Algebraic 𝐾-theory of 𝑇𝐻𝐻(𝔽_{𝕡})

In this work we study the E ∞ E_{\infty } -ring THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) as a graded spectrum. Following an identification at the level of E 2 E_2 -algebras with F p [ Ω S 3 ] \mathbb {F}_p[\Omega S^3] , the group ring of the E 1 E_1 -group Ω S 3 \Omega S^3 over F p \mathbb {F}_p , we show that the grading on THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) arises from decomposition on the cyclic bar construction of the pointed monoid Ω S 3 \Omega S^3 . This allows us to use trace methods to compute the algebraic K K -theory of THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) . We also show that as an E 2 E_2 H F p H\mathbb {F}_p -ring, THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) is uniquely determined by its homotopy groups. These results hold in fact for THH ⁡ ( k ) \operatorname {THH}(k) , where k k is any perfect field of characteristic p 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 K -theory of formal DGAs.