Abstract

Let $Q$ denote Mac Lane’s $Q$-construction and $\otimes$ the smash product of spectra. We construct an equivalence $Q\left(R\right)\simeq \mathbb{Z}\otimes R$ in the category of $A_{\infty }$ 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 $\infty$-categorical sense) extending the usual monoidal structure, for which we prove an equivalence $Q\left(-\right)\simeq \mathbb{Z}\otimes -$ as symmetric monoidal functors. From this, we obtain a new proof of the equivalence $HML\left(R,M\right)\simeq THH\left(R,M\right)$ originally proved by Pirashvili and Waldhausen. This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence $HML\left(R\right)\simeq THH\left(R\right)$ as $E_{\infty }$ ring spectra, when $R$ is a commutative ring.

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