nLab model structure on dg-modules

Idea

The category of dg-modules over a dg-algebra, or more generally a dg-category, admits dg-model structures which present the derived dg-category.

In the case of dg-algebras in an abelian category $\mathcal{A}$, dg-modules are the same as modules over algebras over the associative operad in $Ch(\mathcal{A})$. These admit model structures as described in model structure on modules over an algebra over an operad, transferred from the model structures on chain complexes in $\mathcal{A}$.

More generally there are model structures on dg-modules over a dg-category, analogous to model structures on simplicial presheaves.

Definitions

Theorem

(projective model structure)

There is a cofibrantly generated model structure on the category of dg-modules, called the projective model structure, where

1. the weak equivalences are object-wise quasi-isomorphisms of chain complexes,

2. the fibrations are object-wise epimorphisms.

Moreover, this is a dg-model structure.

Theorem

(injective model structure)

There is a model structure on the category of dg-modules, called the injective model structure, where

1. the weak equivalences are object-wise quasi-isomorphisms of chain complexes,

2. and the cofibrations are object-wise monomorphisms.

See (Toën 04, section 3) and (Keller 06, theorem 3.2).

These model structures present the derived dg-category.

Related concepts

• model structure on chain complexes

• module structure on modules in a monoidal model category?

• model structure on dg-algebras

• model structure on dg-comodules

References

Section 3 of

• Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, arXiv:math/0408337.

Paragraph 3.2 of

• Bernhard Keller, On differential graded categories, arXiv:math/0601185.

For dg-algebras

For the case of dg-algebras, see the references below.

A general account is around section 11.2.5 of

• Benoit Fresse, Modules over operads and functors Springer Lecture Notes in Mathematics, (2009) (pdf)

and in section 3 of

• Vladimir Hinich, Homological algebra of homotopy algebras Communications in algebra, 25(10). 3291-3323 (1997)(arXiv:q-alg/9702015, Erratum (arXiv:math/0309453))

The homotopy category and triangulated category of dg-modules is discussed for instance also in

• Joseph Bernstein, DG-modules and equivariant cohomology (pdf).

See also

• Tobias Barthel, Peter May, Emily Riehl, Six model structures for DG-modules over DGAs: model category theory in homological action, New York J. Math. 20 (2014) 1077–1159 (pdf)

Discussion as a model for rational parameterized stable homotopy theory is due to

• Vincent Braunack-Mayer, Rational parameterized stable homotopy theory, Zurich 2018