Six model structures for DG-modules over DGAs: Model category theory in homological action

TL;DR

This paper develops six new projective-type model structures for differential graded modules over DGAs, introduces novel variants of the small object argument, and explores their theoretical and computational implications in homological algebra.

Contribution

It introduces new relative and mixed model structures for DG-modules over DGAs, expanding the toolkit beyond classical models, and provides methods for cofibrant approximation in these contexts.

Findings

Six model structures on DG-modules over DGAs are described.

New variants of the small object argument are developed.

Cofibrant approximations are constructed and analyzed in different model structures.

Abstract

In Part 1, we describe six projective-type model structures on the category of differential graded modules over a differential graded algebra A over a commutative ring R. When R is a field, the six collapse to three and are well-known, at least to folklore, but in the general case the new relative and mixed model structures offer interesting alternatives to the model structures in common use. The construction of some of these model structures requires two new variants of the small object argument, an enriched and an algebraic one, and we describe these more generally. In Part 2, we present a variety of theoretical and calculational cofibrant approximations in these model categories. The classical bar construction gives cofibrant approximations in the relative model structure, but generally not in the usual one. In the usual model structure, there are two quite different ways to lift cofibrant approximations from the level of homology modules over homology algebras, where they are classical projective resolutions, to the level of DG-modules over DG-algebras. The new theory makes model theoretic sense of earlier explicit calculations based on one of these constructions. A novel phenomenon we encounter is isomorphic cofibrant approximations with different combinatorial structure such that things proven in one avatar are not readily proven in the other.