nLab model structure on dg-modules

model category

Model structures

for ∞-groupoids

for $(\infty,1)$-sheaves / $\infty$-stacks

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Idea

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

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

For $A$ a fixed dg-algebra, a dg-module is then a module in $Ch(\mathcal{A})$ over $A$: a module over an algebra over an operad. Correspondingly the category $A Mod_{\mathcal{A}}$ 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 $Ch^\bullet(k)$ for the category of chain complexes.

Let $A \in Mon(Ch^\bullet(k)) = dgAlg_k$ be a differential graded algebra.

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

Proposition

If $A$ is commutative, then $A Mod$ 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 $Ch^\bullet(k)$ then the transferred model structure along

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

exists.

This appears as (Fresse, prop. 11.2.6).

(…)

Proposition

The homotopy cofibers in $A Mod$ 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

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

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

Revised on April 13, 2012 21:41:13 by Anonymous Coward (128.230.13.225)