Homology over trivial extensions of commutative DG algebras

TL;DR

This paper explores conditions on the Koszul complex of noetherian local rings that ensure infinite non-vanishing of Tor groups for modules with infinite projective dimension, using differential graded algebra techniques.

Contribution

It introduces new conditions derived from trivial extensions of commutative DG algebras that guarantee infinite Tor groups for certain modules.

Findings

Conditions on Koszul complexes imply infinite Tor groups

Results connect DG algebra extensions with homological properties

Provides criteria for non-vanishing of Tor in specific algebraic contexts

Abstract

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.