The Stacks project

Lemma 15.28.12. Let $R$ be a ring. Let $f_1, \ldots , f_ r$, $g_1, \ldots , g_ s$ be elements of $R$. Then there is an isomorphism of Koszul complexes

\[ K_\bullet (R, f_1, \ldots , f_ r, g_1, \ldots , g_ s) = \text{Tot}(K_\bullet (R, f_1, \ldots , f_ r) \otimes _ R K_\bullet (R, g_1, \ldots , g_ s)). \]

Proof. Omitted. Hint: If $K_\bullet (R, f_1, \ldots , f_ r)$ is generated as a differential graded algebra by $x_1, \ldots , x_ r$ with $\text{d}(x_ i) = f_ i$ and $K_\bullet (R, g_1, \ldots , g_ s)$ is generated as a differential graded algebra by $y_1, \ldots , y_ s$ with $\text{d}(y_ j) = g_ j$, then we can think of $K_\bullet (R, f_1, \ldots , f_ r, g_1, \ldots , g_ s)$ as the differential graded algebra generated by the sequence of elements $x_1, \ldots , x_ r, y_1, \ldots , y_ s$ with $\text{d}(x_ i) = f_ i$ and $\text{d}(y_ j) = g_ j$. $\square$


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