On certain DG-algebra resolutions

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).