Suppose that $X$ is a topological space, $F$ is a sheaf (of, say, Abelian groups) on $X$ (in this question, sheaves are "complete presheaves", i.e. they are not étalé spaces), and $G$ is a subsheaf of $F$.
As it is well-known, the presheaf $ U\mapsto F(U)/G(U) $ is not necessarily complete (apparantly it is always separated though), so the quotient sheaf $F/G$ is defined as the sheafification of this latter presheaf: $$ F/G:=\mathscr S\left(U\mapsto F(U)/G(U)\right) $$(here $\mathscr S:\mathsf{PSh}\rightarrow\mathsf{Sh}$ is the sheafification functor).
Pre-question: I have some trouble "imagining" the quotient sheaf $F/G$. I prefer to think of sheaves in terms of their sections rather than étalé spaces. Thus for given $U\subseteq X$ open set I'd like to know how to specify $(F/G)(U)$. Based on the definitions and on some answers to other questions about quotient shaves I think we have $$ (F/G)(U)=F(U)/\sim, $$ where $\sim$ is the following equivalence relation on $F(U)$. Two sections $f,f^\prime\in F(U)$ are $\sim$-equivalent if for each $p\in U$ there is an open set $V\subseteq U$ with $p\in V$ such that $f^\prime|_V=f|_V+g$, where $g\in G(V)$, or in other words, sections of $F/G$ are equivalence classes of $F$-sections which locally differ from each other by $G$-sections.
Is this correct?
Question: I would like to know conditions on the sheaf $F$ and subsheaf $G$ which ensure that we have $$ (F/G)(U)=F(U)/G(U), $$ i.e. the quotient sheaf agrees with the quotient presheaf. Apparantly if $G$ is flasque, then this is true.
What about less restrictive conditions, for example if $G$ or both $F$ and $G$ are soft sheaves?
Best Answer
The short exact sequence of sheaves $$ 0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{G} / \mathcal{F} \rightarrow 0 $$ yields the exact sequence $$ 0 \rightarrow \Gamma (U ,\mathcal{F}) \rightarrow \Gamma (U ,\mathcal{G}) \rightarrow \Gamma (U ,\mathcal{G} / \mathcal{F} ) \rightarrow H^1 ( U , \mathcal{ F} ) . $$ So one might ask that $$ H^1 ( U , \mathcal{ F} ) = 0 $$ for every open set. This is true for flasque sheaves.