New York Journal of Mathematics
Volume 20 (2014) 1077-1159


Tobias Barthel, J.P. May, and Emily Riehl

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

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Published: December 1, 2014
Keywords: Differential homological algebra, differential torsion products, Eilenberg-Moore spectral sequence, Massey product, model category theory, projective resolution, projective class, relative homological algebra
Subject: 16E45, 18G25, 18G55, 55S30, 55T20, 55U35

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.


The third author was supported by a National Science Foundation postdoctoral research fellowship DMS-1103790.

Author information

Tobias Barthel:
Department of Mathematics, Harvard University, Cambridge, MA 02138

J.P. May:
Department of Mathematics, University of Chicago, Chicago, IL 60637

Emily Riehl:
Department of Mathematics, Harvard University, Cambridge, MA 02138