Is the sum of two sub-sheaves a sheaf

Question

Let $X$ be a topological space, let $\mathcal{E}$ be a sheaf of abelian groups on $X$ and let $\mathcal{F}$ and $\mathcal{G}$ be two sub-sheaves of $\mathcal{E}$. Let $\mathcal{F}+\mathcal{G}$ be the presheaf $U\mapsto \mathcal{F}(U)+\mathcal{G}(U)$. Is $\mathcal{F}+\mathcal{G}$ a sheaf ?

Answer 1

No, you need to sheafify. For instance, let $j_U:U\to X$ be the inclusion of an open subset. Then ${j_U}_!\underline{\mathbb{Z}}_U$ is a subsheaf of the constant sheaf $\underline{\mathbb{Z}}_X$. Now, if $U,V$ is an open cover of $X$, then ${j_U}_!\underline{\mathbb{Z}}_U+{j_V}_!\underline{\mathbb{Z}}_V$ is not a sheaf in general and its sheafification is $\underline{\mathbb{Z}}_X$.

To see this, note that the presheaf ${j_U}_!\underline{\mathbb{Z}}_U+{j_V}_!\underline{\mathbb{Z}}_V$ is the presheaf image of ${j_U}_!\underline{\mathbb{Z}}_U\oplus{j_V}_!\underline{\mathbb{Z}}_V\to\underline{\mathbb{Z}}_X$ and that the sheaf morphism is onto as you can see by looking at the stalks, this implies that the sheafification of ${j_U}_!\underline{\mathbb{Z}}_U+{j_V}_!\underline{\mathbb{Z}}_V$ is indeed $\underline{\mathbb{Z}}_X$. In fact, you have a short exact sequence of sheaves : $$0\to {j_{U\cap V}}_!\underline{\mathbb{Z}}_{U\cap V}\to{j_U}_!\underline{\mathbb{Z}}_U\oplus{j_V}_!\underline{\mathbb{Z}}_V\to \underline{\mathbb{Z}}_X\to 0$$ If $X$ is compact, then $H^i(X,{j_U}_!\mathbb{Z}_U)=H^i_c(U,\mathbb{Z})$ is the usual cohomology with compact support, and the above short exact sequence gives rise to the Mayer-Vietoris long exact sequence with compact supports : $$0\to H_c^0(U\cap V,\mathbb{Z})\to H^0_c(U,\mathbb{Z})\oplus H^0_c(V,\mathbb{Z})\to H^0(X,\mathbb{Z})\to H^1_c(U\cap V,\mathbb{Z})\to...$$ It follows that if the boundary map $H^0(X,\mathbb{Z})\to H^1_c(U\cap V,\mathbb{Z})$ is non zero, then ${j_U}_!\underline{\mathbb{Z}}_U+{j_V}_!\underline{\mathbb{Z}}_V$ is different from its sheafification hence not a sheaf. For an actual example, take $X=S^1$ and $U,V$ the open cover you get when removing the north and the south pole.