On certain DG-algebra resolutions

TL;DR

This paper explores DG-algebra resolutions of certain non-Gorenstein rings, establishing conditions for finite projective dimension and unbounded Bass numbers, and provides new proofs for known results in Cohen-Macaulay module theory.

Contribution

It introduces classes of non-Gorenstein rings with specific Ext vanishing properties and offers an alternative proof for Takahashi's result on Cohen-Macaulay modules using DG-algebra techniques.

Findings

Certain non-Gorenstein rings satisfy Ext vanishing implies finite projective dimension.

Over these rings, infinite injective dimension leads to unbounded Bass numbers.

DG-algebra structures on minimal resolutions are key to the proofs.

Abstract

In this paper we give several classes of Non-Gorenstein local rings $A$ which satisfy the property that $\text{Ext}^i_A(M, A) = 0$ for $i \gg 0$ then $\text{projdim}_A M$ is finite. We also show that if $\text{injdim}_A M = \infty$ then over such rings the bass-numbers of $M$ (with respect to $\mathfrak{m}$) are unbounded. When $A$ is a hypersurface ring we give an alternate proof of a result due to Takahashi regarding thick subcategories of the stable category of maximal Cohen-Macaulay $A$-modules. This result of Takahashi implies some results due to Avramov, Buchweitez, Huneke and Wiegand. The technique used to prove our results is that the minimal resolution of the relevant rings have an appropriate DG-algebra structure (philosophically this technique is due to Nasseh, Ono, and Yoshino).