Abstract
IfB is an étale extension of ak-algebraA, we prove for Hochschild homology thatHH *(B)≅HH*(A)⊗AB. For Galois descent with groupG there is a similar result for cyclic homology:HC *≅HC*(B)G if\(\mathbb{Q} \subseteq A\). In the process of proving these results we give a localization result for Hochschild homology without any flatness assumption. We then extend the definition of Hochschild homology to all schemes and show that Hochschild homology satisfies cohomological descent for the Zariski, Nisnevich and étale topologies. We extend the definition of cyclic homology to finite-dimensional noetherian schemes and show that cyclic homology satisfies cohomological descent for the Zariski and Nisnevich topologies, as well as for the étale topology overQ. Finally we apply these results to complete the computation of the algebraicK-theory of seminormal curves in characteristic zero.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
[AKTEC]R. W. Thomason,Algebraic K-theory and étale cohomology, Ann. scient. Éc. Norm. Sup. (Paris),18 (1986), 437–552.
[Am]S. A. Amitsur,Simple algebras and cohomology groups of arbitrary fields, Trans. AMS,90 (1959), 73–112.
[B]J. M. Boardman,Conditionally convergent spectral sequences, preprint, 1981.
[Bl]J. Block, Harvard Ph. D. thesis, 1987.
[BO]S. Bloch andA. Ogus,Gersten's conjecture and the homology of schemes, Ann. Sc. Ec. Norm. Sup. (Paris),7 (1974), 181–202.
[Bry]J.-L. Brylinski,Central localization inHochschild homology, J. Pure and Appl. Math.,57 (1989), 1–4.
[Brylet]J.-L. Brylinski, Letter to C. Weibel, February, 1990.
[BX]N. Bourbaki,Algèbre Chapitre 10, Masson, Paris, 1980.
[CE]H. Cartan andS. Eilenberg,Homological Algebra, Princeton Univ. Press, Princeton, 1956.
[D]E. Davis,On the geometric interpretation of seminormality, Proc. AMS,68 (1978), 1–5.
[EM]S. Eilenberg andJ. Moore,Limits and spectal sequences, Topology1 (1962), 1–23.
[EGA]A. Grothendieck andJ. A. Dieudonné,Eléments de Géométrie Algébrique III (Première Partie), Publ. Math. IHES,11 (1961).
[G]T. G. Goodwillie,Cyclic homology, derivations and the free loop space, Topology24 (1985), 187–215.
[GdR]A. Grothendieck,On the de Rham cohomology of algebraic varieties, Publ. Math. IHES,29 (1966), 351–359.
[GRW]S. Geller, L. Reid andC. Weibel,The cyclic homology and K-theory of curves, J. reine und angew. Math.,393 (1989), 39–90.
[Hart]R. Hartshorne,Residues and Duality, Springer Lecture Notes in Math. #20, New York-Heidelberg-Berlin, 1966.
[KO]M. Knus andM. Ojanguren,Théorie de la Descente et Algèbres d'Azumaya, Springer Lecture Notes in Math. #389, New York-Heidelberg-Berlin, 1974.
[KS]C. Kasel andA. Sletsjoe, Base change, transitivity and Künneth formulas for the Quillen decomposition of Hochschild homology, preprint, 1991.
[L]J.-L. Loday,Cyclic homology: a survey, Banach Center Publications (Warsaw),18 (1986), 285–307.
[LQ]J.-L. Loday andD. Quillen,Cyclic homology and the Lie algebra homology of matrices, Comm. Math Helv.,59 (1984), 565–591.
[M]J. Milne,Étale Cohomology, Princeton Univ. Press, Princeton, 1980.
[Mac]S. MacLane,Homology, Springer-Verlag, New York-Heidelberg-Berlin, 1963.
[Nis]Y. A. Nisnevich The completely decomposed topology on schemes and associated descent spectral sequences in algebraicK-theory,Algebraic K-theory: Connections with Geometry and Topology NATO ASI Series C #279, Kluwer Academic Publishers, Dordrecht-Boston-London, 1989, 241–342.
[TDTE]A. Grothendieck,Technique de descente et théorèmes d'existence en géométrie algébrique, I and II (exposés 190 and 195), Séminaire Bourbaki, 12e année 1959/60, Secrétariat mathématique, 11 rue P Curie, Paris, 1960.
[vdK]W. van der Kallen,Descent for the K-theory of polynomial rings, Math. Zeit.,191 (1986), 405–415.
[WA]C. Weibel, Appendix:K * andHC * of certain gradedk-algebras,Algebraic K-theory: Connections with Geometry and Topology, NATO ASI series C #279, Kluwer Academic Publishers, Dordrech-Boston-London, 1989, 418–424.
Author information
Authors and Affiliations
Additional information
Partially supported by National Science Foundation grant DMS-8803497
Partially supported by National Security Agency grant MDA904-90-H-4019
Rights and permissions
About this article
Cite this article
Weibel, C.A., Geller, S.C. Étale descent for hochschild and cyclic homology. Comment. Math. Helv. 66, 368–388 (1991). https://doi.org/10.1007/BF02566656
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02566656