Let (A,m) be a Noetherian local ring. Let I ⊂ J ⊂ 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. See Lemma Tag 09PW.

Here is the graph of this lemma