Differential graded algebra

This article is about homological algebra. For the algebraic study of differential equations, see Differential algebra.

In mathematics, in particular in homological algebra, algebraic topology, and algebraic geometry, a differential graded algebra (or DG-algebra, or DGA) is a graded associative algebra with an added chain complex structure that respects the algebra structure. In particular, DGAs have applications in rational homotopy theory.

Definition

Let $A=\bigoplus _{i}A_{i}$ be a graded algebra. We say that $A$ is a differential graded algebra if it is equipped with a map $d\colon A\to A$ of degree 1 (cochain complex convention) or degree −1 (chain complex convention). This map is a differential, giving $A$ the structure of a chain complex or cochain complex (depending on the degree of $d$), and satisfies graded Leibniz rule. Explicitly, the map $d$ satisfies

1. $d\circ d=0$, often written $d^{2}=0$.
2. $d(a\cdot b)=(da)\cdot b+(-1)^{\deg(a)}a\cdot (db)$, where $\operatorname {deg} (a)=i$ is the degree of the homogeneous element $a\in A_{i}$.

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).

Categorical Definition

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. A DG morphism between DG-algebras is a graded algebra homomorphism that respects the differential d.

Examples of DG-algebras

Trivial DGAs

First, we note that any graded algebra $A=\bigoplus _{i}A_{i}$ has the structure of a DGA with trivial differential, $d=0$.

Tensor algebra

The tensor algebra is a DGA with differential similar to that of the Koszul complex. For a vector space $V$ over a field $K$ there is a graded vector space $T(V)$ defined as

$T(V)=\bigoplus _{i\geq 0}T^{i}(V)=\bigoplus _{i\geq 0}V^{\otimes i}$

where $V^{\otimes 0}=K$.

If $e_{1},\ldots ,e_{n}$ is a basis for $V$ there is a differential $d$ on the tensor algebra defined component-wise

$d:T^{k}(V)\to T^{k-1}(V)$

sending basis elements to

$d(e_{i_{1}}\otimes \cdots \otimes e_{i_{k}})=\sum _{1\leq j\leq k}e_{i_{1}}\otimes \cdots \otimes d(e_{i_{j}})\otimes \cdots \otimes e_{i_{k}}$

In particular we have $d(e_{i})=(-1)^{i}$ and so

$d(e_{i_{1}}\otimes \cdots \otimes e_{i_{k}})=\sum _{1\leq j\leq k}(-1)^{i_{j}}e_{i_{1}}\otimes \cdots \otimes e_{i_{j-1}}\otimes e_{i_{j+1}}\otimes \cdots \otimes e_{i_{k}}$

De-Rham algebra

Let $M$ be a manifold. Then, the differential forms on $M$, 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. See also de Rham cohomology.

Singular cohomology

The singular cohomology of a topological space with coefficients in $\mathbb {Z} /p\mathbb {Z}$ is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence $0\to \mathbb {Z} /p\mathbb {Z} \to \mathbb {Z} /p^{2}\mathbb {Z} \to \mathbb {Z} /p\mathbb {Z} \to 0$, 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.

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.

Other facts about DG-algebras

The homology $H_{*}(A)=\ker(d)/\operatorname {im} (d)$ of a DG-algebra $(A,d)$ is a graded algebra. The homology of a DGA-algebra is an augmented algebra.

See also

• Differential graded Lie algebra
• Rational homotopy theory
• Homotopy associative algebra
• Differential graded category
• Differential graded scheme
• Differential graded module

References

• Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43583-9, see sections V.3 and V.5.6

• Griffiths, Phillip; Morgan, John (2013), Rational Homotopy Theory and Differential Forms, New York, Heidelberg, Dordrecht, London: Birkhäuser, ISBN 978-1-4614-8467-7, see Chapters 10-12.