Differential graded algebra

Algebraic structure in homological algebra

In mathematics, in particular in homological algebra, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.

Definition

A differential graded algebra (or DG-algebra for short) A is a graded algebra equipped with a map ${\displaystyle d\colon A\to A}$ that has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions:

1. ${\displaystyle d\circ d=0}$.
   This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).
2. ${\displaystyle d(a\cdot b)=(da)\cdot b+(-1)^{\deg(a)}a\cdot (db)}$, where ${\displaystyle \operatorname {deg} }$ is the degree of homogeneous elements.
   This says that the differential d respects the graded Leibniz rule.

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.

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).^[1]

Warning: some sources use the term DGA for a DG-algebra.

Examples of DG-algebras

Tensor algebra

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

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

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

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

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

sending basis elements to

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

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

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

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.

De-Rham algebra

Differential forms on a manifold, together with the exterior derivation and the exterior product form a DG-algebra. 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 ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence ${\displaystyle 0\to \mathbb {Z} /p\mathbb {Z} \to \mathbb {Z} /p^{2}\mathbb {Z} \to \mathbb {Z} /p\mathbb {Z} \to 0}$, 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.^[3][4]

Other facts about DG-algebras

• The homology ${\displaystyle H_{*}(A)=\ker(d)/\operatorname {im} (d)}$ of a DG-algebra ${\displaystyle (A,d)}$ 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

References

1. Cartan, Henri (1954). "Sur les groupes d'Eilenberg-Mac Lane H ( Π , n ) {\displaystyle H(\Pi ,n)} ". Proceedings of the National Academy of Sciences of the United States of America. 40 (6): 467–471. doi:10.1073/pnas.40.6.467. PMC 534072. PMID 16589508.
2. Manetti, Marco. "Differential graded Lie algebras and formal deformation theory" (PDF). Archived (PDF) from the original on 16 Jun 2013.
3. Cartan, Henri (1954–1955). "DGA-algèbres et DGA-modules". Séminaire Henri Cartan. 7 (1): 1–9.
4. Cartan, Henri (1954–1955). "DGA-modules (suite), notion de construction". Séminaire Henri Cartan. 7 (1): 1–11.

• 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