Finite Projective Dimension and the Vanishing of Ext_R(M, M)

Abstract

We show that for a finitely generated module M over a complete intersection R, the vanishing of for a certain number of consecutive values of i starting at n forces the projective dimension of M to be at most n − 1. In particular, if and only if the projective dimension of M over R is at most one. We also verify a conjecture of Auslander and Reiten for modules over commutative rings with certain typical behaviors, which augments the recent literature.

Keywords:

• Betti numbers
• Complete intersection
• Finite projective dimension
• Trivial self-extensions
• Vanishing ext

Mathematics Subject Classification:

• 13D25
• 13D02

Notes

Communicated by A. K. Singh.