\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

20.13 Flasque sheaves

Here is the definition.

Definition 20.13.1. Let $X$ be a topological space. We say a presheaf of sets $\mathcal{F}$ is flasque or flabby if for every $U \subset V$ open in $X$ the restriction map $\mathcal{F}(V) \to \mathcal{F}(U)$ is surjective.

We will use this terminology also for abelian sheaves and sheaves of modules if $X$ is a ringed space. Clearly it suffices to assume the restriction maps $\mathcal{F}(X) \to \mathcal{F}(U)$ is surjective for every open $U \subset X$.

Lemma 20.13.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Then any injective $\mathcal{O}_ X$-module is flasque.

Proof. This is a reformulation of Lemma 20.9.1. $\square$

Lemma 20.13.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Any flasque $\mathcal{O}_ X$-module is acyclic for $R\Gamma (X, -)$ as well as $R\Gamma (U, -)$ for any open $U$ of $X$.

Proof. We will prove this using Derived Categories, Lemma 13.16.6. Since every injective module is flasque we see that we can embed every $\mathcal{O}_ X$-module into a flasque module, see Injectives, Lemma 19.4.1. Thus it suffices to show that given a short exact sequence

\[ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 \]

with $\mathcal{F}$, $\mathcal{G}$ flasque, then $\mathcal{H}$ is flasque and the sequence remains short exact after taking sections on any open of $X$. In fact, the second statement implies the first. Thus, let $U \subset X$ be an open subspace. Let $s \in \mathcal{H}(U)$. We will show that we can lift $s$ to a section of $\mathcal{G}$ over $U$. To do this consider the set $T$ of pairs $(V, t)$ where $V \subset U$ is open and $t \in \mathcal{G}(V)$ is a section mapping to $s|_ V$ in $\mathcal{H}$. We put a partial ordering on $T$ by setting $(V, t) \leq (V', t')$ if and only if $V \subset V'$ and $t'|_ V = t$. If $(V_\alpha , t_\alpha )$, $\alpha \in A$ is a totally ordered subset of $T$, then $V = \bigcup V_\alpha $ is open and there is a unique section $t \in \mathcal{G}(V)$ restricting to $t_\alpha $ over $V_\alpha $ by the sheaf condition on $\mathcal{G}$. Thus by Zorn's lemma there exists a maximal element $(V, t)$ in $T$. We will show that $V = U$ thereby finishing the proof. Namely, pick any $x \in U$. We can find a small open neighbourhood $W \subset U$ of $x$ and $t' \in \mathcal{G}(W)$ mapping to $s|_ W$ in $\mathcal{H}$. Then $t'|_{W \cap V} - t|_{W \cap V}$ maps to zero in $\mathcal{H}$, hence comes from some section $r' \in \mathcal{F}(W \cap V)$. Using that $\mathcal{F}$ is flasque we find a section $r \in \mathcal{F}(W)$ restricting to $r'$ over $W \cap V$. Modifying $t'$ by the image of $r$ we may assume that $t$ and $t'$ restrict to the same section over $W \cap V$. By the sheaf condition of $\mathcal{G}$ we can find a section $\tilde t$ of $\mathcal{G}$ over $W \cup V$ restricting to $t$ and $t'$. By maximality of $(V, t)$ we see that $V \cap W = V$. Thus $x \in V$ and we are done. $\square$

The following lemma does not hold for flasque presheaves.

Lemma 20.13.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $\mathcal{U} : U = \bigcup U_ i$ be an open covering. If $\mathcal{F}$ is flasque, then $\check{H}^ p(\mathcal{U}, \mathcal{F}) = 0$ for $p > 0$.

Proof. The presheaves $\underline{H}^ q(\mathcal{F})$ used in the statement of Lemma 20.12.5 are zero by Lemma 20.13.3. Hence $\check{H}^ p(U, \mathcal{F}) = H^ p(U, \mathcal{F}) = 0$ by Lemma 20.13.3 again. $\square$

Lemma 20.13.5. Let $(X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is flasque, then $R^ pf_*\mathcal{F} = 0$ for $p > 0$.

Proof. Immediate from Lemma 20.8.3 and Lemma 20.13.3. $\square$

The following lemma can be proved by an elementary induction argument for finite coverings, compare with the discussion of Čech cohomology in [FOAG].

Lemma 20.13.6. Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be an open covering. Assume the restriction mappings $\mathcal{F}(U) \to \mathcal{F}(U')$ are surjective for $U'$ an arbitrary union of opens of the form $U_{i_0 \ldots i_ p}$. Then $\check{H}^ p(\mathcal{U}, \mathcal{F})$ vanishes for $p > 0$.

Proof. Let $Y$ be the set of nonempty subsets of $I$. We will use the letters $A, B, C, \ldots $ to denote elements of $Y$, i.e., nonempty subsets of $I$. For a finite nonempty subset $J \subset I$ let

\[ V_ J = \{ A \in Y \mid J \subset A\} \]

This means that $V_{\{ i\} } = \{ A \in Y \mid i \in A\} $ and $V_ J = \bigcap _{j \in J} V_{\{ j\} }$. Then $V_ J \subset V_ K$ if and only if $J \supset K$. There is a unique topology on $Y$ such that the collection of subsets $V_ J$ is a basis for the topology on $Y$. Any open is of the form

\[ V = \bigcup \nolimits _{t \in T} V_{J_ t} \]

for some family of finite subsets $J_ t$. If $J_ t \subset J_{t'}$ then we may remove $J_{t'}$ from the family without changing $V$. Thus we may assume there are no inclusions among the $J_ t$. In this case the minimal elements of $V$ are the sets $A = J_ t$. Hence we can read off the family $(J_ t)_{t \in T}$ from the open $V$.

We can completely understand open coverings in $Y$. First, because the elements $A \in Y$ are nonempty subsets of $I$ we have

\[ Y = \bigcup \nolimits _{i \in I} V_{\{ i\} } \]

To understand other coverings, let $V$ be as above and let $V_ s \subset Y$ be an open corresponding to the family $(J_{s, t})_{t \in T_ s}$. Then

\[ V = \bigcup \nolimits _{s \in S} V_ s \]

if and only if for each $t \in T$ there exists an $s \in S$ and $t_ s \in T_ s$ such that $J_ t = J_{s, t_ s}$. Namely, as the family $(J_ t)_{t \in T}$ is minimal, the minimal element $A = J_ t$ has to be in $V_ s$ for some $s$, hence $A \in V_{J_{t_ s}}$ for some $t_ s \in T_ s$. But since $A$ is also minimal in $V_ s$ we conclude that $J_{t_ s} = J_ t$.

Next we map the set of opens of $Y$ to opens of $X$. Namely, we send $Y$ to $U$, we use the rule

\[ V_ J \mapsto U_ J = \bigcap \nolimits _{i \in J} U_ i \]

on the opens $V_ J$, and we extend it to arbitrary opens $V$ by the rule

\[ V = \bigcup \nolimits _{t \in T} V_{J_ t} \mapsto \bigcup \nolimits _{t \in T} U_{J_ t} \]

The classification of open coverings of $Y$ given above shows that this rule transforms open coverings into open coverings. Thus we obtain an abelian sheaf $\mathcal{G}$ on $Y$ by setting $\mathcal{G}(Y) = \mathcal{F}(U)$ and for $V = \bigcup \nolimits _{t \in T} V_{J_ t}$ setting

\[ \mathcal{G}(V) = \mathcal{F}\left(\bigcup \nolimits _{t \in T} U_{J_ t}\right) \]

and using the restriction maps of $\mathcal{F}$.

With these preliminaries out of the way we can prove our lemma as follows. We have an open covering $\mathcal{V} : Y = \bigcup _{i \in I} V_{\{ i\} }$ of $Y$. By construction we have an equality

\[ \check{C}^\bullet (\mathcal{V}, \mathcal{G}) = \check{C}^\bullet (\mathcal{U}, \mathcal{F}) \]

of Čech complexes. Since the sheaf $\mathcal{G}$ is flasque on $Y$ (by our assumption on $\mathcal{F}$ in the statement of the lemma) the vanishing follows from Lemma 20.13.4. $\square$

Comments (4)

Comment #2058 by Reimundo on

In the proof of 20.13.3 there are some small typos: "We can find a small open neighborhood of and ..." should be

similarly below, should be restricting to and not

Comment #2995 by Sandor Kovacs on

Another typo: in the proof of 20.13.3 "We will show that we can lift s to a sequence of G over U." should be "We will show that we can lift s to a section of G over U."

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