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Central localization in Hochschild homology. (English) Zbl 0669.55001

For an algebra A with unit and an A-bimodule M, the Hochschild homology groups \(H_ n(A,M)\) are modules over the center Z(A) of A. The author shows that localization of modules over Z(A) commutes with the formation of Hochschild homology [cf. S. Geller, L. Reid and C. Weibel, J. Reine Angew. Math. 393, 39-90 (1989; Zbl 0649.14006)]. A Mayer-Vietoris principle for cyclic homology is deduced [cf. J. Block, Harvard thesis (1987) for a general case] and some examples are given.
There should be some localization principle unifying central localization and sheaf theory.
Reviewer: M.Golasiński

MSC:

55N35 Other homology theories in algebraic topology
55U15 Chain complexes in algebraic topology
16Exx Homological methods in associative algebras
18G35 Chain complexes (category-theoretic aspects), dg categories

Citations:

Zbl 0649.14006
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References:

[1] Block, J., Harvard thesis (1987)
[2] Brylinski, J.-L., A differential complex on Poison manifolds, J. Diff. Geom., 28, 93-114 (1988)
[3] Brylinski, J.-L., Some examples of Hochschild and cyclic homology, (Proc. Symposium on Algebraic Groups. Proc. Symposium on Algebraic Groups, Lecture Notes in Mathematics, 1271 (1987), Springer: Springer Berlin), 33-72, in the honor of T.A. Springer (1986)
[4] Brylinski, J.-L.; Getzler, E., The homology of pseudo-differential symbols and the non-commutative residue, \(K\)-Theory, 1, 385-403 (1987) · Zbl 0646.58026
[5] S. Geller, L. Reid and C. Weibel, Analytic isomorphisms for cyclic homology, Appendix to “Cyclic homology and \(K\); S. Geller, L. Reid and C. Weibel, Analytic isomorphisms for cyclic homology, Appendix to “Cyclic homology and \(K\) · Zbl 0616.14005
[6] Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku Math. J., 9, 119-221 (1957) · Zbl 0118.26104
[7] Kassel, C., Hochschild and cyclic homology of enveloping algebras, 303, 797-802 (1986), note C.R.A.S., Paris
[8] J.-L. Loday, Cyclic homology; a survey, Preprint, Strassbourg.; J.-L. Loday, Cyclic homology; a survey, Preprint, Strassbourg. · Zbl 0637.16013
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