What are the stalks of $\Gamma_Z(F)$ for a locally closed subset $Z$ and a flabby sheaf $F$

Question

My references for the following notations are Iversen & Hotta,Takeuchi, Tanisaki.

Let $Z \subset X$ be a locally closed set, and $F \in Sh(X)$. Let $Z \subset W \subset X$ be an open subset of $X$ with $Z$ closed relative to $W$. Define $\Gamma_Z(X,F)=\text{Ker}[F(W) \rightarrow F(W\setminus Z)]$ i.e. sections of $F$ supported in $Z$. This construction is independent of the choice of $W$, and can be done for any sheaf (not just $\textit{flabby}$ ones). We can define a (left exact) functor $\Gamma_Z: Sh(X) \rightarrow Sh(X)$ with $\Gamma_Z(F)(U)=\Gamma_{Z \cap U}(U, F \mid_{U})$. If $F$ is flabby, then we have the short exact sequences:

\begin{align} & 0 \rightarrow \Gamma_{Z'}(F) \rightarrow \Gamma_Z(F) \rightarrow \Gamma_{Z\setminus Z'} \rightarrow 0\\ & 0 \rightarrow \Gamma_{Z_1 \cap Z_2}(F) \rightarrow \Gamma_{Z_1}(F) \oplus \Gamma_{Z_2}(F) \rightarrow \Gamma_{Z_2 \cup Z_2}(F) \end{align} The first is excision, the second is Mayer-Vietoris, and this construction is a way of doing relative homology. I am curious if there is a nice description of the stalks of $\Gamma_Z(F)$ in terms of those of $F$, which might make the exactness of these sequences clearer.

I am asking this because of another construction for doing relative homology. Let $i:Z \hookrightarrow X$ denote the inclusion of $Z$, again a locally closed subset. For any sheaf $F \in Sh(X)$, define $F_Z:=i_! i^{-1} F$. A theorem in Iversen states that \begin{align} (F_Z)_x=\begin{cases} F_x & \text{ if }x \in Z\\ 0 & \text{ if } x \in X \setminus Z \end{cases} \end{align} This description makes the exactness of the sequences: \begin{align} &0 \rightarrow F_{Z'} \rightarrow F_Z \rightarrow F_{Z \setminus Z'} \rightarrow 0\\ & 0 \rightarrow F_{Z_1 \cap Z_2} \rightarrow F_{Z_1} \oplus F_{Z_2} \rightarrow F_{Z_1 \cup Z_2} \rightarrow 0 \end{align} easy to verify at the level of stalks.

Is there a similar verification of exactness of the previous sequences by checking at the level of stalks? If not, how do I see these are exact, and why does this require $F$ to be flabby?

Answer 1

I don't know if there is a nice formula for stalks and I doubt of it because it is not mentionned in reference books. Moreover the exact sequences you mentioned are generally not proved with stalks.

Here is a way to do it. The first thing to notice is that if $F$ is flabby, then $\Gamma_Z(F)$ is flabby. Once you have proved that, you can for example show the exactness of $$0 \rightarrow \Gamma_{Z'}(F) \rightarrow \Gamma_Z(F) \rightarrow \Gamma_{Z\setminus Z'}(F) \rightarrow 0$$ by using the fact that for each open set $U$, $$\Gamma(U,\Gamma_Z(F)) \to \Gamma(U,\Gamma_{Z\backslash Z'(F)}) \simeq \Gamma(U\backslash Z', \Gamma_Z(F))$$ is surjective since $\Gamma_Z(F)$ is flabby.

This is well-explained in Kashiwara and Shapira, Sheaves on manifold, page 99.