The Stacks project

Lemma 52.6.16. Let $\mathcal{C}$ be a site. Assume $\varphi : \mathcal{O} \to \mathcal{O}'$ is a flat homomorphism of sheaves of rings. Let $f_1, \ldots , f_ r$ be global sections of $\mathcal{O}$ such that $\mathcal{O}/(f_1, \ldots , f_ r) \cong \mathcal{O}'/(f_1, \ldots , f_ r)\mathcal{O}'$. Then the map of extended alternating Čech complexes

\[ \xymatrix{ \mathcal{O} \to \prod _{i_0} \mathcal{O}_{f_{i_0}} \to \prod _{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_ r} \ar[d] \\ \mathcal{O}' \to \prod _{i_0} \mathcal{O}'_{f_{i_0}} \to \prod _{i_0 < i_1} \mathcal{O}'_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}'_{f_1\ldots f_ r} } \]

is a quasi-isomorphism.

Proof. Observe that the second complex is the tensor product of the first complex with $\mathcal{O}'$. We can write the first extended alternating Čech complex as a colimit of the Koszul complexes $K_ n = K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n)$, see More on Algebra, Lemma 15.29.6. Hence it suffices to prove $K_ n \to K_ n \otimes _\mathcal {O} \mathcal{O}'$ is a quasi-isomorphism. Since $\mathcal{O} \to \mathcal{O}'$ is flat it suffices to show that $H^ i \to H^ i \otimes _\mathcal {O} \mathcal{O}'$ is an isomorphism where $H^ i$ is the $i$th cohomology sheaf $H^ i = H^ i(K_ n)$. These sheaves are annihilated by $f_1^ n, \ldots , f_ r^ n$, see More on Algebra, Lemma 15.28.6. Hence these sheaves are annihilated by $(f_1, \ldots , f_ r)^ m$ for some $m \gg 0$. Thus $H^ i \to H^ i \otimes _\mathcal {O} \mathcal{O}'$ is an isomorphism by Modules on Sites, Lemma 18.28.15. $\square$


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