Vol. 6, No. 3, 2021

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A multiplicative comparison of Mac Lane homology and topological Hochschild homology

Geoffroy Horel and Maxime Ramzi

Vol. 6 (2021), No. 3, 571–605
Abstract

Let Q denote Mac Lane’s Q-construction and the smash product of spectra. We construct an equivalence Q(R) R in the category of A ring spectra for any ring R, thus proving a conjecture of Fiedorowicz, Pirashvili, Schwänzl, Vogt and Waldhausen. More precisely, we construct a symmetric monoidal structure on Q (in the -categorical sense) extending the usual monoidal structure, for which we prove an equivalence Q() as symmetric monoidal functors. From this, we obtain a new proof of the equivalence HML(R,M) THH(R,M) originally proved by Pirashvili and Waldhausen. This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence HML(R) THH(R) as E ring spectra, when R is a commutative ring.

Keywords
Mac Lane homology, Hochschild homology, topological Hochschild homology, $Q$-construction
Mathematical Subject Classification
Primary: 19D55
Secondary: 18G99
Milestones
Received: 12 December 2020
Revised: 14 January 2021
Accepted: 4 February 2021
Published: 11 September 2021
Authors
Geoffroy Horel
Université Sorbonne Paris Nord
Villetaneuse
France
Maxime Ramzi
Department of Mathematical Sciences
University of Copenhagen
Denmark
École Normale Supérieure
Paris
France