In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
– particularly in
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
,
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, and
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
– a differential graded algebra (or DGA, or DG algebra) is an
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
often used to capture information about a
topological
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
or
geometric
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
space. Explicitly, a differential graded algebra is a
graded associative algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
with a
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 contained in the kernel o ...
structure that is compatible with the
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
structure.
In geometry, the
de Rham algebra
In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geome ...
of
differential forms
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 applications, ...
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 N ...
has the structure of a differential graded algebra, and it encodes the
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 adapte ...
of the manifold. In algebraic topology, the
singular cochains of a topological space form a DGA encoding 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 viewed ...
. Moreover, American mathematician
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the Graduate Center of the City University ...
developed a DGA to encode the
rational homotopy type of topological spaces.
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Definitions
Let
be a
-
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
, with product
, equipped with a map
of
degree (homologically graded) or degree
(cohomologically graded). We say that
is a differential graded algebra if
is a differential, giving
the structure of a
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 contained in the kernel o ...
or
cochain 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 contained in the kernel ...
(depending on the degree), and satisfies a graded
Leibniz rule. In what follows, we will denote the "degree" of a
homogeneous element
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, but ...
by
. Explicitly, the map
satisfies the conditions
Often one omits the differential and multiplication and simply writes
or
to refer to the DGA
.
A
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
between
graded vector spaces is said to be of degree ''n'' if
for all
. When considering (co)chain complexes, we restrict our attention to
chain maps, that is, maps of degree 0 that commute with the differentials
. The morphisms in the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
of DGAs are chain maps that are also
algebra homomorphisms.
Categorical Definition
One can also define DGAs more abstractly using
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
. There is a
category of chain complexes
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
over a
ring
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
, often denoted
, whose objects are chain complexes and whose morphisms are chain maps. We define the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of chain complexes
and
by
:
with differential
:
This operation makes
into a
symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...
. Then, we can equivalently define a differential graded algebra as a
monoid object
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms
* ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'',
* ''η ...
in
. Heuristically, it is an object in
with an associative and unital multiplication.
Homology and Cohomology
Associated to any chain complex
is its
homology. Since
, it follows that
is a subobject of
. Thus, we can form the quotient
:
This is called the
th homology group, and all together they form a graded vector space
. In fact, the homology groups form a DGA with zero differential. Analogously, one can define the
cohomology groups of a cochain complex, which also form a graded algebra with zero differential.
Every chain map
of complexes induces a map on (co)homology, often denoted
(respectively
). If this induced map is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
on all (co)homology groups, the map
is called a
quasi-isomorphism. In many contexts, this is the natural notion of equivalence one uses for (co)chain complexes. We say a morphism of DGAs is a quasi-isomorphism if the chain map on the underlying (co)chain complexes is.
Properties of DGAs
Commutative Differential Graded Algebras
A commutative differential graded algebra (or CDGA) is a differential graded algebra,
, which satisfies a graded version of commutativity. Namely,
:
for homogeneous elements
. Many of the DGAs commonly encountered in math happen to be CDGAs, like the de Rham algebra of differential forms.
Differential graded Lie algebras
A
differential graded Lie algebra
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applica ...
(or DGLA) is a differential graded analogue of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
. That is, it is a differential graded vector space,
, together with an operation
, satisfying the following graded analogues of the Lie algebra axioms.
An example of a DGLA is the de Rham algebra
tensored with a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
, with the bracket given by the
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of ...
of the
differential forms
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 applications, ...
and Lie bracket; elements of this DGLA are known as
Lie algebra–valued differential forms. DGLAs also arise frequently in the study of
deformations of algebraic structures where, over a field of characteristic 0, "nice" deformation problems are described by the space of
Maurer-Cartan elements of some suitable DGLA.
Formal DGAs
A (co)chain complex
is called formal if there is a chain map to its (co)homology
(respectively
), thought of as a complex with 0 differential, that is a quasi-isomorphism. We say that a DGA
is formal if there exists a morphism of DGAs
(respectively
) that is a quasi-isomorphism. This notion is important, for instance, when one wants to consider quasi-isomorphic chain complexes or DGAs as being equivalent, as in the
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
.
Examples
Trivial DGAs
Notice that any
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
has the structure of a DGA with trivial differential, i.e.,
. In particular, as noted above, the (co)homology of any DGA forms a trivial DGA, since it is a graded algebra.
The de-Rham algebra
Let
be 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 N ...
. Then, the
differential forms
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 applications, ...
on
, denoted by
, naturally have the structure of a (cohomologically graded) DGA. The graded vector space is
, where the grading is given by form degree. This vector space has a product, given by the
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of ...
, which makes it into a graded algebra. Finally, the exterior derivative
satisfies
and the graded Leibniz rule. In fact, the exterior product is graded-commutative, which makes the de Rham algebra an example of a CDGA.
Singular Cochains
Let
be a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. Recall that we can associate to
its complex of
singular cochains with coefficients in a
ring
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
, denoted
, whose cohomology is 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 viewed ...
of
. On
, one can define 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 commutative product opera ...
of cochains, which gives this cochain complex the structure of a DGA. In the case where
is a smooth manifold and
, the
de Rham theorem states that the singular cohomology is isomorphic to the
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 adapte ...
and, moreover, the cup product and exterior product of differential forms induce the same operation on cohomology.
Note, however, that while the cup product induces a graded-commutative operation on cohomology, it is not graded commutative directly on cochains. This is an important distinction, and the failure of a DGA to be commutative is referred to as the "commutative cochain problem". This problem is important because if, for any topological space
, one can associate a commutative DGA whose cohomology is the singular cohomology of
over
, then this CDGA determines the
-homotopy type of
.
The Free DGA
Let
be a (non-graded) vector space over a
field . The
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
is defined to be the graded algebra
:
where, by convention, we take
. This vector space can be made into a graded algebra with the multiplication
given by the tensor product
. This is the
free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the ...
on
, and can be thought of as the algebra of all non-commuting polynomials in the elements of
.
One can give the tensor algebra the structure of a DGA as follows. Let
be any linear map. Then, this extends uniquely to a
derivation of
of degree
(homologically graded) by the formula
:
One can think of the minus signs on the right-hand side as coming from "jumping" the map
over the elements
, which are all of degree 1 in
. This is commonly referred to as the
Koszul sign rule.
One can extend this construction to differential graded vector spaces. Let
be a differential graded vector space, i.e.,
and
. Here we work with a homologically graded DG vector space, but this construction works equally well for a cohomologically graded one. Then, we can endow the tensor algebra
with a DGA structure which extends the DG structure on V. The differential is given by
:
This is similar to the previous case, except that now the elements of
can have different degrees, and
is no longer graded by the number of tensor products but instead by the sum of the degrees of the elements of
, i.e.,
.
The Free CDGA
Similar to the previous case, one can also construct the free CDGA. Given a graded vector space
, we define the free graded commutative algebra on it by
:
where
denotes the
symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
and
denotes the
exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
. If we begin with a DG vector space
(either homologically or cohomologically graded), then we can extend
to
such that
is a CDGA in a unique way.
Models for DGAs
As mentioned previously, oftentimes one is most interested in the (co)homology of a DGA. As such, the specific (co)chain complex we use is less important, as long as it has the right (co)homology. Given a DGA
, we say that another DGA
is a model for
if it comes with a
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
DGA morphism
that is a quasi-isomorphism.
Minimal Models
Since one could form arbitrarily large (co)chain complexes with the same cohomology, it is useful to consider the "smallest" possible model of a DGA. We say that a DGA
is a minimal if it satisfies the following conditions.
Note that some conventions, often used in algebraic topology, additionally require that
be simply connected, which means that
and
. This condition on the 0th and 1st degree components of
mirror the (co)homology groups of a
simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoi ...
.
Finally, we say that
is a minimal model for
if it is both minimal and a model for
. The fundamental theorem of minimal models states that if
is simply connected then it admits a minimal model, and that if a minimal model exists it is unique up to (non-unique) isomorphism.
The Sullivan minimal model
Minimal models were used with great success by Dennis Sullivan in his work on
rational homotopy theory. Given a
simplicial complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
, one can define a rational analogue of the (real) de Rham algebra: the DGA
of "piecewise polynomial" differential forms with
-coefficients. Then,
has the structure of a CDGA over the field
, and in fact the cohomology is isomorphic to the singular cohomology of
. In particular, if
is a simply connected topological space then
is simply connected as a DGA, thus there exists a minimal model.
Moreover, since
is a CDGA whose cohomology is the singular cohomology of
with
-coefficients, it is a solution to the commutative cochain problem. Thus, if
is a simply connected
CW complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
with finite dimensional rational homology groups, the minimal model of the CDGA
captures entirely the rational homotopy type of
.
See also
*
Differential graded Lie algebra
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applica ...
*
Rational homotopy theory
*
Homotopy associative algebra
Notes
References
*
*
*
*
*
*
*
{{refend
Algebras
Homological algebra
Algebraic topology
Algebraic geometry
Commutative algebra
Differential algebra