  A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any
finitely generated $R$-module $M$, the vanishing of
$\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ 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 $(R,\mathfrak{m})$ with
$\mathfrak{m}^3=0$ 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.
