Examples of DG-algebras in algebraic geometry, representation theory and abstract algebra

Question

I'm studying DG-algebras at the moment and I'm looking for interesting examples of where they occur. I've been told that they have applications in algebraic geometry and representation theory, but unfortunately I'm not very well versed in these particular topics. I know that they appear in topology, specifically, the chain complex of singular cochains of a topological space is a commutative DG-algebra, but past that I'm not too sure of interesting appearances.

Does anyone know of any interesting examples of where DG-algebras are used in algebraic geometry, representation theory, abstract algebra or something of that kind? Or perhaps a book/paper in which they appear?

For context, a DG-algebra $A$ is a graded algebra with a differential $d:A^n\to A^{n+1}$ that satisfies the Leibniz rule $$ d(ab) = d(a)b + (-1)^{|a|}ad(b) $$ where $a\in A^i, b\in A^j$ and $|a| = i$. So in other words a DG-algebra is a chain complex where the differential satisfies the Leibniz rule.

Answer 1

They arise in many areas. One interesting example is deformation theory of rings and algebras. It is "controlled" by a differential graded algebra. For more details (Kontsevich's lectures) see the reference below.

Reference:

Deformation Theory and DG-algebras.

Another topic are DG-algebras and Hochschild homology, see here.

Answer 2

If $X$ is a scheme and $G \in Perf(X)$ is a generator of the category of perfect complexes on $X$ (which is considered as a DG-enhanced triangulated category) then $$ A = RHom(G,G) $$ is an interesting DG-algebra. It enjoys the property $Perf(X) \cong Perf(A)$.

Answer 3

Expanding upon Mindlack's comment: there are a number of complexes of differential forms that generalize the de Rham complex. In complex differential geometry (which is equally a branch of algebraic geometry) there is the Dolbeaut complex: let $\Omega^{p,q}$ be the sheaf of $(p,q)$-forms on a complex $n$-fold, which locally look like $$ f(z_1,...,z_n,\bar z_1,...,\bar z_n)\, dz_1 \wedge\cdots dz_p \wedge d\bar z_1\wedge\cdots\wedge \bar z_q $$

(or really a sum of similar expressions with indices permuted; the important point is the number of holomorphic and antiholomorphic terms) with $f \in C^\infty(X)$. The differential $\bar\partial:\Omega^{p,q} \to \Omega^{p,q+1}$ is a variant of the exterior derivative $d$, and in fact using local holomorphic coordinates you can easily write $d = \partial + \bar\partial$, where the first term is the sum of the $\partial/\partial z^j$ and the second is the sum of the $\partial/\partial \bar z^j$. So for each fixed $p = 1,...,n$, there is a complex $(\Omega^{p,\bullet},\bar\partial)$. The cohomologies of these complexes yield a bigraded cohomology $H^{p,q}(X)$ called Dolbeaut cohomology. The Dolbeaut theorem says that these are isomorphic to the sheaf cohomology groups $H^q(X,\Omega^p)$ of holomorphic forms. These are the central objects in the Hodge decomposition of de Rham cohomology.

There is also algebraic de Rham cohomology, which is defined by taking sheaf cohomology with coefficients in the de Rham complex, but only using algebraic differential forms (since the coefficients live in a complex of sheaves, one actually takes hypercohomology here).