2014/03/03 by Bjørn Ian Dundas, Dundas, Bjørn Ian, Matthew Morrow +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Topological and Geometric Data Analysis #math.AG #math.AT #math.KT
paper · pdf · doi:10.48550/arxiv.1403.0534
arxiv created 2014/03/03 · openalex publication_date 2014/03/03 · arxiv updated 2014/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 TRr, 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.