In
mathematics, in particular
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
and
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, a differential graded algebra is a
graded associative algebra with an added
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
structure that respects the
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
structure.
__TOC__
Definition
A differential graded algebra (or DG-algebra for short) ''A'' is a graded algebra equipped with a map
which has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions:
A more succinct way to state the same definition is to say that a DG-algebra is a
monoid object in the
monoidal category of chain complexes
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
.
A DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential ''d''.
A differential graded
augmented algebra (also called a DGA-algebra,
an augmented DG-algebra or simply a DGA) is a DG-algebra equipped with a DG morphism to the ground
ring (the terminology is due to
Henri Cartan).
''Warning:'' some sources use the term ''DGA'' for a DG-algebra.
Examples of DG-algebras
Tensor algebra
The
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of bein ...
is a DG-algebra with differential similar to that of the
Koszul complex. For a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
over a
field there is a
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be ...
defined as
:
where
.
If
is a
basis for
there is a differential
on the tensor algebra defined component-wise
:
sending basis elements to
:
In particular we have
and so
:
Koszul complex
One of the foundational examples of a differential graded algebra, widely used in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
and
algebraic geometry, is the Koszul complex. This is because of its wide array of applications, including constructing
flat resolutions of complete intersections, and from a
derived perspective, they give the derived algebra representing a derived critical locus.
De-Rham algebra
Differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
, together with the
exterior derivation and the
exterior product form a DG-algebra. These have wide applications, including in
derived deformation theory. See also
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
.
Singular cohomology
*The
singular cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
with coefficients in
is a DG-algebra: the differential is given by the
Bockstein homomorphism associated to the
short exact sequence , and the product is given by the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
. This differential graded algebra was used to help compute the cohomology of
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this nam ...
s in the Cartan seminar.
Other facts about DG-algebras
* The ''
homology''
of a DG-algebra
is a graded algebra. The homology of a DGA-algebra is an
augmented algebra.
See also
*
Homotopy associative algebra
*
Differential graded category
*
Differential graded Lie algebra
*
Differential graded scheme
*
Differential graded module
In algebra, a differential graded module, or dg-module, is a \mathbb-graded module together with a differential; i.e., a square-zero graded endomorphism of the module of degree 1 or −1, depending on the convention. In other words, it is a chain c ...
References
* {{Citation , last1=Manin , first1=Yuri Ivanovich , author1-link=Yuri Ivanovich Manin , last2=Gelfand , first2=Sergei I. , title=Methods of Homological Algebra , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , isbn=978-3-540-43583-9 , year=2003, see sections V.3 and V.5.6
Algebras
Homological algebra
commutative algebra
Differential algebra