Categorical properties of reduction functors over non-positive DG-rings

TL;DR

This paper systematically studies the categorical properties of reduction and coreduction functors over non-positive DG-rings and applies these findings to derive new descent results for Ext and Tor vanishing over commutative noetherian rings.

Contribution

It provides a detailed analysis of the categorical behavior of reduction functors over non-positive DG-rings and introduces new descent results for Ext and Tor vanishings.

Findings

Categorical properties of reduction and coreduction functors are characterized.

New descent results for Ext and Tor vanishings are established.

The study links DG-ring functors to classical homological algebra over rings.

Abstract

Given a non-positive DG-ring $A$, associated to it are the reduction and coreduction functors $F(-) = \mathrm{H}^0(A)\otimes^{\mathrm{L}}_A -$ and $G(-) = \mathrm{R}\operatorname{Hom}_A(\mathrm{H}^0(A),-)$, considered as functors $\operatorname{\mathsf{D}}(A) \to \operatorname{\mathsf{D}}(\mathrm{H}^0(A))$, as well as the forgetful functor $S:\operatorname{\mathsf{D}}(\mathrm{H}^0(A)) \to \operatorname{\mathsf{D}}(A)$. In this paper we carry a systematic study of the categorical properties of these functors. As an application, a new descent result for vanishing of $\operatorname{Ext}$ and $\operatorname{Tor}$ over ordinary commutative noetherian rings is deduced.