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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.
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aisejohan committed Nov 6, 2018
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  1. +56 −1 etale-cohomology.tex
(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}




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