stacks/stacks-project

Kunneth for etale cohomology

This is obviously insane, but somehow I started writing it this way and
I couldn't get myself to stop. If you have a better way of writing the
proofs in this section, by all means send me suggestions.
 (akin to smooth base change). Finally we use this to get a more general K\"unneth formula. \begin{remark} \label{remark-define-kunneth-map} Consider a cartesian diagram in the category of schemes: $$\xymatrix{ X \times_S Y \ar[d]_p \ar[r]_q \ar[rd]_c & Y \ar[d]^g \\ X \ar[r]^f & S }$$ Let $\Lambda$ be a ring and let $E \in D(X_\etale, \Lambda)$ and $K \in D(Y_\etale, \Lambda)$. Then there is a canonical map $$Rf_*E \otimes_\Lambda^\mathbf{L} Rg_*K \longrightarrow Rc_*(p^{-1}E \otimes_\Lambda^\mathbf{L} q^{-1}K)$$ For example we can define this using the canonical maps $Rf_*E \to Rc_*p^{-1}E$ and $Rg_*K \to Rc_*q^{-1}K$ and the relative cup product defined in Cohomology on Sites, Remark \ref{sites-cohomology-remark-cup-product}. Or you can use the adjoint to the map $$c^{-1}(Rf_*E \otimes_\Lambda^\mathbf{L} Rg_*K) = p^{-1}f^{-1}Rf_*E \otimes_\Lambda^\mathbf{L} q^{-1} g^{-1}Rg_*K \to p^{-1}E \otimes_\Lambda^\mathbf{L} q^{-1}K$$ which uses the adjunction maps $f^{-1}Rf_*E \to E$ and $g^{-1}Rg_*K \to K$. \end{remark} \begin{lemma} \label{lemma-kunneth-one-proper} Let $k$ be a separably closed field. Let $X$ be a proper scheme over $k$. $K$-subalgebras $B_i$. Let $J$ be the set of pairs $(i, g)$ where $i \in I$ and $g \in B_i$ nonzero with ordering $(i', g') \geq (i, g)$ if and only if $i' \geq i$ and $g$ maps to an invertible element of $B_{i'}$. $g$ maps to an invertible element of $(B_{i'})_{g'}$. Then $L = \colim_{(i, g) \in J} (B_i)_g$. For $j = (i, g) \in J$ set $S_j = \Spec(B_i)$ and $U_j = \Spec((B_i)_g)$. proved in Lemma \ref{lemma-kunneth-localize-on-X}. \end{proof} \begin{lemma} \label{lemma-punctual-base-change-upgrade} Let $K$ be a field. Let $X$ be a scheme over $K$. For any commutative diagram $$\xymatrix{ X \ar[d] & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ \Spec(K) & S' \ar[l] & T \ar[l]_g }$$ $X' = X \times_{\Spec(K)} S'$ and $Y = X' \times_{S'} T$ and $g$ quasi-compact and quasi-separated, and every abelian sheaves $\mathcal{G}$ on $X_\etale$ and $\mathcal{F}$ on $T_\etale$ whose stalks are torsion of orders invertible in $K$ the base change map $$\mathcal{G} \otimes (f')^{-1}Rg_*\mathcal{F} \longrightarrow Rh_*(\mathcal{G} \otimes e^{-1}\mathcal{F})$$ is an isomorphism. \end{lemma}