Bökstedt periodicity generator via \(K\)-theory (Q6594743)

In its most basic form, ``Bõkstedt periodicity'' refers to the computation of the Topological Hochschild Homology \(THH_\ast(k)\) of a prime field \(k = \mathbb{F}_p\) with \(p\) elements; the answer is that \(THH_\ast(k) \cong k[v]\) is the polynomial algebra in one generator \(v\) of homological degree \(2\). The computation was done by \textit{M. Bõkstedt} [\url{https://people.math.rochester.edu/faculty/doug/otherpapers/bokstedt1.pdf}] in the same paper where topological Hochschild homology was first defined. Stemming from [Soobshch. Akad. Nauk Gruz. SSR 133, No. 3, 477--480 (1989; Zbl 0676.16024)], \textit{T. I. Pirashvili} and others showed that \(THH_\ast(k)\) can be identified with several other homology theories including Mac Lane Homology \(HM_\ast(k)\). As of now, Bõkstedt periodicity has numerous applications, including some very prominent ones such as [\textit{B. Bhatt} et al., Publ. Math., Inst. Hautes Étud. Sci. 129, 199--310 (2019; Zbl 1478.14039)], and several proofs none of which are elementary.\N\NFor a prime field \(k\) of characteristic \(p> 2\), the authors construct the Bõkstedt periodicity generator \(v\in THH_2(k)\) as an explicit class in the stabilization of \(K\)-theory with coefficients \(K(k,-)\)., and it is shown directly that \(v\) is not nilpotent in \(THH_\ast(k)\). This gives an alternative proof of the ``multiplicative'' part of Bõkstedt periodicity. After this work was finished, an updated version of the preprint [\textit{C. Barwick} et al., ``K-theory and polynomial functors'', Preprint, \url{arXiv:2102.00936}] appeared that contains, among other things, yet another new proof of Bõkstedt periodicity.