论文标题
代数$ k $ - $ \ text {thh}的理论（\ mathbb {f} _p）$
Algebraic $K$-theory of $\text{THH}(\mathbb{F}_p)$
论文作者
论文摘要
在这项工作中，我们将$ e _ {\ infty} $ - ring $ \ text {thh}（\ mathbb {f} _p）$作为分级光谱。 Following an identification at the level of $E_2$-algebras with $\mathbb{F}_p[ΩS^3]$, the group ring of the $E_1$-group $ΩS^3$ over $\mathbb{F}_p$, we show that the grading on $\text{THH}(\mathbb{F}_p)$ arises from decomposition在尖头的$ωs^3 $的循环杆构造上。这允许我们使用跟踪方法来计算$ \ text {thh}的代数$ k $ - 理论（\ mathbb {f} _p）$。我们还表明，作为$ e_2 $ $ h \ mathbb {f} _p $ -ring，$ \ text {thh}（\ mathbb {f} _p）$由其同型组唯一确定。实际上，这些结果适用于$ \ text {thh}（k）$，其中$ k $是特征性$ p $的任何完美字段。在此过程中，我们扩展了Hesselholt-Madsen使用的一些方法，后来是Speirs开发某些工具来研究分级环光谱的THH和代数$ K $ - 正式DGA的理论。
In this work we study the $E_{\infty}$-ring $\text{THH}(\mathbb{F}_p)$ as a graded spectrum. Following an identification at the level of $E_2$-algebras with $\mathbb{F}_p[ΩS^3]$, the group ring of the $E_1$-group $ΩS^3$ over $\mathbb{F}_p$, we show that the grading on $\text{THH}(\mathbb{F}_p)$ arises from decomposition on the cyclic bar construction of the pointed monoid $ΩS^3$. This allows us to use trace methods to compute the algebraic $K$-theory of $\text{THH}(\mathbb{F}_p)$. We also show that as an $E_2$ $H\mathbb{F}_p$-ring, $\text{THH}(\mathbb{F}_p)$ is uniquely determined by its homotopy groups. These results hold in fact for $\text{THH}(k)$, where $k$ is any perfect field of characteristic $p$. Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic $K$-theory of formal DGAs.
