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 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.

Definition

Let $A=\bigoplus _{i\in \mathbb {Z} }A_{i}$ be a $\mathbb {Z}$ 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 $-1$ (homological grading) or degree $1$ (cohomological grading). This map is a differential, giving $A$ the structure of a chain complex or cochain complex (depending on the degree of $d$), and satisfies a graded Leibniz rule. In what follows, we will denote the "degree" of a homogeneous element $a\in A_{i}$ by $|a|=i$.

Explicitly, the map $d$ satisfies

1. $d\circ d=0$, often written $d^{2}=0$.
2. $d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot (db)$.

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

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

$(V\otimes W)_{n}=\bigoplus _{i+j=n}V_{i}\otimes W_{j}$

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 $f(A_{i})\subseteq B_{i+n}$ for all $i$. When considering (co)chain complexes, we restrict our attention to chain maps, that is, those that satisfy $f\circ d_{A}=d_{B}\circ f$. 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 $d\circ d=0$, it follows that $\operatorname {im} (d:A_{i+1}\to A_{i})$ is a subset of $\operatorname {ker} (d:A_{i}\to A_{i-1})$. Thus, we can form the quotient

$H_{i}(A)=\operatorname {ker} (d:A_{i}\to A_{i-1})/\operatorname {im} (d:A_{i+1}\to A_{i})$

This is called the $i$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 $i$th cohomology group is given by

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

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, $(A_{\bullet },d,\cdot )$, which satisfies a graded version of commutativity. Namely,

$a\cdot b=(-1)^{|a||b|}b\cdot a$

for homogeneous elements $a\in A_{i},b\in A_{j}$. 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 $[,]:L_{i}\otimes L_{j}\to L_{i+j}$, satisfying graded analogues of the Lie algebra axioms. Let $[x,y]=-(-1)^{ij}[y,x]$

1. Graded skew-symmetry: $[x,y]=-(-1)^{|x||y|}[y,x]$ for homogeneous elements $x\in L_{i},y\in L_{j}$.
2. Graded Jacobi identity: $(-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0$.
3. Graded Leibniz rule: $d[x,y]=[dx,y]+(-1)^{|x|}[x,dy]$.

An example of a DGLA is the de Rham algebra tensored with an ordinary Lie algebra ${\mathfrak {g}}$. 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 $A$ is formal if there exists a morphism of DGAs $A\to H_{\bullet }(A)$ (respectively $A\to H^{\bullet }(A)$) that is a quasi-isomorphism.

Examples

Trivial DGAs

First, we note that any graded algebra $A=\bigoplus _{i}A_{i}$ 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 $V$ be a (non-graded) vector space over a field $k$. The tensor algebra $T(V)$ is defined to be the graded algebra

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

where, by convention, we take $T^{0}(V)=k$. This vector space can be made into a graded algebra with the multiplication $T^{i}(V)\otimes T^{j}(V)\to T^{i+j}(V)$ given by the tensor product $\otimes$. This is the free algebra on $V$, and can be thought of as the algebra of all non-commuting polynomials in the elements of $V$.

One can give the tensor algebra the structure of a DGA as follows. Let $f:V\to k$ be any linear map. Then, this extends uniquely to a derivation of $T(V)$ of degree $-1$ by the formula

$d_{f}(v_{1}\otimes \cdots \otimes v_{n})=\sum _{i=1}^{n}(-1)^{i-1}v_{1}\otimes \cdots \otimes f(v_{i})\otimes \cdots \otimes v_{n}$

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., $d:V_{i}\to V_{i-1}$ and $d^{2}=0$. 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

$d(v_{1}\otimes \cdots \otimes v_{n})=\sum _{i=1}^{n}(-1)^{|v_{1}|+\ldots +|v_{i-1}|}v_{1}\otimes \cdots \otimes d_{V}(v_{i})\otimes \cdots \otimes v_{n}$

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

$S(V)=\operatorname {Sym} \left(\bigoplus _{i=2k}V_{i}\right)\otimes \bigwedge \left(\bigoplus _{i=2k+1}V_{i}\right)$

where $\operatorname {Sym}$ denotes the symmetric algebra and $\bigwedge$ denotes the exterior algebra. If we begin with a DG vector space $(V_{\bullet },d)$ (either homologically or cohomologically graded), then we can extend $d$ to $S(V)$ such that $(S(V),d)$ is a CDGA in a unique way.

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.

Minimal DGAs

We say that a DGA $({\mathcal {A}}^{\bullet },d,\cdot )$ is minimal if

1. It is free as a graded algebra.
2. ${\mathcal {A}}^{0}=k$ and ${\mathcal {A}}^{1}=0$
3. $d({\mathcal {A}}^{\bullet })\subseteq {\mathcal {A}}^{+}\wedge {\mathcal {A}}^{+}$, where ${\mathcal {A}}^{+}=\bigoplus _{i>0}{\mathcal {A}}_{i}$ consists of all the parts of degree $>0$.

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.

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

• Griffiths, Phillip; Morgan, John (2013), Rational Homotopy Theory and Differential Forms, Progress in Mathematics, vol. 16 (3rd ed.), New York, Heidelberg, Dordrecht, London: Birkhäuser, ISBN 978-1-4614-8467-7

• Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic Operads, Grundlehren der mathematischen Wissenschaften, vol. 346, Heidelberg, New York, Dordrecht, London: Springer, ISBN 978-3-642-30362-3

• Kontsevich, Maxim; Soibelman, Yan, Deformation Theory. I