symmetric monoidal (∞,1)-category of spectra
First notice that we always have the following statement about the situation in degree 1.
The HKR-theorem generalizes this to higher degrees.
For write for the -fold wedge product of with itself: the degree -Kähler forms.
The isomorphism extends to a graded ring morphism
If the -algebra is sufficiently well-behaved, then this morphism is an isomorphism that identifies the Hochschild homology of in degree with for all :
essentially of finite type (finitely presented)
smooth over , meaning:
then there is an isomorphism of graded -algebras
Moreover, dually, there is an isomorphism of Hochschild cohomology with wedge products of derivations:
This is reviewed for instance as theorem 9.4.7 of
or as theorem 9.1.3 in Ginzburg.
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 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 class. This is analogous to the Duflo isomorphism. See Kontsevich and Caldararu.
There is also a noncommutative analogue due to Alain Connes.
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 , the (natural) derivative map
where is the free symmetric algebra triple.
This is due to (McCarthy-Manasian 03).
The original source is
Standard textbook references include
A new approach to the generalized HKR isomorphism is proposed in
The version of the theorem for smooth -algebras is explained in