Algebraic 𝐾-theory of 𝑇𝐻𝐻(𝔽_{𝕡})

Bayındır, Haldun Özgür; Moulinos, Tasos

doi:10.1090/tran/8613

Algebraic $K$-theory of $\operatorname {THH}(\mathbb {F}_p)$
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by Haldun Özgür Bayındır and Tasos Moulinos PDF

Trans. Amer. Math. Soc. 375 (2022), 4177-4207 Request permission

Abstract:

In this work we study the $E_{\infty }$-ring $\operatorname {THH}(\mathbb {F}_p)$ as a graded spectrum. Following an identification at the level of $E_2$-algebras with $\mathbb {F}_p[\Omega S^3]$, the group ring of the $E_1$-group $\Omega S^3$ over $\mathbb {F}_p$, we show that the grading on $\operatorname {THH}(\mathbb {F}_p)$ arises from decomposition on the cyclic bar construction of the pointed monoid $\Omega S^3$. This allows us to use trace methods to compute the algebraic $K$-theory of $\operatorname {THH}(\mathbb {F}_p)$. We also show that as an $E_2$ $H\mathbb {F}_p$-ring, $\operatorname {THH}(\mathbb {F}_p)$ is uniquely determined by its homotopy groups. These results hold in fact for $\operatorname {THH}(k)$, 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.

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