Polygonic spectra and TR with coefficients

TL;DR

This paper introduces polygonic spectra to axiomatize topological Hochschild homology with coefficients, providing a conceptual framework for TR and constructing Frobenius and Verschiebung maps using a new notion of quasifinitely genuine spectra.

Contribution

It defines polygonic spectra for THH, offers a new conceptual definition of TR, and introduces quasifinitely genuine spectra to construct key maps.

Findings

TR agrees with classical definitions for cyclotomic spectra.

Constructs Frobenius and Verschiebung maps via quasifinitely genuine spectra.

Introduces a new notion of quasifinitely genuine $bZ$-spectra.

Abstract

We introduce the notion of a polygonic spectrum which is designed to axiomatize the structure on topological Hochschild homology $\mathrm{THH}(R,M)$ of an $\mathbb{E}_1$-ring $R$ with coefficients in an $R$-bimodule $M$. For every polygonic spectrum $X$, we define a spectrum $\mathrm{TR}(X)$ as the mapping spectrum from the polygonic version of the sphere spectrum $\mathbb{S}$ to $X$. In particular if applied to $X = \mathrm{THH}(R,M)$ this gives a conceptual definition of $\mathrm{TR}(R,M)$. Every cyclotomic spectrum gives rise to a polygonic spectrum and we prove that TR agrees with the classical definition of TR in this case. We construct Frobenius and Verschiebung maps on $\mathrm{TR}(X)$ by exhibiting $\mathrm{TR}(X)$ as the $\mathbb{Z}$-fixedpoints of a quasifinitely genuine $\mathbb{Z}$-spectrum. The notion of quasifinitely genuine $\mathbb{Z}$-spectra is a new notion that we introduce and discuss inspired by a similar notion over $\mathbb{Z}$ introduced by Kaledin. Besides the usual coherences for genuine spectra, this notion additionally encodes that $\mathrm{TR}(X)$ admits certain infinite sums of Verschiebung maps.