Lemma 15.28.4. Let $f_1, \ldots , f_ r \in R$ be a sequence. Let $(x_{ij})$ be an invertible $r \times r$-matrix with coefficients in $R$. Then the complexes $K_\bullet (f_\bullet )$ and

are isomorphic.

Lemma 15.28.4. Let $f_1, \ldots , f_ r \in R$ be a sequence. Let $(x_{ij})$ be an invertible $r \times r$-matrix with coefficients in $R$. Then the complexes $K_\bullet (f_\bullet )$ and

\[ K_\bullet (\sum x_{1j}f_ j, \sum x_{2j}f_ j, \ldots , \sum x_{rj}f_ j) \]

are isomorphic.

**Proof.**
Set $g_ i = \sum x_{ij}f_ j$. The matrix $(x_{ji})$ gives an isomorphism $x : R^{\oplus r} \to R^{\oplus r}$ such that $(g_1, \ldots , g_ r) = (f_1, \ldots , f_ r) \circ x$. Hence this follows from the functoriality of the Koszul complex described in Lemma 15.28.3.
$\square$

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