Finite generation and continuity of topological Hochschild and cyclic homology

TL;DR

This paper proves finite generation and continuity properties of topological Hochschild and cyclic homology for commutative, Noetherian, F-finite rings, with applications to algebraic geometry and proper schemes.

Contribution

It establishes finite generation and continuity of topological Hochschild and cyclic homology for a broad class of rings, extending known results to more general settings.

Findings

Topological Hochschild homology groups are finitely generated modules.

Homotopy groups of fixed point spectra TR^r are finitely generated.

Continuity of these homology theories is proven for any ideal.

Abstract

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 $TR^r$, are finitely generated modules. 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.