Polygonic spectra and TR with coefficients

Krause, Achim; McCandless, Jonas; Nikolaus, Thomas

Abstract

We introduce the notion of a polygonic spectrum which is designed to axiomatize the structure on topological Hochschild homology THH(R,M) of an E1-ring R with coefficients in an R-bimodule M. For every polygonic spectrum X, we define a spectrum TR(X) as the mapping spectrum from the polygonic version of the sphere spectrum S to X. In particular if applied to X=THH(R,M) this gives a conceptual definition of 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 TR(X) by exhibiting TR(X) as the Z-fixedpoints of a quasifinitely genuine Z-spectrum. The notion of quasifinitely genuine Z-spectra is a new notion that we introduce and discuss inspired by a similar notion over Z introduced by Kaledin. Besides the usual coherences for genuine spectra, this notion additionally encodes that TR(X) admits certain infinite sums of Verschiebung maps.