Exterior powers and Tor-persistence

A commutative Noetherian ring R is said to be Tor-persistent if, for any finitely generated R-module M, the vanishing of ToriR(M,M) i?0 implies M has finite projective dimension. An open question Avramov, et al. asks whether such Tor-persistent. In this work, we exploit properties exterior powers modules and complexes provide several partial answers question; in particular, show that every local (R,m) with m3=0 As a consequence our methods, new proof Tachikawa Conjecture positively graded rings over field characteristic different from 2.