Differential graded algebra

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

In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DG algebra, or DGA) is an algebraic structure often used to capture information about a topological or geometric space. For example, the de Rham algebra of differential forms on a manifold is a differential graded algebra, which encodes the cohomology of the underlying topological space. Explicitly, a differential graded algebra is a graded associative algebra with a chain complex structure that is compatible with the algebra structure. American mathematician Dennis Sullivan used DGAs to model the rational homotopy type of topological spaces.

Definitions

Let $A=\bigoplus \nolimits _{i\in \mathbb {Z} }A_{i}$ be a $\mathbb {Z}$-graded algebra equipped with a map $d\colon A\to A$ of degree $-1$ (homologically graded) or degree $1$ (cohomologically graded). We say that $(A,d)$ is a differential graded algebra if $d$ is a differential, giving $A$ the structure of a chain complex or cochain complex (depending on the degree), 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 the conditions

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

Often one omits the differential and simply writes $A$ to refer to the DGA $(A,d)$.

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, maps of degree 0 that commute with the differentials $f\circ d_{A}=d_{B}\circ f$. The morphisms in the category of DGAs are chain maps that are also algebra homomorphisms.

Categorical Definition

One can define a DGA more abstractly using category theory. There is a category of chain complexes over a field $k$, often denoted $\operatorname {Ch} _{k}$, whose objects are chain complexes and whose morphisms are chain maps. We define the tensor product of chain complexes $(V,d_{V})$ and $(W,d_{W})$ by

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

with differential

$d(v\otimes w)=(d_{V}v)\otimes w-(-1)^{|v|}v\otimes (d_{W}w)$

This operation makes $\operatorname {Ch} _{k}$ into a symmetric monoidal category. Then, we can equivalently define a differential graded algebra as a monoid object in $\operatorname {Ch} _{k}$.

Homology and Cohomology

Associated to any chain complex $(A_{\bullet },d)$ 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 $H_{\bullet }(A)$. In fact, the homology groups form a graded algebra 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 $f:(A,d_{A})\to (B,d_{B})$ of complexes induces a map on (co)homology, often denoted $f_{*}:H_{\bullet }(A)\to H_{\bullet }(B)$ (respectively f^*: H^\bullet(B) \to H^\bullet(A)). If this induced map is an isomorphism on all (co)homology groups, the map $f$ is called a quasi-isomorphism. In many contexts, this is the natural notion of equivalence one uses for (co)chain complexes.

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,

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

for homogeneous elements $a\in A_{i},b\in A_{j}$. Many of the DGAs commonly encountered in math happen to be CDGAs, including the de Rham algebra of differential forms.

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. That is,

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

Most generally, for a (co)chain complex $C$ we say that it is formal if there is a chain map to its (co)homology $H_{\bullet }(C)$ (respectively H^\bullet(C)) that is a quasi-isomorphism. Now, 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. 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.

Examples

Trivial DGAs

First, we note that any graded algebra $A=\bigoplus \nolimits _{i}A_{i}$ 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.

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 (cohomologically graded) DGA. The graded vector space is $\Omega ^{\bullet }(M)$, where the grading is given by form degree. This vector space has a product, which is the exterior product, which makes it into a graded algebra. Finally, the exterior derivative $d:\Omega ^{i}(M)\to \Omega ^{i+1}(M)$ satisfies the required conditions for it to be a differential. In fact, the exterior product is graded commutative, which makes the de Rham algebra an example of a CDGA.

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.

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.

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 the (co)homology is the same. Given a DGA $A$, we say that another DGA $M$ is a model for $A$ if it comes with a surjective DGA morphism $p:M\to A$ 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, in some sense. We say that a DGA $(A,d,\cdot )$ is a minimal if it satisfies the following conditions.

1. It is free as a graded algebra, i.e., $A\cong T(V)$ for some graded vector space $V$.
2. The differential satisfies $d(A)\subseteq A^{+}\cdot A^{+}$, where $A^{+}=\bigoplus \nolimits _{i>0}{\mathcal {A}}_{i}$ consists of the positive degree parts of $A$.

Note that some conventions, often used in algebraic topology, additionally require that $A$ be simply connected, which means that $A^{0}=k$ and $A^{1}=0$. This condition on the 0th and 1st degree pieces mirrors what happens for the (co)homology of a simply connected space.

Finally, we say that $M$ is a minimal model for $A$ if it is both minimal and a model for $A$. The fundamental theorem of minimal models states that, for a given DGA $A$, the minimal model is unique up to (non-unique) isomorphism, and that if $A$ is simply connected it admits a minimal model.

The Sullivan Minimal Model

Main article: Rational homotopy theory

Minimal models were used with great success by Dennis Sullivan in his work on rational homotopy theory. Given a simplicial complex $X$, one can define the DGA $A_{PL}(X)$ of "piecewise polynomial" differential forms with $\mathbb {Q}$-coefficients. Then, $A(X)$ has the structure of a DGA over the field $\mathbb {Q}$, and in fact the cohomology is isomorphic to the singular cohomology of $X$. In particular, if $X$ is a simply connected topological space then $A(X)$ is simply connected as a DGA, thus there exists a minimal model.

The main result is the following. For $X$ a simply connected CW complex with finite dimensional rational homology groups, there is a sense in which the minimal model of $A_{PL}(X)$ captures entirely the rational homotopy type of the space $X$.

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

• Sullivan, Dennis (1977), "Infinitesimal computations in topology", Publications Mathématiques de l'IHÉS, 47: 269–331, doi:10.1007/bf02684341, hdl:10338.dmlcz/128041, MR 0646078

• Cartan, Henri (1954–1955). "DGA-algèbres et DGA-modules". Séminaire Henri Cartan. 7 (1): 1–9.

• Cartan, Henri (1954–1955). "DGA-modules (suite), notion de construction". Séminaire Henri Cartan. 7 (1): 1–11.

• Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis (1975). "Real homotopy theory of Kähler manifolds". Inventiones mathematicae. 29. Springer: 245–274. doi:10.1007/BF01389853.