Syzygetic ideals, regular sequences, and a question of Simis
This paper provides conditions for an ideal I of a noetherian local ring to be generated by a regular sequence. The first part investigates the link between that property and the freeness of \(Tor^ A_ 1(I,A/I)\) as an A/I module. More precisely, the authors prove the following result: Let (A,M,K) be a noetherian local ring, char(K)\(\neq 2\). If I is an ideal of A of finite homological dimension, then the following are equivalent: (1) I is generated by a regular sequence; (2) \(Tor^ A_ 1(I,A/I)\) is an A/I-free module. The second part gives a different proof and a globalization of a result by Micali and Roby; moreover, the following theorem is proved: The conditions: (i) I is a syzygetic ideal and the A/I module \(I/I^ 2\) has finite homological dimension; (ii) \(Tor^ A_ 1(I,A/I)\) is an A/I free module; (iii) height \(I>0\) are sufficient to guarantee that I is generated by a regular sequence.
scientific article; zbMATH DE number 4093613