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Abstract
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Let
denote Mac Lane’s
-construction
and
the smash product of spectra. We construct an equivalence
in the category
of
ring spectra
for any ring
,
thus proving a conjecture of Fiedorowicz, Pirashvili, Schwänzl, Vogt and
Waldhausen. More precisely, we construct a symmetric monoidal structure on
(in the
-categorical
sense) extending the usual monoidal structure, for which we prove an equivalence
as
symmetric monoidal functors. From this, we obtain a new proof of the equivalence
originally proved by Pirashvili and Waldhausen. This equivalence is in fact
made symmetric monoidal, and so it also provides a proof of the equivalence
as
ring spectra,
when
is a commutative ring.
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Keywords
Mac Lane homology, Hochschild homology, topological
Hochschild homology, $Q$-construction
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Mathematical Subject Classification
Primary: 19D55
Secondary: 18G99
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Milestones
Received: 12 December 2020
Revised: 14 January 2021
Accepted: 4 February 2021
Published: 11 September 2021
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