symmetric monoidal (∞,1)-category of spectra
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Hochschild-Kostant-Rosenberg theorem identifies the Hochschild homology and cohomology of certain algebras with their modules of Kähler differentials and derivations, respectively.
First notice that we always have the following statement about the situation in degree 1.
For a $k$-algebra $R$, its module of Kähler differentials coincides with its first Hochschild homology
Write $\Omega^0(R/k) := R \simeq HH_0(R,R)$.
The HKR-theorem generalizes this to higher degrees.
For $n \geq 2$ write $\Omega^n(R/k) = \wedge^n_R \Omega(R/k)$ for the $n$-fold wedge product of $\Omega(R/k)$ with itself: the degree $n$-Kähler forms.
The isomorphism $\Omega^1(R/k) \simeq H_1(R,R)$ extends to a graded ring morphism
If the $k$-algebra $R$ is sufficiently well-behaved, then this morphism is an isomorphism that identifies the Hochschild homology of $R$ in degree $n$ with $\Omega^n(R/k)$ for all $n$:
(Hochschild-Kostant-Rosenberg theorem)
If $k$ is a field and $A$ a commutative $k$-algebra which is
essentially of finite type (finitely presented)
smooth over $k$, meaning:
then there is an isomorphism of graded $k$-algebras
Moreover, dually, there is an isomorphism of Hochschild cohomology with wedge products of derivations:
Actually, the HKR theorem holds on the level of chains: there is a quasi-isomorphism of chain complexes from polyvector fields (with zero differential) to the Hochschild cochain complex (with Hochschild differential).
The HKR map is a map of dg vector spaces, but not a map of dg-algebras nor a map of dg-Lie algebras. However, the formality theorem of Maxim Kontsevich states that nevertheless the HKR map can be extended to an $L_\infty$ quasi-isomorphism. See this MO post for details.
The HKR map is only an isomorphism of vector spaces, not an isomorphism of algebras. In order to make it an isomorphism of algebras, one must add a “correction” by the square root of the $\hat{A}$ class. This is analogous to the Duflo isomorphism. See Kontsevich and Caldararu.
There is also a noncommutative analogue due to Alain Connes.
(…)
Discussion for dg-algebras is in (Cattaneo-Fiorenza-Longoni 05).
Randy McCarthy and Vahagn Minasian have also proven an HKR theorem in the setting of higher algebra in stable homotopy theory, where associative algebras are generalized to A-∞ algebras, where the role of Hochschild homology is played by topological Hochschild homology and that of Kähler differentials by topological André-Quillen homology Again, this works under a certain smoothness property:
For a connective smooth E-∞ ring $A$, the (natural) derivative map
from topological Hochschild homology to topological André-Quillen homology has a section in the (∞,1)-category of ∞-modules over $A$ which induces an equivalence of $A$-algebras
where $\mathbb{P}$ is the free symmetric algebra triple.
This is due to (McCarthy-Manasian 03).
The original source is
Standard textbook references include
Charles Weibel, An Introduction to Homological Algebra section 9.4
Victor Ginzburg, Lectures on noncommutative geometry (arXiv:math/0506603) section 4, Theorem 9.1.3
Discussion in positive characteristic is in
A new approach to the generalized HKR isomorphism is proposed in
The version for dg-algebras is discussed in
The version for $E_\infty$-algebras is discussed in