symmetric monoidal (∞,1)-category of spectra
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
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.
Let 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.
A left (resp. right) dg-module over is a left (resp. right) module over the monoid .
Spelling this out in components, it means the following:
A dg-module over is a chain complex equipped with a chain map
out of the tensor product of chain complexes, which satisfies the action property. Explicitly this means that
preserves degrees: for all and of homogeneous degrees and , respectively, then ;
for all , with in degree , then
for all , then
For two dg-modules over , then a homomorphism between them is a chain map
such that for all and then
This defines a category of dg-modules over .
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 category with weak equivalences.
More generally let be a dg-category.
A left (resp. right) dg-module over is a module over in the sense of enriched category theory. That is, it is a functor of dg-categories
(resp. ) where is the base commutative ring.
Note that when is a dg-category with a single object, a dg-module over is the same thing as a dg-module over the dg-algebra of endomorphisms of the unique object.
The shift of a dg-module over by an integer is the dg-module defined by composing with the shift endofunctor on .
By general enriched category theory, dg-modules themselves form a dg-category which we denote .
A dg-module is called representable if it is in the essential image of the dg-Yoneda embedding.
is called free if it is equivalent to a direct sum of shifts of representable dg-modules.
is called semi-free if it admits a filtration whose associated graded objects are free dg-modules.
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Last revised on July 19, 2021 at 21:27:05. See the history of this page for a list of all contributions to it.