nLab dg-module

Idea

The notion of dg-module over a dg-algebra is the specialization of module over a monoid to the category of chain complexes. More generally one can consider dg-modules over dg-algebras with many objects, i.e. dg-categories.

Definition

Over dg-algebras

Let $A$ be a dg-algebra in an abelian category, hence a monoid in the symmetric monoidal category of chain complexes with respect to the tensor product of chain complexes.

Definition

A left (resp. right) dg-module over $A$ is a left (resp. right) module over the monoid $A$.

Spelling this out in components, it means the following:

A dg-module over $A$ is a chain complex $V$ equipped with a chain map

$$\rho \;\colon\; A \otimes V \longrightarrow V$$

out of the tensor product of chain complexes, which satisfies the action property. Explicitly this means that

1. $\rho$ preserves degrees: for all $a \in A$ and $v \in V$ of homogeneous degrees ${\vert a \vert}$ and ${\vert v \vert}$, respectively, then $\vert \rho(a,v)\vert = {\vert a \vert} + {\vert v \vert}$;

2. for all $a \in A$, $v \in V$ with $a$ in degree $\vert a \vert$, then

   $$d_V (\rho(a,v)) = \rho(d_A a, v) + (-1)^{\vert a\vert} \rho(a,d_V v)$$

3. for all $a,b \in A$, $v \in V$ then

   $$\rho(a,\rho(b,v)) = \rho(a \cdot b,v) \,.$$

For $(V_i, \rho_i)$ two dg-modules over $A$, then a homomorphism between them is a chain map

$$\phi \;\colon\; V_1 \longrightarrow V_2$$

such that for all $a \in A$ and $v \in V$ then

$$\rho_2(a, \phi(v)) = \phi(\rho_1(a,v)) \,.$$

This defines a category $A Mod$ of dg-modules over $A$.

We say that a morphism of dg-modules is a quasi-isomorphism if its underlying chain map is, hence if it induces an isomorphism on chain homology. This makes $A Mod$ a category with weak equivalences.

Over dg-categories

More generally let $T$ be a dg-category.

Definition

A left (resp. right) dg-module over $T$ is a module over $T$ in the sense of enriched category theory. That is, it is a functor of dg-categories

$$M : T^{op} \longrightarrow dg-mod_k$$

(resp. $M : T \longrightarrow dg-mod_k$) where $k$ is the base commutative ring.

Note that when $T$ is a dg-category with a single object, a dg-module over $T$ is the same thing as a dg-module over the dg-algebra of endomorphisms of the unique object.

Definition

The shift of a dg-module $M$ over $T$ by an integer $n$ is the dg-module $M[n]$ defined by composing $M$ with the shift endofunctor $-[n]$ on $dg-mod_k$.

Properties

By general enriched category theory, dg-modules themselves form a dg-category which we denote $dg-mod_T$.

Definition

A dg-module $M \in dg-mod_T$ is called representable if it is in the essential image of the dg-Yoneda embedding.

$M$ is called free if it is equivalent to a direct sum of shifts of representable dg-modules.

$M$ is called semi-free if it admits a filtration whose associated graded objects are free dg-modules.

Related concepts

• minimal dg-module

• dg-comodule

References

• Bertrand Toen, section 3 in The homotopy theory of dg-categories and derived Morita theory, arXiv:math/0408337.

Paragraph 2.2 of

• Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, arXiv:1402.7364.