 

Tobias Barthel, J.P. May, and Emily Riehl
Six model structures for DGmodules over DGAs: model category theory in homological action view print


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


Abstract
In Part 1, we describe six projectivetype 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 wellknown, 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 DGmodules over DGalgebras.
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.


Acknowledgements
The third author was supported by a National Science Foundation postdoctoral research fellowship DMS1103790.


Author information
Tobias Barthel:
Department of Mathematics, Harvard University, Cambridge, MA 02138
tbarthel@math.harvard.edu
J.P. May:
Department of Mathematics, University of Chicago, Chicago, IL 60637
may@math.uchicago.edu
Emily Riehl:
Department of Mathematics, Harvard University, Cambridge, MA 02138
eriehl@math.harvard.edu

