pi'}"', from=1-1, to=3-1] \arrow["{f'}", from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYJyJdLFsyLDAsIlgiXSxbMiwyLCJTIl0sWzAsMiwiVCJdLFszLDIsImYiXSxbMSwyLCJcXHBpIl0sWzAsMywiXFxwaSciLDJdLFswLDEsImYnIl0sWzAsMiwiIiwwLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d) For any $F\in \Sh(X_\et; \zmod)$ there is a natural map \[ f^* (\RR^i \pi_* F) \to \RR^i\pi'_* (f')^* F .\] If $\pi$ is proper and $F$ is torsion, this is an isomorphism. ::: :::{.remark} Note that if $T$ is a point, this reduces to the previous corollary since $f^*$ is taking the stalk. ::: :::{.proof title="?"} Construct this map using adjointness and check on stalks. This works for constructible sheaves, and one needs to use dévissage to make this work for torsion sheaves as well. ::: :::{.remark} In real life, most torsion sheaves one runs into are constructible! :::