Let $X:=S^1 \times S^1$ be the torus and let $\mathcal{F}$ denote the sheaf of locally constant functions with values in $\mathbb{Z}$. Determine the cohomology group $H^1(X,\mathcal{F})$.
Here I try to workout my approach which fails:
In order to apply Leray's theorem, we need to find a open cover $U_1, U_2, U_3$ of $X$ such that $H^1(U_i,\mathcal{F})=0$. Note that is fulfilled if $U_1, U_2, U_3$ is path-connected. Let $U_1$ be the red open, $U_2$ be the blue open and $U_3$ be the black open in the picture below. Clearly all this open sets are path-connected.
Let $f:=(f_{ij})_{i,j=1,2,3}$ be a 1-cochain, then f is 1-cocycle iff $f_{ii}=0, f_{ij}=-f_{ji}, f_{ij} + f_{ki}=f_{kj} $ where $k,i,j$ ranges from $1$ to $3$. This implies that 1-cocycle is determined by giving the values $f_{13}, f_{23}$, therefore 1-cocycles are isomorphic to $(\mathbb{Z}^3)^2$. Note that here I use that $U_1 \cap U_3$ and $U_2 \cap U_3$ has precisely 3 path-connected components. Moreover note that the 0-cochain group is isomorphic to $\mathbb{Z}^3$ and the kernel of the zero-coboundary map is $\mathbb{Z}$. Basic Algebra implies that then $H^1(X,\mathcal{F})$ has rank 4. But this wrong, because we know that this cohomology group should coincide with the first singular cohomology group, which is $\mathbb{Z}^2$.
What did I wrong?
