# Cech cohomology of a quasi-coherent sheaf on an affine scheme and Leray acyclicity Theorem.

Let $$X$$ be an affine scheme, $$\mathcal{F}$$ a quasi-coherent sheaf on $$X$$. Let $$\mathcal{U}=\{U_i\}_{i \in I}$$ be an affine covering of $$U$$ (not necessarely made up of principal open subsets). Moreover, let $$\mathcal{V}=\{V_j\}_{j \in J}$$ be a covering of $$X$$ made up of principal open subsets. Of course we can assume that both $$J$$ and $$I$$ are finite. Consider the covering $$\mathcal{V}_i:=\{V_j \cap U_i\}_{j \in J}$$ of $$U_i$$. Let $$H^q(\mathcal{V}_i,\mathcal{F}|_{U_i})$$ be the $$q$$-th Cech cohomology group of $$\mathcal{F}|_{U_i}$$ with respect to the covering $$\mathcal{V}_i$$. Can we say that $$H^q(\mathcal{V}_i,\mathcal{F}|_{U_i})=0$$ for each $$q \geq 1$$?

If the covering $$\mathcal{V_i}$$ is made up of principal open subset, then the answer is yes. But i think that $$\mathcal{V}_i$$ is not necessarely such. I am trying to understand Theorem 5.2.19 of Liu's Algebraic Geometry and Arithmetic Curves, which uses Theorem 5.2.12 (Leray's Theorem) of the same book.