The Stacks project

Lemma 23.7.4. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset J \subset A$ be proper ideals. If $A/J$ has finite tor dimension over $A/I$, then $I/\mathfrak m I \to J/\mathfrak m J$ is injective.

Proof. Let $f \in I$ be an element mapping to a nonzero element of $I/\mathfrak m I$ which is mapped to zero in $J/\mathfrak mJ$. We can choose an ideal $I'$ with $\mathfrak mI \subset I' \subset I$ such that $I/I'$ is generated by the image of $f$. Set $R = A/I$ and $R' = A/I'$. Let $J = (a_1, \ldots , a_ m)$ for some $a_ j \in A$. Then $f = \sum b_ j a_ j$ for some $b_ j \in \mathfrak m$. Let $r_ j, f_ j \in R$ resp. $r'_ j, f'_ j \in R'$ be the image of $b_ j, a_ j$. Then we see we are in the situation of Lemma 23.7.3 (with the ideal $I$ of that lemma equal to $\mathfrak m_ R$) and the lemma is proved. $\square$

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