2020/07/17 by Lyle, Justin, Montaño, Jonathan, Sather-Wagstaff, Keri · 1 citation
#Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2007.09174
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) for i≫ 0 implies M has finite projective dimension. An open question of Avramov, et. al. asks whether any such R is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring (R,\mathfrakm) with \mathfrakm3=0 is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2.