HKR theorem for smooth \(S\)-algebras. (English) Zbl 1051.55005
The “HKR” theorem of the title is an extension of the Hochschild-Kostant-Rosenberg theorem which says that, if \(k\) is a commutative ring, the Hochschild homology of a smooth \(k\)-algebra is given by the differential forms. The context of this paper is that of commutative ring spectra, or commutative \(S\)-algebras.
The authors give a definition of (thh-)étale and smooth maps extending the usual definitions, and prove the HKR-theorem: If \(f\colon R\to A\) is a thh-smooth map in the category of connective commutative \(S\)-algebras, then the derivative map \[ THH(A| R)\to \Sigma TAQ(A| R) \] has a section in the category of \(A\)-modules which induces an equivalence of \(A\)-algebras \[ \mathbb P_A\Sigma\, {TAQ}(A| R)@>\sim>>THH(A| R) \] where \(\mathbb P_A\) is the symmetric algebra triple.
A key result in the paper is the “étale descent formula”, i.e. the authors give conditions implying that the claim that a map \(A\to B\) of commutative \(R\)-algebras is thh-étale is equivalent to claiming that the map \[ (A\otimes_RX)\wedge_AB\to B\otimes_RX \] is an equivalence.
The authors give a definition of (thh-)étale and smooth maps extending the usual definitions, and prove the HKR-theorem: If \(f\colon R\to A\) is a thh-smooth map in the category of connective commutative \(S\)-algebras, then the derivative map \[ THH(A| R)\to \Sigma TAQ(A| R) \] has a section in the category of \(A\)-modules which induces an equivalence of \(A\)-algebras \[ \mathbb P_A\Sigma\, {TAQ}(A| R)@>\sim>>THH(A| R) \] where \(\mathbb P_A\) is the symmetric algebra triple.
A key result in the paper is the “étale descent formula”, i.e. the authors give conditions implying that the claim that a map \(A\to B\) of commutative \(R\)-algebras is thh-étale is equivalent to claiming that the map \[ (A\otimes_RX)\wedge_AB\to B\otimes_RX \] is an equivalence.
Reviewer: Bjørn Dundas (Bergen)
MSC:
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
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