Teleman localization of Hochschild homology in a singular setting. (English) Zbl 1184.55002
Generalizations of the Hochschild-Kostant-Rosenberg theorem to singular spaces are considered.
In particular, given a singular variety \(X\) with isolated singularities, the authors define a smooth intersection function algebra \(IC^\infty(X)\). This is defined as an algebra of smooth functions on the regular part of \(X\) which satisfy certain growth conditions near the singularities. The main theorem of the paper states that that the closed Hochschild homology of this algebra with coefficients in a suitable module (depending on a parameter \(\delta\)) is isomorphic to a certain “intersection” complex of differential forms, and the cohomology of the latter is the intersection cohomology of \(X\) (with perversity determined by \(\delta\)).
Closed Hochschild homology here means that one divides the space of Hochschild cycles by the closure of the space of Hochschild boundaries. Another often used terminology for this construction is reduced cohomology.
The identification of Hochschild homology with a suitable complex of differential forms is called “Teleman localization” in the paper. In addition to the above localization result, the paper also shows that such localization results hold for suitable algebras of functions on a manifold with boundary.
In particular, given a singular variety \(X\) with isolated singularities, the authors define a smooth intersection function algebra \(IC^\infty(X)\). This is defined as an algebra of smooth functions on the regular part of \(X\) which satisfy certain growth conditions near the singularities. The main theorem of the paper states that that the closed Hochschild homology of this algebra with coefficients in a suitable module (depending on a parameter \(\delta\)) is isomorphic to a certain “intersection” complex of differential forms, and the cohomology of the latter is the intersection cohomology of \(X\) (with perversity determined by \(\delta\)).
Closed Hochschild homology here means that one divides the space of Hochschild cycles by the closure of the space of Hochschild boundaries. Another often used terminology for this construction is reduced cohomology.
The identification of Hochschild homology with a suitable complex of differential forms is called “Teleman localization” in the paper. In addition to the above localization result, the paper also shows that such localization results hold for suitable algebras of functions on a manifold with boundary.
Reviewer: Thomas Schick (Göttingen)
MSC:
| 55N33 | Intersection homology and cohomology in algebraic topology |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
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