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Technical lemmas on base change

Examples of diagrams of topoi for which the base change map is
automatically an isomorphism (trivial base change).

Thanks to David Hansen for several discussions.
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aisejohan committed Oct 13, 2019
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  1. +4 −14 etale-cohomology.tex
  2. +179 −0 sites-cohomology.tex
  3. +91 −0 sites-modules.tex
  4. +104 −0 sites.tex
  5. +4 −14 spaces-more-cohomology.tex
  6. +1 −2 spaces-simplicial.tex
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (5) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.
Part (5) is Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-localize-cartesian-square}.

\medskip\noindent
Part (6): Observe that $g_{big}$ and $g'_{big}$ are localizations
and hence $g_{big}^{-1} = g_{big}^*$ and $(g'_{big})^{-1} = (g'_{big})^*$
are the restriction functors. Hence (6) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of modules representing $K$.
Part (6) is Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-localize-cartesian-square-modules}.

\medskip\noindent
Part (2) can be proved as follows. Above we have seen
@@ -4102,6 +4102,43 @@ \section{Localization and cohomology}
Sites, Lemma \ref{sites-lemma-localize-cartesian-square}.
\end{proof}

\noindent
If we have a ringed site $(\mathcal{C}, \mathcal{O})$
and a morphism $f : X \to Y$ of $\mathcal{C}$, then $j_{X/Y}$
becomes a morphism of ringed topoi
$$
j_{X/Y} :
(\Sh(\mathcal{C}/X), \mathcal{O}_X)
\longrightarrow
(\Sh(\mathcal{C}/Y), \mathcal{O}_Y)
$$
See Modules on Sites, Lemma \ref{sites-modules-lemma-relocalize}.

\begin{lemma}
\label{lemma-localize-cartesian-square-modules}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let
$$
\xymatrix{
X' \ar[d] \ar[r] & X \ar[d] \\
Y' \ar[r] & Y
}
$$
be a cartesian diagram of $\mathcal{C}$. Then we have
$j_{Y'/Y}^* \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^*$
as functors
$D(\mathcal{O}_X) \to D(\mathcal{O}_{Y'})$.
\end{lemma}

\begin{proof}
Since $j_{Y'/Y}^{-1}\mathcal{O}_Y = \mathcal{O}_{Y'}$ we have
$j_{Y'/Y}^* = Lj_{Y'/Y}^* = j_{Y'/Y}^{-1}$. Similarly we have
$j_{X'/X}^* = Lj_{X'/X}^* = j_{X'/X}^{-1}$. Thus by
Lemma \ref{lemma-modules-abelian-unbounded} it suffices
to prove the result on derived categories of abelian sheaves
which we did in
Lemma \ref{lemma-localize-cartesian-square}.
\end{proof}




@@ -8286,6 +8323,148 @@ \section{Derived lower shriek}
Lemma \ref{lemma-existence-derived-lower-shriek}.
\end{proof}

\begin{lemma}
\label{lemma-special-square-cocontinuous}
Assume given a commutative diagram
$$
\xymatrix{
(\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'})
\ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} &
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\
(\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} &
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})
}
$$
of ringed topoi. Assume
\begin{enumerate}
\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors
$u$, $u'$, $v$, and $v'$ as in
Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi},
\item $v \circ u' = u \circ v'$,
\item $v$ and $v'$ are continuous as well as cocontinuous,
\item for any object $V'$ of $\mathcal{D}'$ the functor
${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$
given by $v$ is cofinal,
\item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$
and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$, and
\item $g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$
is exact\footnote{Holds if fibre products and equalizers exist in
$\mathcal{C}'$ and $v'$ commutes with them, see
Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek}.}.
\end{enumerate}
Then we have $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors
$D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_{\mathcal{D}'})$.
\end{lemma}

\begin{proof}
We have $g^* = Lg^* = g^{-1}$ and $(g')^* = L(g')^* = (g')^{-1}$
by condition (5).
By Lemma \ref{lemma-modules-abelian-unbounded} it suffices
to prove the result on the derived category $D(\mathcal{C})$
of abelian sheaves. Choose an object $K \in D(\mathcal{C})$.
Let $\mathcal{I}^\bullet$ be a K-injective complex of abelian
sheaves on $\mathcal{C}$ representing $K$. By
Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}
and assumption (6) we find that $(g')^{-1}\mathcal{I}^\bullet$
is a K-injective complex of abelian sheaves on $\mathcal{C}'$.
By Modules on Sites, Lemma
\ref{sites-modules-lemma-special-square-cocontinuous}
we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$.
Since $f_*\mathcal{I}^\bullet$ represents $Rf_*K$ and since
$f'_*(g')^{-1}\mathcal{I}^\bullet$ represents $Rf'_*(g')^{-1}K$
we conclude.
\end{proof}

\begin{lemma}
\label{lemma-special-square-continuous}
Consider a commutative diagram
$$
\xymatrix{
(\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}
\ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} &
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\
(\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} &
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})
}
$$
of ringed topoi and suppose we have functors
$$
\xymatrix{
\mathcal{C}' \ar[r]_{v'} &
\mathcal{C} \\
\mathcal{D}' \ar[r]^v \ar[u]^{u'} &
\mathcal{D} \ar[u]_u
}
$$
such that (with notation as in
Sites, Sections \ref{sites-section-morphism-sites} and
\ref{sites-section-cocontinuous-morphism-topoi}) we have
\begin{enumerate}
\item $u$ and $u'$ are continuous and give rise to the morphisms
$f$ and $f'$,
\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$,
\item $u \circ v = v' \circ u'$,
\item $v$ and $v'$ are continuous as well as cocontinuous, and
\item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$
and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$.
\end{enumerate}
Then $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors
$D^+(\mathcal{O}_\mathcal{C}) \to D^+(\mathcal{O}_{\mathcal{D}'})$.
If in addition
\begin{enumerate}
\item[(6)] $g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$
is exact\footnote{Holds if fibre products and equalizers exist in
$\mathcal{C}'$ and $v'$ commutes with them, see
Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek}.},
\end{enumerate}
then $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors
$D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_{\mathcal{D}'})$.
\end{lemma}

\begin{proof}
We have $g^* = Lg^* = g^{-1}$ and $(g')^* = L(g')^* = (g')^{-1}$
by condition (5).
By Lemma \ref{lemma-modules-abelian-unbounded} it suffices
to prove the result on the derived category $D^+(\mathcal{C})$ or
$D(\mathcal{C})$ of abelian sheaves.

\medskip\noindent
Choose an object $K \in D^+(\mathcal{C})$.
Let $\mathcal{I}^\bullet$ be a bounded below complex of injective abelian
sheaves on $\mathcal{C}$ representing $K$. By
Lemma \ref{lemma-pullback-injective-pre-limp}
we see that $H^p(U', (g')^{-1}\mathcal{I}^q) = 0$ for
all $p > 0$ and any $q$ and any $U' \in \Ob(\mathcal{C}')$.
Recall that $R^pf'_*(g')^{-1}\mathcal{I}^q$ is the sheaf
associated to the presheaf $V' \mapsto H^p(u'(V'), (g')^{-1}\mathcal{I}^q)$,
see Lemma \ref{lemma-higher-direct-images}.
Thus we see that $(g')^{-1}\mathcal{I}^q$ is right acyclic
for the functor $f'_*$. By Leray's acyclicity lemma
(Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity})
we find that $f'_*(g')^*\mathcal{I}^\bullet$
represents $Rf'_*(g')^{-1}K$.
By Modules on Sites, Lemma
\ref{sites-modules-lemma-special-square-continuous}
we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$.
Since $g^{-1}f_*\mathcal{I}^\bullet$ represents $g^{-1}Rf_*K$
we conclude.

\medskip\noindent
Choose an object $K \in D(\mathcal{C})$.
Let $\mathcal{I}^\bullet$ be a K-injective complex of abelian
sheaves on $\mathcal{C}$ representing $K$. By
Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}
and assumption (6) we find that $(g')^{-1}\mathcal{I}^\bullet$
is a K-injective complex of abelian sheaves on $\mathcal{C}'$.
By Modules on Sites, Lemma
\ref{sites-modules-lemma-special-square-continuous}
we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$.
Since $f_*\mathcal{I}^\bullet$ represents $Rf_*K$ and since
$f'_*(g')^{-1}\mathcal{I}^\bullet$ represents $Rf'_*(g')^{-1}K$
we conclude.
\end{proof}





@@ -7372,6 +7372,97 @@ \section{Lower shriek for modules}
is an isomorphism for all objects $U$ of $\mathcal{C}$.
\end{remark}

\noindent
The following two results are of a slightly different nature.

\begin{lemma}
\label{lemma-special-square-cocontinuous}
Assume given a commutative diagram
$$
\xymatrix{
(\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'})
\ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} &
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\
(\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} &
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})
}
$$
of ringed topoi. Assume
\begin{enumerate}
\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors
$u$, $u'$, $v$, and $v'$ as in
Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi},
\item $v \circ u' = u \circ v'$,
\item $v$ and $v'$ are continuous as well as cocontinuous,
\item for any object $V'$ of $\mathcal{D}'$ the functor
${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$
given by $v$ is cofinal, and
\item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$
and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$.
\end{enumerate}
Then we have $f'_* \circ (g')^* = g^* \circ f_*$ and
$g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ on modules.
\end{lemma}

\begin{proof}
We have $(g')^*\mathcal{F} = (g')^{-1}\mathcal{F}$ and
$g^*\mathcal{G} = g^{-1}\mathcal{G}$ because of condition (5).
Thus the first equality follows immediately from the corresponding
equality in Sites, Lemma \ref{sites-lemma-special-square-cocontinuous}.
Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$
exist by Lemma \ref{lemma-lower-shriek-modules}
we see that the second equality follows by
uniqueness of adjoint functors.
\end{proof}

\begin{lemma}
\label{lemma-special-square-continuous}
Consider a commutative diagram
$$
\xymatrix{
(\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'})
\ar[r]_{(g', (g')^\sharp)} \ar[d]_{(f', (f')^\sharp)} &
(\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^{(f, f^\sharp)} \\
(\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp)} &
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})
}
$$
of ringed topoi and suppose we have functors
$$
\xymatrix{
\mathcal{C}' \ar[r]_{v'} &
\mathcal{C} \\
\mathcal{D}' \ar[r]^v \ar[u]^{u'} &
\mathcal{D} \ar[u]_u
}
$$
such that (with notation as in
Sites, Sections \ref{sites-section-morphism-sites} and
\ref{sites-section-cocontinuous-morphism-topoi}) we have
\begin{enumerate}
\item $u$ and $u'$ are continuous and give rise to the morphisms
$f$ and $f'$,
\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$,
\item $u \circ v = v' \circ u'$,
\item $v$ and $v'$ are continuous as well as cocontinuous, and
\item $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$
and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$.
\end{enumerate}
Then $f'_* \circ (g')^* = g^* \circ f_*$ and
$g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ on modules.
\end{lemma}

\begin{proof}
We have $(g')^*\mathcal{F} = (g')^{-1}\mathcal{F}$ and
$g^*\mathcal{G} = g^{-1}\mathcal{G}$ because of condition (5).
Thus the first equality follows immediately from the corresponding
equality in Sites, Lemma \ref{sites-lemma-special-square-continuous}.
Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$
exist by Lemma \ref{lemma-lower-shriek-modules}
we see that the second equality follows by
uniqueness of adjoint functors.
\end{proof}




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