The Stacks project

Lemma 15.30.8. Let $A$ be a ring. Let $I \subset A$ be an ideal. Let $g_1, \ldots , g_ m$ be a sequence in $A$ whose image in $A/I$ is $H_1$-regular. Then $I \cap (g_1, \ldots , g_ m) = I(g_1, \ldots , g_ m)$.

Proof. Consider the exact sequence of complexes

\[ 0 \to I \otimes _ A K_\bullet (A, g_1, \ldots , g_ m) \to K_\bullet (A, g_1, \ldots , g_ m) \to K_\bullet (A/I, g_1, \ldots , g_ m) \to 0 \]

Since the complex on the right has $H_1 = 0$ by assumption we see that

\[ \mathop{\mathrm{Coker}}(I^{\oplus m} \to I) \longrightarrow \mathop{\mathrm{Coker}}(A^{\oplus m} \to A) \]

is injective. This is equivalent to the assertion of the lemma. $\square$

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