The Stacks project

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 18.28.16. $\square$