多边形谱和带系数的TR
Polygonic spectra and TR with coefficients
摘要：
我们引入了多边形谱的概念，它的目的是公理化系数在$R$ -双模$M$中的$\mathbb{E}_1$ -环$R$的拓扑Hochschild同调$\mathrm{THH}(R,M)$上的结构.对于每个多边形谱$X$，我们定义一个谱$\mathrm{TR}(X)$作为从球面谱$\mathbb{S}$到$X$的多边形版本的映射谱。特别地，如果应用于$X = \mathrm{THH}(R,M)$，这给出了$\mathrm{TR}(R,M)$的概念定义。每一个分圆谱产生一个多边形谱，我们证明了TR同意在这种情况下的TR的经典定义。通过将$\mathrm{TR}(X)$表示为一个拟伪真$\mathbb{Z}$ -谱的$\mathbb{Z}$ -不动点，我们构造了$\mathrm{TR}(X)$上的Frobenius和Verdiebung映射.准真$\mathbb{Z}$ -谱的概念是受Kaledin关于$\mathbb{Z}$的类似概念的启发而提出并讨论的一个新概念。除了真正谱的通常的相干性之外，这个概念还编码了$\mathrm{TR}(X)$允许某些无穷和的Vernebung映射。
英文摘要：
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.
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