# The Stacks Project

## Tag: 04CB

This tag has label etale-cohomology-section-exactness-lower-shriek, it is called Exactness of big lower shriek in the Stacks project and it points to

The corresponding content:

### 41.49. Exactness of big lower shriek

This is just the following technical result. Note that the functor $f_{big!}$ has nothing whatsoever to do with cohomology with compact support in general.

Lemma 41.49.1. Let $\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$. Let $f : X \to Y$ be a morphism of schemes. Let $$f_{big} : \mathop{\textit{Sh}}\nolimits((\textit{Sch}/X)_\tau) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}/Y)_\tau)$$ be the corresponding morphism of topoi as in Topologies, Lemma 30.3.15, 30.4.15, 30.5.10, 30.6.10, or 30.7.12.

1. The functor $f_{big}^{-1} : \textit{Ab}((\textit{Sch}/Y)_\tau) \to \textit{Ab}((\textit{Sch}/X)_\tau)$ has a left adjoint $$f_{big!} : \textit{Ab}((\textit{Sch}/X)_\tau) \to \textit{Ab}((\textit{Sch}/Y)_\tau)$$ which is exact.
2. The functor $f_{big}^* : \textit{Mod}((\textit{Sch}/Y)_\tau, \mathcal{O}) \to \textit{Mod}((\textit{Sch}/X)_\tau, \mathcal{O})$ has a left adjoint $$f_{big!} : \textit{Mod}((\textit{Sch}/X)_\tau, \mathcal{O}) \to \textit{Mod}((\textit{Sch}/Y)_\tau, \mathcal{O})$$ which is exact.
Moreover, the two functors $f_{big!}$ agree on underlying sheaves of abelian groups.

Proof. Recall that $f_{big}$ is the morphism of topoi associated to the continuous and cocontinuous functor $u : (\textit{Sch}/X)_\tau \to (\textit{Sch}/Y)_\tau$, $U/X \mapsto U/Y$. Moreover, we have $f_{big}^{-1}\mathcal{O} = \mathcal{O}$. Hence the existence of $f_{big!}$ follows from Modules on Sites, Lemma 17.16.2, respectively Modules on Sites, Lemma 17.38.1. Note that if $U$ is an object of $(\textit{Sch}/X)_\tau$ then the functor $u$ induces an equivalence of categories $$u' : (\textit{Sch}/X)_\tau/U \longrightarrow (\textit{Sch}/Y)_\tau/U$$ because both sides of the arrow are equal to $(\textit{Sch}/U)_\tau$. Hence the agreement of $f_{big!}$ on underlying abelian sheaves follows from the discussion in Modules on Sites, Remark 17.38.2. The exactness of $f_{big!}$ follows from Modules on Sites, Lemma 17.16.3 as the functor $u$ above which commutes with fibre products and equalizers. $\square$

Next, we prove a technical lemma that will be useful later when comparing sheaves of modules on different sites associated to algebraic stacks.

Lemma 41.49.2. Let $X$ be a scheme. Let $\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$. Let $\mathcal{C}_1 \subset \mathcal{C}_2 \subset (\textit{Sch}/X)_\tau$ be full subcategories with the following properties:

1. For an object $U/X$ of $\mathcal{C}_t$,
1. if $\{U_i \to U\}$ is a covering of $(\textit{Sch}/X)_\tau$, then $U_i/X$ is an object of $\mathcal{C}_t$,
2. $U \times \mathbf{A}^1/X$ is an object of $\mathcal{C}_t$.
2. $X/X$ is an object of $\mathcal{C}_t$.
We endow $\mathcal{C}_t$ with the structure of a site whose coverings are exactly those coverings $\{U_i \to U\}$ of $(\textit{Sch}/X)_\tau$ with $U \in \mathop{\rm Ob}\nolimits(\mathcal{C}_t)$. Then
1. The functor $\mathcal{C}_1 \to \mathcal{C}_2$ is fully faithful, continuous, and cocontinuous.
Denote $g : \mathop{\textit{Sh}}\nolimits(\mathcal{C}_1) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C}_2)$ the corresponding morphism of topoi. Denote $\mathcal{O}_t$ the restriction of $\mathcal{O}$ to $\mathcal{C}_t$. Denote $g_!$ the functor of Modules on Sites, Definition 17.16.1.
1. The canonical map $g_!\mathcal{O}_1 \to \mathcal{O}_2$ is an isomorphism.

Proof. Assertion (\romannumeral1) is immediate from the definitions. In this proof all schemes are schemes over $X$ and all morphisms of schemes are morphisms of schemes over $X$. Note that $g^{-1}$ is given by restriction, so that for an object $U$ of $\mathcal{C}_1$ we have $\mathcal{O}_1(U) = \mathcal{O}_2(U) = \mathcal{O}(U)$. Recall that $g_!\mathcal{O}_1$ is the sheaf associated to the presheaf $g_{p!}\mathcal{O}_1$ which associates to $V$ in $\mathcal{C}_2$ the group $$\mathop{\rm colim}\nolimits_{V \to U} \mathcal{O}(U)$$ where $U$ runs over the objects of $\mathcal{C}_1$ and the colimit is taken in the category of abelian groups. Below we will use frequently that if $$V \to U \to U'$$ are morphisms with $U, U' \in \mathop{\rm Ob}\nolimits(\mathcal{C}_1)$ and if $f' \in \mathcal{O}(U')$ restricts to $f \in \mathcal{O}(U)$, then $(V \to U, f)$ and $(V \to U', f')$ define the same element of the colimit. Also, $g_!\mathcal{O}_1 \to \mathcal{O}_2$ maps the element $(V \to U, f)$ simply to the pullback of $f$ to $V$.

Surjectivity. Let $V$ be a scheme and let $h \in \mathcal{O}(V)$. Then we obtain a morphism $V \to X \times \mathbf{A}^1$ induced by $h$ and the structure morphism $V \to X$. Writing $\mathbf{A}^1 = \mathop{\rm Spec}(\mathbf{Z}[x])$ we see the element $x \in \mathcal{O}(X \times \mathbf{A}^1)$ pulls back to $h$. Since $X \times \mathbf{A}^1$ is an object of $\mathcal{C}_1$ by assumptions (1)(b) and (2) we obtain the desired surjectivity.

Injectivity. Let $V$ be a scheme. Let $s = \sum_{i = 1, \ldots, n} (V \to U_i, f_i)$ be an element of the colimit displayed above. For any $i$ we can use the morphism $f_i : U_i \to X \times \mathbf{A}^1$ to see that $(V \to U_i, f_i)$ defines the same element of the colimit as $(f_i : V \to X \times \mathbf{A}^1, x)$. Then we can consider $$f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n$$ and we see that $s$ is equivalent in the colimit to $$\sum\nolimits_{i = 1, \ldots, n} (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_i) = (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_1 + \ldots + x_n)$$ Now, if $x_1 + \ldots + x_n$ restricts to zero on $V$, then we see that $f_1 \times \ldots \times f_n$ factors through $X \times \mathbf{A}^{n - 1} = V(x_1 + \ldots + x_n)$. Hence we see that $s$ is equivalent to zero in the colimit. $\square$

\section{Exactness of big lower shriek}
\label{section-exactness-lower-shriek}

\noindent
This is just the following technical result. Note that the functor $f_{big!}$
has nothing whatsoever to do with cohomology with compact support in
general.

\begin{lemma}
\label{lemma-exactness-lower-shriek}
Let $\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$.
Let $f : X \to Y$ be a morphism of schemes. Let
$$f_{big} : \Sh((\Sch/X)_\tau) \longrightarrow \Sh((\Sch/Y)_\tau)$$
be the corresponding morphism of topoi as in
Topologies, Lemma
\ref{topologies-lemma-morphism-big},
\ref{topologies-lemma-morphism-big-etale},
\ref{topologies-lemma-morphism-big-smooth},
\ref{topologies-lemma-morphism-big-syntomic}, or
\ref{topologies-lemma-morphism-big-fppf}.
\begin{enumerate}
\item The functor
$f_{big}^{-1} : \textit{Ab}((\Sch/Y)_\tau) \to \textit{Ab}((\Sch/X)_\tau)$
$$f_{big!} : \textit{Ab}((\Sch/X)_\tau) \to \textit{Ab}((\Sch/Y)_\tau)$$
which is exact.
\item The functor
$f_{big}^* : \textit{Mod}((\Sch/Y)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/X)_\tau, \mathcal{O})$
$$f_{big!} : \textit{Mod}((\Sch/X)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/Y)_\tau, \mathcal{O})$$
which is exact.
\end{enumerate}
Moreover, the two functors $f_{big!}$ agree on underlying sheaves
of abelian groups.
\end{lemma}

\begin{proof}
Recall that $f_{big}$ is the morphism of topoi associated to the
continuous and cocontinuous functor
$u : (\Sch/X)_\tau \to (\Sch/Y)_\tau$, $U/X \mapsto U/Y$.
Moreover, we have $f_{big}^{-1}\mathcal{O} = \mathcal{O}$.
Hence the existence of $f_{big!}$ follows from
respectively
Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}.
Note that if $U$ is an object of $(\Sch/X)_\tau$ then the functor
$u$ induces an equivalence of categories
$$u' : (\Sch/X)_\tau/U \longrightarrow (\Sch/Y)_\tau/U$$
because both sides of the arrow are equal to $(\Sch/U)_\tau$.
Hence the agreement of $f_{big!}$ on underlying abelian sheaves
follows from the discussion in
Modules on Sites, Remark \ref{sites-modules-remark-when-shriek-equal}.
The exactness of $f_{big!}$ follows from
Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek}
as the functor $u$ above which commutes with fibre products and equalizers.
\end{proof}

\noindent
Next, we prove a technical lemma that will be useful later when comparing
sheaves of modules on different sites associated to algebraic stacks.

\begin{lemma}
\label{lemma-compare-structure-sheaves}
Let $X$ be a scheme. Let
$\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$.
Let $\mathcal{C}_1 \subset \mathcal{C}_2 \subset (\Sch/X)_\tau$ be full
subcategories with the following properties:
\begin{enumerate}
\item For an object $U/X$ of $\mathcal{C}_t$,
\begin{enumerate}
\item if $\{U_i \to U\}$ is a covering of $(\Sch/X)_\tau$, then
$U_i/X$ is an object of $\mathcal{C}_t$,
\item $U \times \mathbf{A}^1/X$ is an object of $\mathcal{C}_t$.
\end{enumerate}
\item $X/X$ is an object of $\mathcal{C}_t$.
\end{enumerate}
We endow $\mathcal{C}_t$ with the structure of a site whose coverings are
exactly those coverings $\{U_i \to U\}$ of $(\Sch/X)_\tau$ with
$U \in \Ob(\mathcal{C}_t)$. Then
\begin{enumerate}
\item[(\romannumeral1)] The functor $\mathcal{C}_1 \to \mathcal{C}_2$
is fully faithful, continuous, and cocontinuous.
\end{enumerate}
Denote $g : \Sh(\mathcal{C}_1) \to \Sh(\mathcal{C}_2)$ the corresponding
morphism of topoi. Denote $\mathcal{O}_t$ the restriction of $\mathcal{O}$
to $\mathcal{C}_t$. Denote $g_!$ the functor of
Modules on Sites, Definition \ref{sites-modules-definition-g-shriek}.
\begin{enumerate}
\item[(\romannumeral2)] The canonical map $g_!\mathcal{O}_1 \to \mathcal{O}_2$
is an isomorphism.
\end{enumerate}
\end{lemma}

\begin{proof}
Assertion (\romannumeral1) is immediate from the definitions.
In this proof all schemes are schemes over $X$ and all morphisms of
schemes are morphisms of schemes over $X$. Note that $g^{-1}$ is
given by restriction, so that for an object $U$ of $\mathcal{C}_1$
we have $\mathcal{O}_1(U) = \mathcal{O}_2(U) = \mathcal{O}(U)$.
Recall that $g_!\mathcal{O}_1$ is the sheaf associated to the presheaf
$g_{p!}\mathcal{O}_1$ which associates to $V$ in $\mathcal{C}_2$ the group
$$\colim_{V \to U} \mathcal{O}(U)$$
where $U$ runs over the objects of $\mathcal{C}_1$ and the colimit is
taken in the category of abelian groups. Below we will use frequently
that if
$$V \to U \to U'$$
are morphisms with $U, U' \in \Ob(\mathcal{C}_1)$
and if $f' \in \mathcal{O}(U')$ restricts to $f \in \mathcal{O}(U)$,
then $(V \to U, f)$ and $(V \to U', f')$ define the same element of the
colimit. Also, $g_!\mathcal{O}_1 \to \mathcal{O}_2$ maps the element
$(V \to U, f)$ simply to the pullback of $f$ to $V$.

\medskip\noindent
Surjectivity. Let $V$ be a scheme and let $h \in \mathcal{O}(V)$.
Then we obtain a morphism $V \to X \times \mathbf{A}^1$ induced by $h$
and the structure morphism $V \to X$. Writing
$\mathbf{A}^1 = \Spec(\mathbf{Z}[x])$ we see the element
$x \in \mathcal{O}(X \times \mathbf{A}^1)$ pulls
back to $h$. Since $X \times \mathbf{A}^1$ is an object of $\mathcal{C}_1$
by assumptions (1)(b) and (2) we obtain the desired surjectivity.

\medskip\noindent
Injectivity. Let $V$ be a scheme. Let
$s = \sum_{i = 1, \ldots, n} (V \to U_i, f_i)$ be an element of the colimit
displayed above. For any $i$ we can use the morphism
$f_i : U_i \to X \times \mathbf{A}^1$
to see that $(V \to U_i, f_i)$ defines the same element of the colimit as
$(f_i : V \to X \times \mathbf{A}^1, x)$. Then we can consider
$$f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n$$
and we see that $s$ is equivalent in the colimit to
$$\sum\nolimits_{i = 1, \ldots, n} (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_i) = (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_1 + \ldots + x_n)$$
Now, if $x_1 + \ldots + x_n$ restricts to zero on $V$, then we see
that $f_1 \times \ldots \times f_n$ factors through
$X \times \mathbf{A}^{n - 1} = V(x_1 + \ldots + x_n)$. Hence we see
that $s$ is equivalent to zero in the colimit.
\end{proof}

%9.29.09


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/04CB}{Tag 04CB}]{stacks-project}


In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).