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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.