Diagonal tensor algebra and naive liftings

TL;DR

This paper investigates the properties of naive liftings of DG modules using tensor algebra, providing characterizations and applications in homological algebra, especially concerning Ext vanishing conditions.

Contribution

It offers new characterizations of naive liftability of DG modules via tensor algebra and addresses an open question in the field.

Findings

Characterizations of naive liftability under Ext vanishing

Application to affirmatively answer an open question

Deep analysis of naive lifting property using diagonal tensor algebra

Abstract

The notion of naive lifting of DG modules was introduced by the authors in [16,17] for the purpose of studying problems in homological commutative algebra that involve self-vanishing of Ext. Our goal in this paper is to deeply study the naive lifting property using the tensor algebra of the shift of the diagonal ideal (or, diagonal tensor algebra, as is phrased in the title of this paper). Our main result provides several characterizations of naive liftability of DG modules under certain Ext vanishing conditions. As an application, we affirmatively answer [19, Question 4.10] under the same assumptions.