nLab
model structure on dg-modules

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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

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

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

Properties

Let kk be a field of characteristic 0.

Write Ch (k)Ch^\bullet(k) for the category of chain complexes.

Let AMon(Ch (k))=dgAlg kA \in Mon(Ch^\bullet(k)) = dgAlg_k be a differential graded algebra.

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

Proposition

If AA is commutative, then AModA Mod is a closed symmetric monoidal category.

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

Proposition

If AA is cofibrant as an object in Ch (k)Ch^\bullet(k) then the transferred model structure along

(FU):AModUFCh (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 AModA 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)