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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.

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