Bökstedt periodicity and quotients of DVRs

Abstract

In this paper we compute the topological Hochschild homology of quotients of discrete valuation rings (DVRs). Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of $\operatorname {THH}$ (respectively, logarithmic $\operatorname {THH}$ ) of complete DVRs originally due to Lindenstrauss and Madsen (respectively, Hesselholt and Madsen).

A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology

“…At its time of writing, the second-named author was unaware of the algebro-geometric log diagonal interpretation of these results. Versions of Proposition 1.4 and Theorem 1.6 hold for log topological Hochschild homology of discrete pre-log rings in the sense of Krause-Nikolaus [27] and Rognes-Sagave-Schlichtkrull [45], [46].…”

“…Remark 1.2. In the context of discrete pre-log rings, the log Hochschild homology in the statement of Theorem 1.1 is equivalent to Rognes' definition [44], see Proposition 1.4, and also used by Krause-Nikolaus [27]. See also Leip [29,Page 886] for a discussion of this definition of log THH.…”