A dg-algebra, or differential graded algebra, is equivalently
For the case of chain complexes we also speak of chain algebras.
For the case of cochain complexes we also speak of cochain algebras.
that the standard tensor product on (co)chain complexes is given by
with the differential .
that is unital and associative in the obvious sense.
This implies that a dg-algebra is, more concretely, a graded algebra equipped with a linear map with the property that
for homogeneous of degree , the element is of degree (for monoids in cochain complexes) or of degree (for monoids in chain complexes)
for all with homogeneous of degree k the graded Leibniz rule holds
A pre-graded algebra- (pre-ga) or -graded algebra is a pre-gvs, , together with an algebra multiplication satisfying for any . The relevant morphisms are pre-gvs morphisms which respect the multiplication. This gives a category .
An augmentation of a pre-ga, , is a homomorphism . The augmentation ideal of is and will also be denoted . The pair is called an augmented pre-ga.
A morphism of augmented pre-gas is a homomorphism (thus of degree zero) such that . The resulting category will be written .
If , are two pre-gas, then is a pre-ga with
for homogeneous , .
If are augmentations of and respectively, then is an augmentation of .
Let be a pre-ga. An (algebra) derivation of degree is a linear map such that
for homogeneous .
A derivation of an augmented algebra, , is an algebra derivation which, in addition, satisfies .
Let be the vector space of derivations of degree of , then is a pre-gvs.
In the case of upper gradings, an element of sends into .
A differential on an (augmented) pre-ga is a derivation of the (augmented) algebra of degree -1 such that .
The pair is called a pre-differential graded algebra (pre-dga). If is augmented, then will be called an augmented pre-dga .
If and are pre-dgas, then , with the conventions already noted, is one as well.
A morphism of pre-dgas (or pre--dgas) is a morphism which is both of pre-gdvs and of pre-gas (with as well if used). This gives categories and .
A pre-ga is said to be graded commutative if for each pair, , of elements of of homogeneous degree.
Commutativity is preserved by tensor product.
We get obvious full subcategories and corresponding to the case with differentials.
A cdga is a negatively graded pre-cdga (in upper grading),
There is an augmented variant, of course. These definitions give categories , etc.
An cdga is -connected (resp. cohomologically -connected if for , (resp. for ). This gives subcategories and .
A filtration on a pre-ga, , is a filtration on , so that , (and, if is differential, also ).
Let be an augmented pre-ga and denote by
the iterated multiplication. The decreasing word length filtration, is given by:
is the space of indecomposables of A.
If is an augmented pre-dga, is stable under and we get is a differential on and hence we get a functor
Given a pre-gvs, , the tensor algebra generated by is given by .
The augmentation sends to 0. is given the tensor product grading, and the multiplication is given by the tensor product.
If is a pre-ga and , a morphism to the underlying pre-gvs of , there is a unique extension , which is a morphism of pre-gvs.
This is the tensor product of the exterior algebra on the odd elements and the symmetric algebra on the even ones:
It satisfies .
If is a pre-cga, any morphism, , to the underlying pre-gvs of , has a unique extension to a pre-cga morphism .
If is a homogeneous basis for , and may be written and respectively.
is a non-commutative polynomial algebra,
is a commutative polynomial algebra.
On (resp. ) write
where is the subspace generated by all with . Then , resp. and ).
If is a pre-cdga, which is free as a pre-cga on a fixed , then is the sum of derivations defined by the condition . There is an isomorphism between and , which identifies with . The derivation (resp. ) is called the linear part (resp. quadratic part) of .
If and are two cdgas, their (categorical) sum (i.e. coproduct) is their tensor product, , whilst their product is the ‘direct sum’, .
Given a permutation of a graded object , the Koszul sign, is defined by
in . We note that although we write , does not suffice to define it as it depends also on the degrees of the various .
Baues (in his book on Algebraic Homotopy) has suggested using the terminology chain algebra for positively graded differential algebras and cochain algebras for the negatively graded ones. This seems to be a very useful convention.
The monoidal Dold-Kan correspondence effectively identifies non-negatively graded chain complex algebras with simplicial algebras, and non-negatively graded cochain complex algebras with cosimplicial algebras.
Since cosimplicial algebras have a fundamental interpretation dual to ∞-space, as described at ∞-quantity, this can be understood as explaining the great role differential graded algebras are playing in various context, suchh as notably in
A dga is homologically smooth if as a dg-bimodule over itself it has a bounded resolution by finitely generated projective dg-bimodules.
A dg-algebra is a formal dg-algebra if there exists a morphism