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Hochschild-Kostant-Rosenberg theorem

Context

Algebra

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Hochschild-Kostant-Rosenberg theorem identifies the Hochschild homology and cohomology of certain algebras with their modules of Kähler differentials and derivations, respectively.

Details

For commutative kk-algebras

First notice that we always have the following statement about the situation in degree 1.

Proposition

For a kk-algebra RR, its module of Kähler differentials coincides with its first Hochschild homology

Ω(R/k)H 1(R,R). \Omega(R/k) \simeq H_1(R,R) \,.

Write Ω 0(R/k):=RHH 0(R,R)\Omega^0(R/k) := R \simeq HH_0(R,R).

The HKR-theorem generalizes this to higher degrees.

As an isomorphism of chain complexes

For n2n \geq 2 write Ω n(R/k)= R nΩ(R/k)\Omega^n(R/k) = \wedge^n_R \Omega(R/k) for the nn-fold wedge product of Ω(R/k)\Omega(R/k) with itself: the degree nn-Kähler forms.

Theorem

The isomorphism Ω 1(R/k)H 1(R,R)\Omega^1(R/k) \simeq H_1(R,R) extends to a graded ring morphism

ψ:Ω (R/k)H (R,R). \psi : \Omega^\bullet(R/k) \to H_\bullet(R,R) \,.

If the kk-algebra RR is sufficiently well-behaved, then this morphism is an isomorphism that identifies the Hochschild homology of RR in degree nn with Ω n(R/k)\Omega^n(R/k) for all nn:

Theorem

(Hochschild-Kostant-Rosenberg theorem)

If kk is a field and AA a commutative kk-algebra which is

then there is an isomorphism of graded kk-algebras

ψ:Ω (A/k)H (A,A). \psi : \Omega^\bullet(A/k) \stackrel{\simeq}{\to} H_\bullet(A,A) \,.

Moreover, dually, there is an isomorphism of Hochschild cohomology with wedge products of derivations:

A Der k(A,A)HH (A,A). \wedge^\bullet_A Der_k(A,A) \simeq HH^\bullet(A,A) \,.
Proof

This is reviewed for instance as theorem 9.4.7 of

or as theorem 9.1.3 in Ginzburg.

As an isomorphism of \infty-algebras

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 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 A^\hat{A} class. This is analogous to the Duflo isomorphism. See Kontsevich and Caldararu.

For non-commutative algebras

There is also a noncommutative analogue due to Alain Connes.

(…)

For ring spectra in homotopy theory

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:

Proposition

For a connective smooth E-∞ ring AA, the (natural) derivative map

THH(A)ΣTAQ(A) THH(A)\to \Sigma TAQ(A)

from topological Hochschild homology to topological André-Quillen homology has a section in the (∞,1)-category of ∞-modules over AA which induces an equivalence of AA-algebras

AΣTAQ(A)THH(A), \mathbb{P}_A\Sigma TAQ(A)\simeq THH(A),

where \mathbb{P} is the free symmetric algebra triple.

This is due to (McCarthy-Manasian 03).

References

The original source is

Standard textbook references include

A new approach to the generalized HKR isomorphism is proposed in

  • Dima Arinkin, Andrei Caldararu, When is the self-intersection of a subvariety a fibration?, arxiv/1007.1671

The version of the theorem for smooth SS-algebras is explained in

Revised on April 8, 2014 03:26:21 by Tim Porter (2.26.40.160)