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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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Showing with 330 additions and 10 deletions.
  1. +256 −10 etale-cohomology.tex
  2. +74 −0 sites-cohomology.tex
@@ -17716,13 +17716,14 @@ \section{K\"unneth in \'etale cohomology}
\begin{lemma}
\label{lemma-punctual-base-change}
Let $K$ be a field. Let $X$ be a scheme over $K$. For any commutative diagram
Let $K$ be a field. For any commutative diagram
$$
\xymatrix{
X \ar[d] & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\
\Spec(K) & S' \ar[l] & T \ar[l]_g
}
$$
of schemes over $K$ with
$X' = X \times_{\Spec(K)} S'$ and $Y = X' \times_{S'} T$ and
$g$ quasi-compact and quasi-separated, and every abelian sheaf
$\mathcal{F}$ on $T_\etale$ whose stalks are torsion of orders
@@ -17824,27 +17825,272 @@ \section{K\"unneth in \'etale cohomology}
\begin{lemma}
\label{lemma-punctual-base-change-upgrade}
Let $K$ be a field. Let $X$ be a scheme over $K$.
For any commutative diagram
Let $K$ be a field. Let $n \geq 1$ be invertible in $K$.
Consider a commutative diagram
$$
\xymatrix{
X \ar[d] & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\
X \ar[d] & X' \ar[l]^p \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\
\Spec(K) & S' \ar[l] & T \ar[l]_g
}
$$
of schemes with
$X' = X \times_{\Spec(K)} S'$ 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
$g$ quasi-compact and quasi-separated. The canonical map
$$
\mathcal{G} \otimes (f')^{-1}Rg_*\mathcal{F}
p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} (f')^{-1}Rg_*F
\longrightarrow
Rh_*(\mathcal{G} \otimes e^{-1}\mathcal{F})
Rh_*(h^{-1}p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} e^{-1}F)
$$
is an isomorphism.
is an isomorphism if $E$ in $D^+(X_\etale, \mathbf{Z}/n\mathbf{Z})$
has tor amplitude in $[a, \infty]$ for some $a \in \mathbf{Z}$ and
$F$ in $D^+(T_\etale, \mathbf{Z}/n\mathbf{Z})$.
\end{lemma}
\begin{proof}
This lemma is a generalization of Lemma \ref{lemma-punctual-base-change}
to objects of the derived category; the assertion of our lemma is true because
in Lemma \ref{lemma-punctual-base-change} the scheme $X$ over $K$
is arbitrary. We strongly urge the reader to skip the laborious proof
(alternative: read only the last paragraph).
\medskip\noindent
We may represent $E$ by a bounded below K-flat complex
$\mathcal{E}^\bullet$ consisting of flat $\mathbf{Z}/n\mathbf{Z}$-modules.
See Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-bounded-below-tor-amplitude}.
Choose an integer $b$ such that $H^i(F) = 0$ for $i < b$.
Choose a large integer $N$ and consider the short exact sequence
$$
0 \to \sigma_{\geq N + 1}\mathcal{E}^\bullet \to
\mathcal{E}^\bullet \to
\sigma_{\leq N}\mathcal{E}^\bullet \to 0
$$
of stupid truncations. This produces a distinguished triangle
$E'' \to E \to E' \to E''[1]$ in $D(X_\etale, \mathbf{Z}/n\mathbf{Z})$.
For fixed $F$ both sides of the arrow
in the statement of the lemma are exact functors in $E$. Observe that
$$
p^{-1}E'' \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} (f')^{-1}Rg_*F
\quad\text{and}\quad
Rh_*(h^{-1}p^{-1}E'' \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} e^{-1}F)
$$
are sitting in degrees $\geq N + b$. Hence, if we can prove the lemma
for the object $E'$, then we see that the lemma holds in degrees
$\leq N + b$ and we will conclude. Some details omitted.
Thus we may assume $E$ is represented
by a bounded complex of flat $\mathbf{Z}/n\mathbf{Z}$-modules.
Doing another argument of the same nature, we may assume
$E$ is given by a single flat $\mathbf{Z}/n\mathbf{Z}$-module
$\mathcal{E}$.
\medskip\noindent
Next, we use the same arguments for the variable $F$
to reduce to the case where $F$ is given by a single
sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules $\mathcal{F}$.
Say $\mathcal{F}$ is annihilated by an integer $m | n$.
If $\ell$ is a prime number dividing $m$ and $m > \ell$,
then we can look at the short exact sequence
$0 \to \mathcal{F}[\ell] \to \mathcal{F} \to
\mathcal{F}/\mathcal{F}[\ell] \to 0$
and reduce to smaller $m$. This finally reduces us to
the case where $\mathcal{F}$ is annihilated by a prime
number $\ell$ dividing $n$.
In this case observe that
$$
p^{-1}\mathcal{E}
\otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L}
(f')^{-1}Rg_*\mathcal{F}
=
p^{-1}(\mathcal{E}/\ell \mathcal{E})
\otimes_{\mathbf{F}_\ell}^\mathbf{L}
(f')^{-1}Rg_*\mathcal{F}
$$
by the flatness of $\mathcal{E}$. Similarly for the other term.
This reduces us to the case where we are working with sheaves
of $\mathbf{F}_\ell$-vector spaces which is discussed
\medskip\noindent
Assume $\ell$ is a prime number invertible in $K$.
Assume $\mathcal{E}$, $\mathcal{F}$ are sheaves of
$\mathbf{F}_\ell$-vector spaces on $X_\etale$ and $T_\etale$.
We want to show that
$$
p^{-1}\mathcal{E} \otimes_{\mathbf{F}_\ell} (f')^{-1}R^qg_*\mathcal{F}
\longrightarrow
R^qh_*(h^{-1}p^{-1}\mathcal{E} \otimes_{\mathbf{F}_\ell} e^{-1}\mathcal{F})
$$
is an isomorphism for every $q \geq 0$. This question is local on $X$
hence we may assume $X$ is affine. We can write $\mathcal{E}$
as a filtered colimit of constructible sheaves of
$\mathbf{F}_\ell$-vector spaces on $X_\etale$, see
Lemma \ref{lemma-torsion-colimit-constructible}.
Since tensor products commute
with filtered colimits and since
higher direct images do too (Lemma \ref{lemma-relative-colimit})
we may assume $\mathcal{E}$ is a constructible sheaf of
$\mathbf{F}_\ell$-vector spaces on $X_\etale$.
Then we can choose an integer $m$ and finite and finitely presented morphisms
$\pi_i : X_i \to X$, $i = 1, \ldots, m$
such that there is an injective map
$$
\mathcal{E} \to
\bigoplus\nolimits_{i = 1, \ldots, m}
\pi_{i, *}\mathbf{F}_\ell
$$
See Lemma \ref{lemma-constructible-maps-into-constant-general}.
Observe that the direct sum is a constructible sheaf as well
(Lemma \ref{lemma-finite-pushforward-constructible}).
Thus the cokernel is constructible too
(Lemma \ref{lemma-constructible-abelian}).
By dimension shifting, i.e., induction on $q$,
on the category of constructible sheaves of
$\mathbf{F}_\ell$-vector spaces on $X_\etale$, it suffices to prove the
result for the sheaves $\pi_{i, *}\mathbf{F}_\ell$
(details omitted; hint: start with proving injectivity for $q = 0$
for all constructible $\mathcal{E}$).
To prove this case we extend the diagram of the lemma to
$$
\xymatrix{
X_i \ar[d]^{\pi_i} &
X'_i \ar[l]^{p_i} \ar[d]^{\pi'_i} &
Y_i \ar[l]^{h_i} \ar[d]^{\rho_i} \\
X \ar[d] & X' \ar[l]^p \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\
\Spec(K) & S' \ar[l] & T \ar[l]_g
}
$$
with all squares cartesian. In the equations below we are
going to use that $R\pi_{i, *} = \pi_{i, *}$ and similarly
for $\pi'_i$, $\rho_i$, we are going to use the Leray spectral sequence,
we are going to use
Lemma \ref{lemma-finite-pushforward-commutes-with-base-change}, and
we are going to use
Lemma \ref{lemma-projection-formula-proper-mod-n}
(although this lemma is almost trivial for finite morphisms) for
$\pi_i$, $\pi'_i$, $\rho_i$.
Doing so we see that
\begin{align*}
p^{-1}\pi_{i, *}\mathbf{F}_\ell
\otimes_{\mathbf{F}_\ell} (f')^{-1}R^qg_*\mathcal{F}
& =
\pi'_{i, *}\mathbf{F}_\ell
\otimes_{\mathbf{F}_\ell} (f')^{-1}R^qg_*\mathcal{F} \\
& =
\pi'_{i, *}((\pi'_i)^{-1} (f')^{-1}R^qg_*\mathcal{F})
\end{align*}
Similarly, we have
\begin{align*}
R^qh_*(h^{-1}p^{-1} \pi_{i, *}\mathbf{F}_\ell
\otimes_{\mathbf{F}_\ell} e^{-1}\mathcal{F})
& =
R^qh_*(\rho_{i, *}\mathbf{F}_\ell
\otimes_{\mathbf{F}_\ell} e^{-1}\mathcal{F}) \\
& =
R^qh_*(\rho_i^{-1}e^{-1}\mathcal{F}) \\
& =
\pi'_{i, *}R^qh_{i, *} \rho_i^{-1}e^{-1}\mathcal{F})
\end{align*}
Simce
$R^qh_{i, *} \rho_i^{-1}e^{-1}\mathcal{F} =
(\pi'_i)^{-1} (f')^{-1}R^qg_*\mathcal{F}$ by
Lemma \ref{lemma-punctual-base-change}
we conclude.
\end{proof}
\begin{lemma}
\label{lemma-punctual-base-change-upgrade-unbounded}
Let $K$ be a field. Let $n \geq 1$ be invertible in $K$.
Consider a commutative diagram
$$
\xymatrix{
X \ar[d] & X' \ar[l]^p \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\
\Spec(K) & S' \ar[l] & T \ar[l]_g
}
$$
of schemes of finite type over $K$ with
$X' = X \times_{\Spec(K)} S'$ and $Y = X' \times_{S'} T$.
The canonical map
$$
p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} (f')^{-1}Rg_*F
\longrightarrow
Rh_*(h^{-1}p^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} e^{-1}F)
$$
is an isomorphism for $E$ in $D(X_\etale, \mathbf{Z}/n\mathbf{Z})$
and $F$ in $D(T_\etale, \mathbf{Z}/n\mathbf{Z})$.
\end{lemma}
\begin{proof}
We will reduce this to
Lemma \ref{lemma-punctual-base-change-upgrade}
using that our functors commute with direct sums.
We suggest the reader skip the proof.
Recall that derived tensor product commutes with
direct sums. Recall that (derived) pullback commutes with direct sums.
Recall that $Rh_*$ and $Rg_*$ commute with direct sums, see
Lemmas \ref{lemma-finite-cd} and
\ref{lemma-finite-cd-mod-n-direct-sums}
(this is where we use our schemes are of finite type
over $K$).
\medskip\noindent
To finish the proof we can argue as follows.
First we write $E = \text{hocolim} \tau_{\leq N} E$.
Since our functors commute with direct sums, they commute
with homotopy colimits. Hence it suffices to prove
the lemma for $E$ bounded above. Similarly for $F$ we
may assume $F$ is bounded above.
Then we can represent $E$ by a bounded above complex
$\mathcal{E}^\bullet$ of sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules.
Then
$$
\mathcal{E}^\bullet = \colim \sigma_{\geq -N}\mathcal{E}^\bullet
$$
(stupid truncations).
Thus we may assume $\mathcal{E}^\bullet$ is a bounded
complex of sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules.
For $F$ we choose a bounded above complex of
flat(!) sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules.
Then we reduce to the case where $F$ is represented
by a bounded complex of flat sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules.
At this point Lemma \ref{lemma-punctual-base-change-upgrade}
kicks in and we conclude.
\end{proof}
\begin{lemma}
\label{lemma-kunneth}
Let $k$ be a separably closed field. Let $X$ and $Y$ be
finite type schemes over $k$. Let $n \geq 1$ be an integer
invertible in $k$. Then for
$E \in D(X_\etale, \mathbf{Z}/n\mathbf{Z})$ and
$K \in D(Y_\etale, \mathbf{Z}/n\mathbf{Z})$
we have
$$
R\Gamma(X \times_{\Spec(k)} Y,
\text{pr}_1^{-1}E
\otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L}
\text{pr}_2^{-1}K
)
=
R\Gamma(X, E)
\otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L}
R\Gamma(Y, K)
$$
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-punctual-base-change-upgrade-unbounded} we have
$$
R\text{pr}_{1, *}(
\text{pr}_1^{-1}E
\otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L}
\text{pr}_2^{-1}K) =
E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L}
\underline{R\Gamma(Y, K)}
$$
We conclude by
Lemma \ref{lemma-pull-out-constant-mod-n}
which we may use because
$\text{cd}(X) < \infty$ by Lemma \ref{lemma-finite-cd}.
\end{proof}
@@ -2971,6 +2971,7 @@ \section{Flat resolutions}
there exists a $K$-flat complex
$\mathcal{K}^\bullet$ and a quasi-isomorphism
$\mathcal{K}^\bullet \to \mathcal{G}^\bullet$.
Moreover, each $\mathcal{K}^n$ is a flat $\mathcal{O}$-module.
\end{lemma}
\begin{proof}
@@ -10468,6 +10469,79 @@ \section{Tor dimension}
and the definitions.
\end{proof}
\begin{lemma}
\label{lemma-bounded-below-tor-amplitude}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of
$D(\mathcal{O})$. Let $a \in \mathbf{Z}$. The following
are equivalent
\begin{enumerate}
\item $E$ has tor-amplitude in $[a, \infty]$.
\item $E$ can be represented by a K-flat complex $\mathcal{E}^\bullet$
of flat $\mathcal{O}$-modules with $\mathcal{E}^i = 0$ for
$i \not \in [a, \infty]$.
\end{enumerate}
Moreover, we can choose $\mathcal{E}^\bullet$ such that any pullback
by a morphism of ringed sites is a K-flat complex with flat terms.
\end{lemma}
\begin{proof}
The implication (2) $\Rightarrow$ (1) is immediate. Assume (1) holds.
First we choose a K-flat complex $\mathcal{K}^\bullet$
with flat terms representing $E$, see Lemma \ref{lemma-K-flat-resolution}.
For any $\mathcal{O}$-module $\mathcal{M}$ the cohomology of
$$
\mathcal{K}^{n - 1} \otimes_\mathcal{O} \mathcal{M} \to
\mathcal{K}^n \otimes_\mathcal{O} \mathcal{M} \to
\mathcal{K}^{n + 1} \otimes_\mathcal{O} \mathcal{M}
$$
computes $H^n(E \otimes_\mathcal{O}^\mathbf{L} \mathcal{M})$.
This is always zero for $n < a$. Hence if we apply
Lemma \ref{lemma-last-one-flat} to the complex
$\ldots \to \mathcal{K}^{a - 1} \to \mathcal{K}^a \to \mathcal{K}^{a + 1}$
we conclude that $\mathcal{N} = \Coker(\mathcal{K}^{a - 1} \to \mathcal{K}^a)$
is a flat $\mathcal{O}$-module. We set
$$
\mathcal{E}^\bullet = \tau_{\geq a}\mathcal{K}^\bullet =
(\ldots \to 0 \to \mathcal{N} \to \mathcal{K}^{a + 1} \to \ldots )
$$
The kernel $\mathcal{L}^\bullet$ of
$\mathcal{K}^\bullet \to \mathcal{E}^\bullet$ is the complex
$$
\mathcal{L}^\bullet = (\ldots \to \mathcal{K}^{a - 1} \to
\mathcal{I} \to 0 \to \ldots)
$$
where $\mathcal{I} \subset \mathcal{K}^a$ is the image of
$\mathcal{K}^{a - 1} \to \mathcal{K}^a$.
Since we have the short exact sequence
$0 \to \mathcal{I} \to \mathcal{K}^a \to \mathcal{N} \to 0$
we see that $\mathcal{I}$ is a flat $\mathcal{O}$-module.
Thus $\mathcal{L}^\bullet$ is a bounded
above complex of flat modules, hence K-flat by
Lemma \ref{lemma-bounded-flat-K-flat}.
It follows that $\mathcal{E}^\bullet$ is K-flat by
Lemma \ref{lemma-K-flat-two-out-of-three}.
\medskip\noindent
Proof of the final assertion. Let
$f : (\mathcal{C}', \mathcal{O}') \to (\mathcal{C}, \mathcal{O})$
be a morphism of ringed sites. The proof of
Lemma \ref{lemma-pullback-K-flat} shows that the complex
$\mathcal{K}^\bullet$ (as constructed in Lemma \ref{lemma-K-flat-resolution})
has the property that $f^*\mathcal{K}^\bullet$ is K-flat.
The complex $f^*\mathcal{L}^\bullet$ is K-flat as it is a bounded
above complex of flat $\mathcal{O}'$-modules. We have a short exact
sequence of complexes of $\mathcal{O}'$-modules
$$
0 \to f^*\mathcal{L}^\bullet \to f^*\mathcal{K}^\bullet \to
f^*\mathcal{E}^\bullet \to 0
$$
because the short exact sequence
$0 \to \mathcal{I} \to \mathcal{K}^a \to \mathcal{N} \to 0$
of flat modules pulls back to a short exact sequence.
Then we can use the 2-out-of-3 property for K-flat
complexes to conclude that $f^*\mathcal{E}^\bullet$ is K-flat.
\end{proof}
\begin{lemma}
\label{lemma-tor-amplitude-pullback}
Let $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to

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