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model structure on dg-modules

Idea

For $𝒜$ an abelian category, a dg-algebra in $𝒜$ is a monoid in the category of chain complexes $\mathrm{Ch}(𝒜)$.

Equivalently this is an algebra over an operad over the associative operad in $\mathrm{Ch}(𝒜)$.

For $A$ a fixed dg-algebra, a dg-module is then a module in $\mathrm{Ch}(𝒜)$ over $A$: a module over an algebra over an operad. Correspondingly the category $A{\mathrm{Mod}}_{𝒜}$ of all $A$-modules carries a model structure on modules over an algebra over an operad. This is a model structure on dg-modules

Properties

Let $k$ be a field of characteristic 0.

Write ${\mathrm{Ch}}^{•}(k)$ for the category of chain complexes.

Let $A\in \mathrm{Mon}({\mathrm{Ch}}^{•}(k))={\mathrm{dgAlg}}_k$ be a differential graded algebra.

Write $A\mathrm{Mod}$ for the category of dg-modules over $A$: modules in ${\mathrm{Ch}}^{•}(k)$ over $A$:

Proposition

If $A$ is commutative, then $A\mathrm{Mod}$ is a closed symmetric monoidal category.

This is a standard construction, for instance Bernstein, p. 5.

Proposition

If $A$ is cofibrant as an object in ${\mathrm{Ch}}^{•}(k)$ then the transferred model structure along

$$(F⊣U):A\mathrm{Mod}\stackrel{\stackrel{F}{\leftarrow }}{\underset{U}{\to }}{\mathrm{Ch}}^{•}(k)$$(F \dashv U) : A Mod \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Ch^\bullet(k)

exists.

This appears as (Fresse, prop. 11.2.6).

(…)

Proposition

The homotopy cofibers in $A\mathrm{Mod}$ are given by the usual mapping cones of dg-modules in the model structure on chain complexes.

This follows from (Bernstein, 10.3.5).

References

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).