Leray's theorem

In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology.

Let $$\mathcal F$$ be a sheaf on a topological space $$X$$ and  $$\mathcal U$$ an open cover of $$X.$$ If  $$\mathcal F$$ is acyclic on every finite intersection of elements of  $$\mathcal U$$, then


 * $$ \check H^q(\mathcal U,\mathcal F)= \check H^q(X,\mathcal F), $$

where $$\check H^q(\mathcal U,\mathcal F)$$ is the $$q$$-th Čech cohomology group of  $$\mathcal F$$ with respect to the open cover $$\mathcal U.$$