Lemma 15.89.7. Let $R$ be a ring. Let $I = (f_1, \ldots , f_ n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ the Koszul homology group $H_1(N, f_\bullet )$ defined in Lemma 15.89.6 is annihilated by $I$.

**Proof.**
Let $(x_1, \ldots , x_ n) \in N^{\oplus n}$ with $f_ i x_ j = f_ j x_ i$. Then we have $f_ i(x_1, \ldots , x_ n) = (f_ i x_ i, \ldots , f_ i x_ n)$. In other words $f_ i$ annihilates $H_1(N, f_\bullet )$.
$\square$

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