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 $\mathcal{F}(K) = \mathop{Mor}\nolimits_{\mathop{\mathit{Sh}}\nolimits(\mathcal{C})}(K, \mathcal{F})$. 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 $\mathop{Mor}\nolimits_{\mathop{\mathit{Sh}}\nolimits(\mathcal{C})}(h_U^\#, \mathcal{F}) = \mathcal{F}(U)$, 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 $j : \mathop{\mathit{Sh}}\nolimits(\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits(\mathcal{C})$, 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 $U \in \mathop{\mathrm{Ob}}\nolimits(\mathcal{C})$, 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 $\mathop{\mathit{Sh}}\nolimits(\mathcal{C})$ 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}

Comments (0)

There are no comments yet for this tag.

Add a comment on tag 079X

Your email address will not be published. Required fields are marked.

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).

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?