# Quotient presheaf is a sheaf for flasque subsheaf

I think the following is true. (Sheaves are of abelian groups) Let $\mathscr{F}$ be a sheaf and $\mathscr{F'}$ is a subsheaf of $\mathscr{F}$ such that $\mathscr{F'}$ is flasque. (The restriction maps of $\mathscr{F'}$ are surjective. ) Show that the presheaf $U \mapsto \mathscr{F}(U)/\mathscr{F'}(U)$ is a sheaf.

Please prove it directly without using the exactness properties.(I want to use this to prove one of them) I am having problems proving sheaf property (II).

What I have tried is this (for showing sheaf property II):

Let $V_i$, $i \in I$, be an open covering for an open subset $U$ of $X$. We have to show that if there are $x_i \in \mathscr{F}(V_i)$ for every $i \in I$ such that $x_i \mid_{V_i \cap V_j} - x_j\mid_{V_i \cap V_j} \in \mathscr{F'}({V_i \cap V_j})$ for every $i,j \in I$. Then we have to show that there exists an $s \in \mathscr{F}(U)$ such that $s\mid _{V_i} - x_i \in \mathscr{F}'(V_i)$ for every $i \in I$.

I will be done if I can prove that there exists $t_i \in \mathscr{F}'(V_i)$ such that $x_i \mid_{V_i \cap V_j} - x_j\mid_{V_i \cap V_j} = t_i \mid_{V_i \cap V_j} - t_j \mid_{V_i \cap V_j}$ for all $i,j$. I'm unable to prove this.

Let us write $\mathcal{G} = \mathcal{F}/\mathcal{F}^{'}$ an remember that it is a presheaf so far.
We want to show that it is a sheaf. Take two intersecting open sets $U_{1}, U_{2}$ and take sections $s_{i} \in \mathcal{G}(U_{i}), i =1,2$ which agree on $\mathcal{G}(U_{1} \cap U_{2})$. Now we can lift these sections to $\widetilde{s_{i}} \in \mathcal{F}(U_{i})$, but they will in general not agree on $U_{1}\cap U_{2}$. Their difference restricted to $U_{1}\cap U_{2}$, $\widetilde{s}_{12} =\widetilde{s_{1}} - \widetilde{s_{2}} |_{$U_{1}\cap U_{2}} \in \mathcal{F}^{'}(U_{1} \cap U_{2})$. Using flasque property you can find a section$t \in \mathcal{F}^{'}(U_{1} \cup U_{2})$that restricts to$\widetilde{s}_{12}$. Now the sections$\widetilde{s_{i}} -t \in \mathcal{F}(U_{i})$patch nicely to a section of$\mathcal{F}(U_{1} \cup U_{2})$and its image in$\mathcal{G}(U_{1} \cup U_{2})$is the section extending$s_{i} \in \mathcal{G}(U_{i})$. The uniqueness follows from the fact that everything is unique mod$\mathcal{F}^{'}(U_{1} \cup U_{2})$. • How do you show that it is compatible with any open covering of$U$? – Sami Jul 9 '13 at 19:07 • You will need to use some commutative diagrams for that.If you consider an open cover$\{U_{i} \}_{i \in I}$of$U$, then you want to show that $$0 \rightarrow\mathcal{G}(U) \rightarrow \prod_{i} \mathcal{G}(U_{i}) \rightarrow \prod_{ij}\mathcal{G}_{U_{ij}}$$ is exact. But the arrows are defined component wise. So to show exactness you argue at each component$U_{i}, U_{j}, U_{i}\cap U_{j}$and that is the argument above (non intersecting sets cause no problem). If you have seen Mayer Vietoris in topology this is exactly the same idea. – DBS Jul 9 '13 at 19:29 • Thanks for your reply. I will try to look into your comment in detail. I've edited my post adding my work. Could you please tell me how to construct the$t_i$s in my post? – Sami Jul 9 '13 at 19:44 • Because in your notation$(x_{i} -x_{j}) |_{V_{i} \cap V_{j}} \in \mathcal{F}^{'}$and$\mathcal{F}^{'}$is flasque so we can lift (x_{i} -x_{j}) |_{V_{i} \cap V_{j}} it to$t \in\mathcal{F}^{'}{V_{i} \cup V_{j}}$. Then$t_{i}$= restriction of$t$. – DBS Jul 9 '13 at 19:52 • I realized my formatting is sub-par above. I hope it is not too confusing. – DBS Jul 9 '13 at 20:14 Let$\mathscr{F}$be a sheaf and$\mathscr{F'}$is a subsheaf of$\mathscr{F}$such that$\mathscr{F'}$is flasque. Show that the presheaf$U \mapsto \mathscr{F}(U)/\mathscr{F'}(U)$is a sheaf. Pf. Let$\{U_i\}_{i\in I}$cover$U$and let$s_i\in\mathscr F(U_i)/\mathscr F(U_i)$such that$\left.s_i\right|_{U_i\cap U_j}=\left.s_j\right|_{U_i\cap U_j}$. This is equivalent to the existence of$t_i\in\mathscr F(U_i)$and$q_{ij}\in\mathscr F'(U_i\cap U_j)$such that$\left.t_i\right|_{U_i\cap U_j}-\left.t_j\right|_{U_i\cap U_j}=q_{ij}$for all$i,j\in I$. Since$\mathscr F'$is flasque, there exists a non-canonical distinguished section$\gamma\in\mathscr F(Q)$over$Q:=\bigcup_{i,j\in I}U_i\cap U_j$such that$\left.\gamma\right|_{U_i\cap U_j}/\mathscr F'(U_i\cap U_j) = \left.s_i\right|_{U_i\cap U_j}\forall i$, and sections$\ell_{ij}\in\mathscr F'(U_i\cap U_j)$such that$\left.t_i\right|_{U_i\cap U_j}+\ell_{ij}=\left.\gamma\right|_{U_i\cap U_j}$. Notice that, for each fixed$i$, the sections$\{\ell_{ij}\}_{j\in I}$collate to a section$\ell_i\in\mathscr F'(Q\cap U_i)$. Replace$t_i$by$t_i+\tilde\ell_i$where$\tilde\ell_i$is a section in$\mathscr F'(U_i)$restricting to$\ell_i$on$Q\cap U_i$, whose existence is guaranteed as$\mathscr F'$is flasque. Then$\left\{t_i+\tilde \ell_i\right\}_{i\in I}$collate to an element$t\in\mathscr F(U)$such that$\overline t\in\mathscr F(U)/\mathscr F'(U)$restricts to$s_i$on$U_i.\quad\square$A note about the distinguished section$\gamma$. Why do we need$\mathscr F'$to be flasque to assure the existence of$\gamma$? To see this, fix some$i\in I$and suppose the restriction of$\gamma$to$U_i\cap U_j$is specified. It may be that$U_i\cap U_j\cap U_k\ne\emptyset$, in which case the restriction of$\gamma$to$U_i\cap U_k$is already specified on an open subset. But now, starting with$t_i$(or$t_k$), there is an element$\eta\in\mathscr F'(U_i\cap U_j\cap U_k)$such that$\left.t_i\right|_{U_i\cap U_j\cap U_k}+\eta=\left.\gamma\right|_{U_i\cap U_j\cap U_k}$. We require that$\mathscr F'$be flasque so that we are assured of an element$\tilde\eta\in\mathscr F'(U_i\cap U_k)$restricting to$\eta$; we then define$\left.\gamma\right|_{U_i\cap U_k}:=\left.t_i\right|_{U_i\cap U_k}+\tilde\eta$if$j$is the sole index such that$U_i\cap U_k\cap U_j\ne\emptyset$, or, more generally, we can ensure that$\gamma$is defined on$U_i\cap U_k$in a manner compatible with any previous specified restrictions on an open cover of$U_i\cap U_k$by sets of the form$U_i\cap U_k\cap U_j$, again since$\mathscr F'$is flasque (in such a case,$\tilde\eta$will be already specified on a possibly-partial open cover of$U_i\cap U_k$). (Note that we won't need to look over intersections finer than the intersection of three different$U_i$, as if$U_i\cap U_j$is covered by sets of the form$U_i\cap U_j\cap_{s\in S}U_s$with$|S|<|\mathbf N|$, it is covered by sets of the form$\{U_i\cap U_j\cap U_s\}_{s\in S}$.) This is enough to establish the existence of the distinguished section$\gamma$in the case$\mathscr F'\$ is flasque.