The Stacks project

Remark 35.3.10. Let $R$ be a ring. Let $f_1, \ldots , f_ n\in R$ generate the unit ideal. The ring $A = \prod _ i R_{f_ i}$ is a faithfully flat $R$-algebra. We remark that the cosimplicial ring $(A/R)_\bullet $ has the following ring in degree $n$:

\[ \prod \nolimits _{i_0, \ldots , i_ n} R_{f_{i_0}\ldots f_{i_ n}} \]

Hence the results above recover Algebra, Lemmas 10.24.2, 10.24.1 and 10.24.5. But the results above actually say more because of exactness in higher degrees. Namely, it implies that Čech cohomology of quasi-coherent sheaves on affines is trivial. Thus we get a second proof of Cohomology of Schemes, Lemma 30.2.1.

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