Lemma 30.2.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{U} : U = \bigcup _{i = 1}^ n D(f_ i)$ be a standard open covering of an affine open of $X$. Then $\check{H}^ p(\mathcal{U}, \mathcal{F}) = 0$ for all $p > 0$.

**Proof.**
Write $U = \mathop{\mathrm{Spec}}(A)$ for some ring $A$. In other words, $f_1, \ldots , f_ n$ are elements of $A$ which generate the unit ideal of $A$. Write $\mathcal{F}|_ U = \widetilde{M}$ for some $A$-module $M$. Clearly the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is identified with the complex

We are asked to show that the extended complex

(whose truncation we have studied in Algebra, Lemma 10.24.1) is exact. It suffices to show that (30.2.1.1) is exact after localizing at a prime $\mathfrak p$, see Algebra, Lemma 10.23.1. In fact we will show that the extended complex localized at $\mathfrak p$ is homotopic to zero.

There exists an index $i$ such that $f_ i \not\in \mathfrak p$. Choose and fix such an element $i_{\text{fix}}$. Note that $M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}$. Similarly for a localization at a product $f_{i_0} \ldots f_{i_ p}$ and $\mathfrak p$ we can drop any $f_{i_ j}$ for which $i_ j = i_{\text{fix}}$. Let us define a homotopy

by the rule

(This is “dual” to the homotopy in the proof of Cohomology, Lemma 20.10.4.) In other words, $h : \prod _{i_0} M_{f_{i_0}, \mathfrak p} \to M_\mathfrak p$ is projection onto the factor $M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}$ and in general the map $h$ equal projection onto the factors $M_{f_{i_{\text{fix}}} f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p} = M_{f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p}$. We compute

This proves the identity map is homotopic to zero as desired. $\square$

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