In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a slightly more general setting (so don't be afraid to either generalize or concretize the statements if that helps).
My question is whether the following is somehow a familiar/general construction concerning Čech cohomology.
Let $M$ be a smooth manifold with a Čech cover of coordinate neighborhoods $\mathcal{U}=(U_i)_{i\in I}$. Suppose we have sheaves $C, K, F$ and $G$ such that
- $C\hookrightarrow F\hookrightarrow G$
- $C\hookrightarrow K\hookrightarrow G$
- $G(U_i)=F(U_i)\cdot K(U_i)$ for all $i\in I$
- $K\cap F = C$
Where the inclusions and intersection hold on every open of $M$. Suppose now we have another sheaf $Z$ and an exact sequence of sheaves
$$0\rightarrow K\hookrightarrow G\rightarrow Z\rightarrow 0$$
Such that the subsequence
$$0\rightarrow C\hookrightarrow F\rightarrow Z\rightarrow 0$$ is also exact. I wish to figure out what the image of $G(M)$ in $Z(M)$ is assuming I know the image of $F(M)$ in $Z(M)$. Any contributions other then the ones coming from $F(M)$ must then be from non-trivial classes in $H^1(\mathcal{U},C)$ which are trivial in $H^1(\mathcal{U}, F)$ and in $H^1(\mathcal{U},K)$.
Since if $s\in G(M)$ decomposes as $s=f\cdot k$, for $f\in F(M)$, $k\in K(M)$ then the image of $s$ is simply the image of $f$. However if $s$ does not decompose, it still decomposes locally as $s|_{U_i}=f_i\cdot k_i$, with $f_i\in F(U_i)$ and $k_i\in K(U_i)$. This forces $\frac{f_j}{f_i}=\frac{k_i}{k_j}\in C(U_i\cap U_j)$ and thus defines a cocycle in $C$ with respect to $\mathcal{U}$.
So in conclusion we should see that if we name the maps:
- $i:C\hookrightarrow K$
- $I:C\hookrightarrow F$
- $d:F\rightarrow Z$
- $D:G\rightarrow Z$
Then $(\mbox{Im}\hspace{0.1cm} D_M)/(\mbox{Im}\hspace{0.1cm} d_M)=(\ker i^*\cap \ker I^*)$, where the stars mean the induced maps on $H^1$, of course this equals the kernel of the first connecting map.
The challenge is still to find nice expressions for this image in specific cases. That is why I was wondering whether there is some general construction which deals with situations like these.
