Lemma 23.7.5. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset J \subset A$ be proper ideals. Assume

1. $A/J$ has finite tor dimension over $A/I$, and

2. $J$ is generated by a regular sequence.

Then $I$ is generated by a regular sequence and $J/I$ is generated by a regular sequence.

Proof. By Lemma 23.7.4 we see that $I/\mathfrak m I \to J/\mathfrak m J$ is injective. Thus we can find $s \leq r$ and a minimal system of generators $f_1, \ldots , f_ r$ of $J$ such that $f_1, \ldots , f_ s$ are in $I$ and form a minimal system of generators of $I$. The lemma follows as any minimal system of generators of $J$ is a regular sequence by More on Algebra, Lemmas 15.30.15 and 15.30.7. $\square$

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