Ausoni-Bokstedt duality for topological Hochschild homology

TL;DR

This paper demonstrates that topological Hochschild homology (THH) often satisfies the Gorenstein condition, providing new insights and proofs for dualities in THH calculations for certain ring spectra.

Contribution

It establishes the Gorenstein property of THH for a broad class of ring spectra using minimal calculations, extending known dualities without complex computations.

Findings

THH(R;k) is Gorenstein when R is a regular local ring and k is a field of characteristic p.

Provides a non-calculational proof of dualities in THH based on Bokstedt's calculations.

Shows THH(R;k) is highly computable via Dundas's lemma.

Abstract

We consider the Gorenstein condition for topological Hochschild homology, and show that it holds remarkably often. More precisely, if R is a commutative ring spectrum and and R----->k is a ring map to a field of characteristic p then, provided k is small as an R-module, THH(R;k) is Gorenstein in the sense of Dwyer-Greenlees-Iyengar. In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p. Using only Bokstedt's calculation of THH(k), this gives a non-calculational proof of dualities observed in calculations by Bokstedt, McClure-Staffeldt, Ausoni-Rognes, Ausoni, Lindenstrauss-Madsen, Angeltweit-Rognes and others. A lemma of Dundas shows that THH(R;k) is remarkably computable.