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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.