# The Stacks Project

## Tag 079X

### 21.14. Limp sheaves

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a sheaf of sets on $\mathcal{C}$ (we intentionally use a roman capital here to distinguish from abelian sheaves). Given an abelian sheaf $\mathcal{F}$ we denote $\mathcal{F}(K) = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(\mathcal{C})}(K, \mathcal{F})$. The functor $\mathcal{F} \mapsto \mathcal{F}(K)$ is a left exact functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}$ hence we have its right derived functors. We will denote these $H^p(K, \mathcal{F})$ so that $H^0(K, \mathcal{F}) = \mathcal{F}(K)$.

We mention two special cases. The first is the case where $K = h_U^\#$ for some object $U$ of $\mathcal{C}$. In this case $H^p(K, \mathcal{F}) = H^p(U, \mathcal{F})$, because $\mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(\mathcal{C})}(h_U^\#, \mathcal{F}) = \mathcal{F}(U)$, see Sites, Section 7.12. The second is the case $\mathcal{O} = \mathbf{Z}$ (the constant sheaf). In this case the cohomology groups are functors $H^p(K, - ) : \textit{Ab}(\mathcal{C}) \to \textit{Ab}$. Here is the analogue of Lemma 21.13.4.

Lemma 21.14.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a sheaf of sets on $\mathcal{C}$. Let $\mathcal{F}$ be an $\mathcal{O}$-module and denote $\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then $H^p(K, \mathcal{F}) = H^p(K, \mathcal{F}_{ab})$.

Proof. Note that both $H^p(K, \mathcal{F})$ and $H^p(K, \mathcal{F}_{ab})$ depend only on the topos, not on the underlying site. Hence by Sites, Lemma 7.28.5 we may replace $\mathcal{C}$ by a ''larger'' site such that $K = h_U$ for some object $U$ of $\mathcal{C}$. In this case the result follows from Lemma 21.13.4. $\square$

Lemma 21.14.2. Let $\mathcal{C}$ be a site. Let $K' \to K$ be a surjective map of sheaves of sets on $\mathcal{C}$. Set $K'_p = K' \times_K \ldots \times_K K'$ ($p + 1$-factors). For every abelian sheaf $\mathcal{F}$ there is a spectral sequence with $E_1^{p, q} = H^q(K'_p, \mathcal{F})$ converging to $H^{p + q}(K, \mathcal{F})$.

Proof. After replacing $\mathcal{C}$ by a ''larger'' site as in Sites, Lemma 7.28.5 we may assume that $K, K'$ are objects of $\mathcal{C}$ and that $\mathcal{U} = \{K' \to K\}$ is a covering. Then we have the Čech to cohomology spectral sequence of Lemma 21.11.6 whose $E_1$ page is as indicated in the statement of the lemma. $\square$

Lemma 21.14.3. Let $\mathcal{C}$ be a site. Let $K$ be a sheaf of sets on $\mathcal{C}$. Consider the morphism of topoi $j : \mathop{\textit{Sh}}\nolimits(\mathcal{C}/K) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C})$, see Sites, Lemma 7.29.3. Then $j^{-1}$ preserves injectives and $H^p(K, \mathcal{F}) = H^p(\mathcal{C}/K, j^{-1}\mathcal{F})$ for any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$.

Proof. By Sites, Lemmas 7.29.1 and 7.29.3 the morphism of topoi $j$ is equivalent to a localization. Hence this follows from Lemma 21.8.1. $\square$

Keeping in mind Lemma 21.14.1 we see that the following definition is the ''correct one'' also for sheaves of modules on ringed sites.

Definition 21.14.4. Let $\mathcal{C}$ be a site. We say an abelian sheaf $\mathcal{F}$ is limp1 if for every sheaf of sets $K$ we have $H^p(K, \mathcal{F}) = 0$ for all $p \geq 1$.

It is clear that being limp is an intrinsic property, i.e., preserved under equivalences of topoi. A limp sheaf has vanishing higher cohomology on all objects of the site, but in general the condition of being limp is strictly stronger. Here is a characterization of limp sheaves which is sometimes useful.

Lemma 21.14.5. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian sheaf. If

1. $H^p(U, \mathcal{F}) = 0$ for $p> 0$ and $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$, and
2. for every surjection $K' \to K$ of sheaves of sets the extended Čech complex $$0 \to H^0(K, \mathcal{F}) \to H^0(K', \mathcal{F}) \to H^0(K' \times_K K', \mathcal{F}) \to \ldots$$ is exact,

then $\mathcal{F}$ is limp (and the converse holds too).

Proof. By assumption (1) we have $H^p(h_U^\#, g^{-1}\mathcal{I}) = 0$ for all $p > 0$ and all objects $U$ of $\mathcal{C}$. Note that if $K = \coprod K_i$ is a coproduct of sheaves of sets on $\mathcal{C}$ then $H^p(K, g^{-1}\mathcal{I}) = \prod H^p(K_i, g^{-1}\mathcal{I})$. For any sheaf of sets $K$ there exists a surjection $$K' = \coprod h_{U_i}^\# \longrightarrow K$$ see Sites, Lemma 7.12.5. Thus we conclude that: (*) for every sheaf of sets $K$ there exists a surjection $K' \to K$ of sheaves of sets such that $H^p(K', \mathcal{F}) = 0$ for $p > 0$. We claim that (*) and condition (2) imply that $\mathcal{F}$ is limp. Note that conditions (*) and (2) only depend on $\mathcal{F}$ as an object of the topos $\mathop{\textit{Sh}}\nolimits(\mathcal{C})$ and not on the underlying site. (We will not use property (1) in the rest of the proof.)

We are going to prove by induction on $n \geq 0$ that (*) and (2) imply the following induction hypothesis $IH_n$: $H^p(K, \mathcal{F}) = 0$ for all $0 < p \leq n$ and all sheaves of sets $K$. Note that $IH_0$ holds. Assume $IH_n$. Pick a sheaf of sets $K$. Pick a surjection $K' \to K$ such that $H^p(K', \mathcal{F}) = 0$ for all $p > 0$. We have a spectral sequence with $$E_1^{p, q} = H^q(K'_p, \mathcal{F})$$ covering to $H^{p + q}(K, \mathcal{F})$, see Lemma 21.14.2. By $IH_n$ we see that $E_1^{p, q} = 0$ for $0 < q \leq n$ and by assumption (2) we see that $E_2^{p, 0} = 0$ for $p > 0$. Finally, we have $E_1^{0, q} = 0$ for $q > 0$ because $H^q(K', \mathcal{F}) = 0$ by choice of $K'$. Hence we conclude that $H^{n + 1}(K, \mathcal{F}) = 0$ because all the terms $E_2^{p, q}$ with $p + q = n + 1$ are zero. $\square$

1. This is probably nonstandard notation. In [SGA4, V, Definition 4.1] this property is dubbed ''flasque'', but we cannot use this because it would clash with our definition of flasque sheaves on topological spaces. Please email stacks.project@gmail.com if you have a better suggestion.

The code snippet corresponding to this tag is a part of the file sites-cohomology.tex and is located in lines 2036–2204 (see updates for more information).

\section{Limp sheaves}
\label{section-limp}

\noindent
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $K$ be a sheaf of sets on $\mathcal{C}$ (we intentionally use a
roman capital here to distinguish from abelian sheaves).
Given an abelian sheaf $\mathcal{F}$ we denote
$\mathcal{F}(K) = \Mor_{\Sh(\mathcal{C})}(K, \mathcal{F})$.
The functor $\mathcal{F} \mapsto \mathcal{F}(K)$ is a left exact functor
$\textit{Mod}(\mathcal{O}) \to \textit{Ab}$ hence we have its
right derived functors. We will denote these $H^p(K, \mathcal{F})$
so that $H^0(K, \mathcal{F}) = \mathcal{F}(K)$.

\medskip\noindent
We mention two special cases. The first is the case where
$K = h_U^\#$ for some object $U$ of $\mathcal{C}$. In this case
$H^p(K, \mathcal{F}) = H^p(U, \mathcal{F})$, because
$\Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F}) = \mathcal{F}(U)$, see
Sites, Section \ref{sites-section-representable-sheaves}.
The second is the case $\mathcal{O} = \mathbf{Z}$ (the constant
sheaf). In this case the cohomology groups are functors
$H^p(K, - ) : \textit{Ab}(\mathcal{C}) \to \textit{Ab}$.
Here is the analogue of
Lemma \ref{lemma-cohomology-modules-abelian-agree}.

\begin{lemma}
\label{lemma-compute-cohomology-on-sheaf-sets}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.
Let $K$ be a sheaf of sets on $\mathcal{C}$.
Let $\mathcal{F}$ be an $\mathcal{O}$-module and denote
$\mathcal{F}_{ab}$ the underlying sheaf of abelian groups.
Then $H^p(K, \mathcal{F}) = H^p(K, \mathcal{F}_{ab})$.
\end{lemma}

\begin{proof}
Note that both $H^p(K, \mathcal{F})$ and $H^p(K, \mathcal{F}_{ab})$
depend only on the topos, not on the underlying site. Hence by
Sites, Lemma \ref{sites-lemma-topos-good-site}
we may replace $\mathcal{C}$ by a larger'' site such
that $K = h_U$ for some object $U$ of $\mathcal{C}$.
In this case the result follows from
Lemma \ref{lemma-cohomology-modules-abelian-agree}.
\end{proof}

\begin{lemma}
\label{lemma-cech-to-cohomology-sheaf-sets}
Let $\mathcal{C}$ be a site. Let $K' \to K$ be a surjective
map of sheaves of sets on $\mathcal{C}$. Set
$K'_p = K' \times_K \ldots \times_K K'$ ($p + 1$-factors).
For every abelian sheaf $\mathcal{F}$ there is a spectral sequence
with $E_1^{p, q} = H^q(K'_p, \mathcal{F})$ converging to
$H^{p + q}(K, \mathcal{F})$.
\end{lemma}

\begin{proof}
After replacing $\mathcal{C}$ by a larger'' site as in
Sites, Lemma \ref{sites-lemma-topos-good-site}
we may assume that $K, K'$ are objects of $\mathcal{C}$ and that
$\mathcal{U} = \{K' \to K\}$ is a covering. Then we have the {\v C}ech
to cohomology spectral sequence of Lemma \ref{lemma-cech-spectral-sequence}
whose $E_1$ page is as indicated in the statement of the lemma.
\end{proof}

\begin{lemma}
\label{lemma-cohomology-on-sheaf-sets}
Let $\mathcal{C}$ be a site. Let $K$ be a sheaf of sets on $\mathcal{C}$.
Consider the morphism of topoi
$j : \Sh(\mathcal{C}/K) \to \Sh(\mathcal{C})$, see
Sites, Lemma \ref{sites-lemma-localize-topos-site}.
Then $j^{-1}$ preserves injectives and
$H^p(K, \mathcal{F}) = H^p(\mathcal{C}/K, j^{-1}\mathcal{F})$
for any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$.
\end{lemma}

\begin{proof}
By
Sites, Lemmas \ref{sites-lemma-localize-topos} and
\ref{sites-lemma-localize-topos-site}
the morphism of topoi $j$ is
equivalent to a localization. Hence this follows from
Lemma \ref{lemma-cohomology-of-open}.
\end{proof}

\noindent
Keeping in mind Lemma \ref{lemma-compute-cohomology-on-sheaf-sets}
we see that the following definition is the correct one'' also
for sheaves of modules on ringed sites.

\begin{definition}
\label{definition-limp}
Let $\mathcal{C}$ be a site.
We say an abelian sheaf $\mathcal{F}$ is
{\it limp}\footnote{This is probably nonstandard notation.
In \cite[V, Definition 4.1]{SGA4} this property is dubbed flasque'', but
we cannot use this because it would clash with our definition
of flasque sheaves on topological spaces. Please email
\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}
if you have a better suggestion.}
if for every sheaf of sets $K$ we have $H^p(K, \mathcal{F}) = 0$
for all $p \geq 1$.
\end{definition}

\noindent
It is clear that being limp is an intrinsic property, i.e.,
preserved under equivalences of topoi.
A limp sheaf has vanishing higher cohomology on all objects of the site,
but in general the condition of being limp is strictly stronger.
Here is a characterization of limp sheaves which is sometimes useful.

\begin{lemma}
\label{lemma-characterize-limp}
Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian sheaf. If
\begin{enumerate}
\item $H^p(U, \mathcal{F}) = 0$ for $p> 0$ and $U \in \Ob(\mathcal{C})$, and
\item for every surjection $K' \to K$ of sheaves of sets the
extended {\v C}ech complex
$$0 \to H^0(K, \mathcal{F}) \to H^0(K', \mathcal{F}) \to H^0(K' \times_K K', \mathcal{F}) \to \ldots$$
is exact,
\end{enumerate}
then $\mathcal{F}$ is limp (and the converse holds too).
\end{lemma}

\begin{proof}
By assumption (1) we have $H^p(h_U^\#, g^{-1}\mathcal{I}) = 0$ for all
$p > 0$ and all objects $U$ of $\mathcal{C}$. Note that if
$K = \coprod K_i$ is a coproduct of sheaves of sets on $\mathcal{C}$
then $H^p(K, g^{-1}\mathcal{I}) = \prod H^p(K_i, g^{-1}\mathcal{I})$.
For any sheaf of sets $K$ there exists a surjection
$$K' = \coprod h_{U_i}^\# \longrightarrow K$$
see Sites, Lemma \ref{sites-lemma-sheaf-coequalizer-representable}.
Thus we conclude that: (*) for every sheaf of sets $K$ there exists a
surjection $K' \to K$ of sheaves of sets such that $H^p(K', \mathcal{F}) = 0$
for $p > 0$. We claim that (*) and condition (2) imply that $\mathcal{F}$
is limp. Note that conditions (*) and (2) only depend on $\mathcal{F}$ as an
object of the topos $\Sh(\mathcal{C})$ and not on the underlying site.
(We will not use property (1) in the rest of the proof.)

\medskip\noindent
We are going to prove by induction on $n \geq 0$ that (*) and (2)
imply the following induction hypothesis $IH_n$:
$H^p(K, \mathcal{F}) = 0$ for all $0 < p \leq n$ and
all sheaves of sets $K$. Note that $IH_0$ holds. Assume $IH_n$. Pick
a sheaf of sets $K$. Pick a surjection $K' \to K$ such that
$H^p(K', \mathcal{F}) = 0$ for all $p > 0$. We have a
spectral sequence with
$$E_1^{p, q} = H^q(K'_p, \mathcal{F})$$
covering to $H^{p + q}(K, \mathcal{F})$, see
Lemma \ref{lemma-cech-to-cohomology-sheaf-sets}.
By $IH_n$ we see that $E_1^{p, q} = 0$ for $0 < q \leq n$ and by
assumption (2) we see that $E_2^{p, 0} = 0$ for $p > 0$. Finally, we have
$E_1^{0, q} = 0$ for $q > 0$ because $H^q(K', \mathcal{F}) = 0$ by
choice of $K'$. Hence we conclude that $H^{n + 1}(K, \mathcal{F}) = 0$
because all the terms $E_2^{p, q}$ with $p + q = n + 1$ are zero.
\end{proof}

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