Finite generation and continuity  of topological Hochschild  and cyclic homology

Dundas, Bjørn Ian; Morrow, Matthew

doi:10.24033/asens.2319

Finite generation and continuity of topological Hochschild and cyclic homology
[La génération finie et continuité en homologies de Hochschild et cyclique]

Dundas, Bjørn Ian ; Morrow, Matthew

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 201-238.

Résumé
Abstract

Le but de cet article est d'établir des propriétés fondamentales des homologies de Hochschild, de Hochschild topologique et cyclique topologique d'anneaux commutatifs et noethériens, qu'on ne suppose être que F-finis pour la majorité de nos résultats. Cette hypothèse faible est satisfaite en tous cas d'intérêts en géométrie algébrique en caractéristique finie et mixte. Nous démontrons d'abord que les groupes d'homologie de Hochschild topologique, ainsi que les groupes d'homotopie du spectre des points fixés $T R^{r}$ , sont des modules de type fini (après la $p$ -complétion dans le cadre de caractéristique mixte). En l'utilisant, nous établissons la continuité de ces homologies pour n'importe quel idéal. Une conséquence de ces résultats de continuité est le théorème de Hochschild-Kostant-Rosenberg pro pour les homologies de Hochschild topologique et cyclique topologique. Finalement, nous démontrons que ces résultats de génération finie et ces propriétés de continuité sont toujours valables pour les schémas propres et lisses sur un tel anneau.

The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our results. This mild hypothesis is satisfied in all cases of interest in finite and mixed characteristic algebraic geometry. We prove firstly that the topological Hochschild homology groups, and the homotopy groups of the fixed point spectra $T R^{r}$ , are finitely generated modules (after $p$ -completion in the mixed characteristic setting). We use this to establish the continuity of these homology theories for any given ideal. A consequence of such continuity results is the pro Hochschild-Kostant-Rosenberg theorem for topological Hochschild and cyclic homology. Finally, we show more generally that the aforementioned finite generation and continuity properties remain true for any proper scheme over such a ring.

Publié le : 2017-11-12

MR Zbl | 1 citation dans Numdam

DOI : 10.24033/asens.2319

Classification : 19D55, 13D03.
Keywords: $K$-theory, topological cyclic homology.
Mot clés : $K$-théorie, homologie cyclique topologique.

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     author = {Dundas, Bj{\o}rn Ian and Morrow, Matthew},
     title = {Finite generation and continuity  of topological {Hochschild}  and cyclic homology},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {201--238},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
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     year = {2017},
     doi = {10.24033/asens.2319},
     mrnumber = {3621430},
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Dundas, Bjørn Ian; Morrow, Matthew. Finite generation and continuity  of topological Hochschild  and cyclic homology. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 201-238. doi : 10.24033/asens.2319. http://www.numdam.org/articles/10.24033/asens.2319/

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