Lemma 15.29.3. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$. Let $M$ be an $R$-module. Let $R \to S$ be a ring map, denote $g_1, \ldots , g_ r \in S$ the images of $f_1, \ldots , f_ r$, and set $N = M \otimes _ R S$. The extended alternating Čech complex constructed using $S$, $g_1, \ldots , g_ r$, and $N$ is the tensor product of the extended alternating Čech complex of $M$ with $S$ over $R$.

**Proof.**
Omitted.
$\square$

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