Polygonic spectra and TR with coefficients
| Autor: | Krause, Achim, McCandless, Jonas, Nikolaus, Thomas |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 61 pages, comments are welcome |
| Databáze: | arXiv |
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