Differential graded algebra explained

In mathematics, in particular in homological algebra, algebraic topology, and algebraic geometry, a differential graded algebra (or DG algebra, or DGA) is an algebraic structure often used to model topological spaces. In particular, it is a graded associative algebra with a chain complex structure that is compatible with the algebra structure. A noteworthy example is the de Rham alegbra of differential forms on a manifold. DGAs have also been used extensively in the development of rational homotopy theory.

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Definition

Let

A=oplus_i\inA_i

be a

graded algebra. We say that

is a differential graded algebra if it is equipped with a map

d\colonA\toA

of degree

-1

(homological grading) or degree

(cohomological grading). This map is a differential, giving

the structure of a chain complex or cochain complex (depending on the degree of

), and satisfies a graded Leibniz rule. In what follows, we will denote the "degree" of a homogeneous element

a\inA_i

|a|=i

Explicitly, the map

satisfies

A differential graded augmented algebra (or augmented DGA) is a DG algebra equipped with a DG morphism to the ground ring (the terminology is due to Henri Cartan).^[1]

Categorical Definition

One can define a DGA more abstractly using category theory. There is a category of chain complexes over

, often denoted

\operatorname{Ch}_k

, whose objects are chain complexes and whose morphisms are chain maps, i.e., maps compatible with the differential. We can define a tensor product on chain complexes by

(V ⊗ W)_n=oplus_i+j=nV_i ⊗ W_j

which makes

\operatorname{Ch}_k

into a symmetric monoidal category. Then, a DGA is simply a monoid object in the category of chain complexes.

Maps of DGAs

A linear map

f:(A_\bullet,d_A)\to(B_\bullet,d_B)

between graded vector spaces is said to be of degree n if

f(A_i)\subseteqB_i+n

for all

. When considering (co)chain complexes, we restrict our attention to chain maps, that is, those that satisfy

f\circd_A=d_B\circf

. The morphisms in the category of DGAs are those chain maps which are of degree 0.

Homology and Cohomology

Associated to any chain complex

A_\bullet

is its homology. Since

d\circd=0

, it follows that

\operatorname{im}(d:A_i+1\toA_i)

is a subset of

\operatorname{ker}(d:A_i\toA_i-1)

. Thus, we can form the quotient

H_i(A)=\operatorname{ker}(d:A_i\toA_i-1)/\operatorname{im}(d:A_i+1\toA_i)

This is called the

th homology group, and all together they form a graded vector space

H_\bullet(A)

, and in fact this is a graded algebra.

Similarly, one can associate to any cochain complex

A^\bullet

its cohomology, i.e., the

th cohomology group is given by

H^i(A)=\operatorname{ker}(d:Aⁱ\toAⁱ⁺¹)/\operatorname{im}(d:A^i-1\toAⁱ⁾

These once again form a graded vector space

H^\bullet(A)

Kinds of DGAs

Commutative Differential Graded Algebras

A commutative differential graded algebra (or CDGA) is a differential graded algebra,

(A_\bullet,d, ⋅ )

, which satisfies a graded version of commutativity. Namely,

a ⋅ b=(-1)^|a|b ⋅ a

for homogeneous elements

a\inA_i,b\inA_j

. Many of the DGAs commonly encountered in math happen to be CDGAs.

Differential Graded Lie Algebras

A differential graded Lie algebra (or DGLA) is a DG analogue of a Lie algebra. That is, it is a differential graded vector space,

(L_\bullet,d)

, together with an operation

[,]:L_i ⊗ L_j\toL_i+j

, satisfying graded analogues of the Lie algebra axioms. Let

[x,y]=-(-1)^ij[y,x]

An example of a DGLA is the de Rham algebra tensored with an ordinary Lie algebra

ak{g}

. DGLAs arise frequently in deformation theory where, over a field of characteristic 0, "nice" deformation problems are described by Maurer-Cartan elements of some suitable DGLA.

Formal DGAs

We say that a DGA

is formal if there exists a morphism of DGAs

A\toH_\bullet(A)

(respectively

A\toH^\bullet(A)

) that is a quasi-isomorphism.

Examples

Trivial DGAs

A=oplus_iA_i

has the structure of a DGA with trivial differential, i.e.,

d=0

. In particular, the homology/cohomology of any DGA forms a trivial DGA, since it is still a graded algebra.

The Free DGA

Let

be a (non-graded) vector space over a field

. The tensor algebra

T(V)

is defined to be the graded algebra

T(V)=oplus_i\geqT^i(V)=oplus_iV^⊗

where, by convention, we take

T^0(V)=k

. This vector space can be made into a graded algebra with the multiplication

T^i(V) ⊗ T^j(V)\toT^i+j(V)

given by the tensor product

⊗

. This is the free algebra 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

f:V\tok

be any linear map. Then, this extends uniquely to a derivation of

T(V)

of degree

-1

by the formula

d_f(v₁ ⊗ … ⊗ v_n)=

	n
\sum
	i=1

(-1)^i-1v₁ ⊗ … ⊗ f(v_i) ⊗ … ⊗ v_n

One can think of the minus signs on the right-hand side as occurring because

d_f

"jumps" over the elements

v_1,\ldots,v_i-1

, which are all of degree 1 in

T(V)

. This is commonly referred to as the Koszul sign rule.

One can extend this construction to differential graded vector spaces. Let

(V_\bullet,d_V)

be a differential graded vector space, i.e.,

d:V_i\toV_i-1

and

d²⁼⁰

. 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

T(V)

with a DGA structure which extends the DG structure on V. This is given by

d(v₁ ⊗ … ⊗ v_n)=

	n
\sum
	i=1

	\|v_1\|+\ldots+\|v_i-1\|
(-1)

v₁ ⊗ … ⊗ d_V(v_i) ⊗ … ⊗ v_n

This is analogous to the previous case, except that now elements of

are not restricted to degree 1 in

T(V)

, but can be of any degree.

The Free CDGA

Similar to the previous case, one can also construct a free CDGA on a vector space. Given a graded vector space

V_\bullet

, we define the free graded commutative algebra on it by

S(V)=\operatorname{Sym}\left(oplus_i=2kV_i\right) ⊗ wedge\left(oplus_i=2k+1V_i\right)

where

\operatorname{Sym}

denotes the symmetric algebra and

wedge

denotes the exterior algebra. If we begin with a DG vector space

(V_\bullet,d)

(either homologically or cohomologically graded), then we can extend

S(V)

such that

(S(V),d)

is a CDGA in a unique way.

de-Rham algebra

Let

be a manifold. Then, the differential forms on

, denoted by

\Omega^\bullet(M)

, naturally have the structure of a DGA. The grading is given by form degree, the multiplication is the wedge product, and the exterior derivative becomes the differential.

These have wide applications, including in derived deformation theory.^[2] See also de Rham cohomology.

Singular cohomology

The singular cohomology of a topological space with coefficients in

\Z/p\Z

is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence

0\to\Z/p\Z\to\Z/p^2\Z\to\Z/p\Z\to0

, and the product is given by the cup product. This differential graded algebra was used to help compute the cohomology of Eilenberg–MacLane spaces in the Cartan seminar.^[3] ^[4]

Koszul complex

One of the foundational examples of a differential graded algebra, widely used in commutative algebra 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.

Minimal DGAs

We say that a DGA

(l{A}^{\bullet,d, ⋅ )}

is minimal if

Minimal Models

Oftentimes, the important information contained in a chain complex is its cohomology. Thus, the natural maps to consider are those which induce isomorphisms on cohomology, but may not be isomorphisms on the entire DGA. We call such maps quasi-isomorphisms.

Every simply connected DGA admits a minimal model.

When a DGA admits a minimal model, it is unique up to a non-unique isomorphism.

Notes and References

Henri. Cartan. Henri Cartan. Sur les groupes d'Eilenberg-Mac Lane
H(\Pi,n)

. Proceedings of the National Academy of Sciences of the United States of America. 40. 1954. 6. 467–471. 10.1073/pnas.40.6.467. 16589508. 534072. free.
Web site: Manetti. Marco. Differential graded Lie algebras and formal deformation theory. live. https://web.archive.org/web/20130616054459/http://www1.mat.uniroma1.it/people/manetti/DT2011/ManSea.pdf. 16 Jun 2013.
Cartan. Henri. 1954–1955. DGA-algèbres et DGA-modules. Séminaire Henri Cartan. en. 7. 1. 1–9.
Cartan. Henri. 1954–1955. DGA-modules (suite), notion de construction. Séminaire Henri Cartan. en. 7. 1. 1–11.