The Stacks project

Lemma 15.30.12. Let $A$ be a ring. Let $f_1, \ldots , f_ n, g_1, \ldots , g_ m \in A$ be an $H_1$-regular sequence. Then the images $\overline{g}_1, \ldots , \overline{g}_ m$ in $A/(f_1, \ldots , f_ n)$ form an $H_1$-regular sequence.

Proof. Set $I = (f_1, \ldots , f_ n)$. We have to show that any relation $\sum _{j = 1, \ldots , m} \overline{a}_ j \overline{g}_ j$ in $A/I$ is a linear combination of trivial relations. Because $I = (f_1, \ldots , f_ n)$ we can lift this relation to a relation

\[ \sum \nolimits _{j = 1, \ldots , m} a_ j g_ j + \sum \nolimits _{i = 1, \ldots , n} b_ if_ i = 0 \]

in $A$. By assumption this relation in $A$ is a linear combination of trivial relations. Taking the image in $A/I$ we obtain what we want. $\square$

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