Inclusion of quotient sheaves restricted to open subset

Question

When introducing sheaf cohomology following for example Chapter 8 of Kempf's book on Algebraic Varieties, we make the following standard definitions.

If $\mathcal{F}$ is a sheaf of abelian groups on top space $X$, we let the sheaf $D\mathcal{F}$ be defined on open $U\subset X$ by $$D\mathcal{F}(U):=\{s:U\to \coprod_{P\in U} \mathcal{F}_P | s(P)\in\mathcal{F}_P\}.$$

For $W$ open in $X$, we also make the standard definition $_W \mathcal{F}(U) := \mathcal{F}(W \cap U)$ and hence we have a sheaf $_W \mathcal{F}$ on $X.$

Now the lecture notes of the course I am revising for claim that there is an injection $$_W D\mathcal{F}/ _W\mathcal{F} \hookrightarrow \sideset{_W}{}( D\mathcal{F}/\mathcal{F}).$$

This statement really confuses me. Are these to sheaves not identical since they come from the sheafification of the same presheaf? Both my lecture notes and Kempf's book indirectly imply that these two sheafs are not identical but one is contained in the other.

Answer 1

They are not identical:

If $U$ is open in $X$ then (with a slight abuse of notation):

\begin{align} _W (D\mathcal{F}/\mathcal{F})(U)= & D\mathcal{F}/\mathcal{F}(U\cap W) \\ =&\{s:U\cap W\to \coprod_{p\in U\cap W} (D\mathcal{F}/\mathcal{F})_p | s(p)\in(D\mathcal{F}/\mathcal{F})_p, \text{and s is locally constant}\}\\ (_W D\mathcal{F}/_W\mathcal{F})(U)=&\{s:U\to \coprod_{p\in U}(_W D\mathcal{F}/_W\mathcal{F})_p | s(p)\in(_W D\mathcal{F}/_W\mathcal{F})_p, \text{and s is locally constant}\}\\ \end{align}

Which are in general different, and not isomorphic.

The inclusion map $\chi$ is probably best obtained from the universal property of cokernels applied to the map $\varphi:_WD\mathcal{F}\rightarrow_W(D\mathcal{F}/\mathcal{F})$. This has kernel $_W\mathcal{F}$ and by looking at stalks this implies that $\chi$ is injective. But $\chi$ is not surjective since $\varphi$ won't be in general. In fact, given injectivity of $\varphi$, one can check surjectivity on sections and then see that this happens exactly when the presheaf of $D\mathcal{F}/\mathcal{F}$ is a sheaf without the need to sheafify.

You can also look at how $\chi$ acts on the explicit descriptions of the sheaves above and convince yourself that it need not be surjective that way.