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dg-module

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Algebra

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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 AA 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 AA is a left (resp. right) module over the monoid AA.

Spelling this out in components, it means the following:

A dg-module over AA is a chain complex VV equipped with a chain map

ρ:AVV \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. ρ\rho preserves degrees: for all aAa \in A and vVv \in V of homogeneous degees |a|{\vert a \vert} and |v|{\vert v \vert}, respectively, then |ρ(a,v)|=|a|+|v|\vert \rho(a,v)\vert = {\vert a \vert} + {\vert v \vert};

  2. for all aAa \in A, vVv \in V with aa in degree |a|\vert a \vert, then

    d V(ρ(a,v))=ρ(d Aa,v)+(1) |a|ρ(a,d Vv) d_V (\rho(a,v)) = \rho(d_A a, v) + (-1)^{\vert a\vert} \rho(a,d_V v)
  3. for all a,bAa,b \in A, vVv \in V then

    ρ(a,ρ(b,v))=ρ(ab,v). \rho(a,\rho(b,v)) = \rho(a \cdot b,v) \,.

For (V i,ρ i)(V_i, \rho_i) two dg-modules over AA, then a homomorphism between them is a chain map

ϕ:V 1V 2 \phi \;\colon\; V_1 \longrightarrow V_2

such that for all aAa \in A and vVv \in V then

ρ 2(a,ϕ(v))=ϕ(ρ 1(a,v)). \rho_2(a, \phi(v)) = \phi(\rho_1(a,v)) \,.

This defines a category AModA Mod of dg-modules over AA.

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 AModA Mod a category with weak equivalences.

Over dg-categories

More generally let TT be a dg-category.

Definition

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

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

(resp. M:Tdgmod kM : T \longrightarrow dg-mod_k) where kk is the base commutative ring.

Note that when TT is a dg-category with a single object, a dg-module over TT 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 MM over TT by an integer nn is the dg-module M[n]M[n] defined by composing MM with the shift endofunctor [n]-[n] on dgmod kdg-mod_k.

Properties

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

Definition

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

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

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

References

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Last revised on March 7, 2017 at 14:32:29. See the history of this page for a list of all contributions to it.