The Stacks project

Lemma 15.30.5. Let $\varphi : R \to S$ be a flat ring map. Let $f_1, \ldots , f_ r \in R$. Let $M$ be an $R$-module and set $N = M \otimes _ R S$.

  1. If $f_1, \ldots , f_ r$ in $R$ is an $M$-$H_1$-regular sequence, then $\varphi (f_1), \ldots , \varphi (f_ r)$ is an $N$-$H_1$-regular sequence in $S$.

  2. If $f_1, \ldots , f_ r$ is an $M$-Koszul-regular sequence in $R$, then $\varphi (f_1), \ldots , \varphi (f_ r)$ is an $N$-Koszul-regular sequence in $S$.

Proof. This is true because $K_\bullet (f_1, \ldots , f_ r) \otimes _ R S = K_\bullet (\varphi (f_1), \ldots , \varphi (f_ r))$ and therefore $(K_\bullet (f_1, \ldots , f_ r) \otimes _ R M) \otimes _ R S = K_\bullet (\varphi (f_1), \ldots , \varphi (f_ r)) \otimes _ S N$. $\square$

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