= Čech cohomology of an affine cover can differ from sheaf cohomology
{c}
{title2=$\check H^1(\{X\},\mathcal F)=0,\quad H^1(X,\mathcal F)=k(t)$}
On $X=\mathbb A^1_k$, let $K=k(t)$ and let $\mathcal K$ be its constant sheaf. For the two closed rational points $0,1$, take skyscraper sheaves $S_0,S_1$ with value $K$, and put $\mathcal F=\ker(\mathcal K\to S_0\oplus S_1)$. The maps on the two stalks are identities, so this is a sheaf surjection. All three displayed outer sheaves are flasque. The induced global map is the diagonal $K\to K^2$, so $H^1(X,\mathcal F)=K^2/\Delta K\simeq K$. The one-member affine cover has zero positive Čech cochains. This even gives an $\mathcal O_X$-module example: let regular functions act on all copies of $K$ by multiplication as rational functions, not by evaluation at the closed point. The kernel is not quasi-coherent.
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