Naive liftings of DG modules

TL;DR

This paper investigates conditions under which differential graded modules over polynomial and free extensions can be naively lifted to their base DG algebras, connecting to the Auslander-Reiten Conjecture.

Contribution

It establishes criteria for naive liftability of DG modules in polynomial and free extensions, linking homological vanishing conditions to liftability and summand properties.

Findings

If Ext_B^i(N, N)=0 for all i>0, then N is naively liftable to A.

N is a direct summand of a liftable DG B-module.

The work relates naive liftability to the Auslander-Reiten Conjecture.

Abstract

Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B=A[X_1,...,X_n] is a polynomial extension of A, where X_1,...,X_n are variables of positive degrees; or (b) A is a divided power DG R-algebra and B=A<X_1,...,X_n> is a free extension of A obtained by adjunction of variables X_1,...,X_n of positive degrees. In this paper, we study naive liftability of DG modules along the natural injection A-->B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that Ext_B^i(N, N)=0 for all i>0, then N is naively liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naive liftability of DG modules and the Auslander-Reiten Conjecture has been described.