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.

There are also:

• 3 comment(s) on Section 35.3: Descent for modules

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).