I am trying to find a Leray covering for the 2-sphere with respect to the sheaf $\mathcal{F}=\mathbb{Z}$. I am also assuming that a contractible open covering satisfies $H^{i}(U,\mathbb{Z})$ for all $i>0$.
We recall that an open set $U$ is acyclic for a sheaf $\mathcal{F}$ if $H^{q}(U,\mathcal{F})=0$ for all $q>0$. Then we say that a covering $\mathfrak{U}$ of $(X,\mathcal{F})$ is Leray if $U_{I}$ is acyclic for all indices $I$.
I played around with this a little, and I think I have found a suitable covering $\mathfrak{U}$. Simply take $\mathbb{H}=U_{0}$ for our first open set, and the cover the lower hemisphere with three open sets $U_{1},U_{2},U_{3}$. I suspect that this is Leray due to the neatness of the cohomology calculations from the resulting Cech complex $C^{0}(\mathfrak{U},\mathbb{Z})\rightarrow C^{1}(\mathfrak{U},\mathbb{Z}) \rightarrow C^{2}(\mathfrak{U},\mathbb{Z})$. However, I get rather lost when I try to directly prove that this covering is Leray. Could someone show me how to do this for my given surface and sheaf? Any help would be appreciated. Thanks.
By your construction (assuming your $U_1,U_2,U_3$ are taken to be 120° rotations around the vertical axis of one another), all pairwise intersections are contractible; among the triple intersections, only $U_1\cap U_2\cap U_3$ is nonempty, and it can be arranged to be a disk, hence contractible as well.