A slice refinement of Bökstedt periodicity

Abstract

Let $R$ be a perfectoid ring. Hesselholt and Bhatt-Morrow-Scholze have identified the Postnikov filtration on $\mathrm{THH}(R;{\mathbb{Z}}_p)$: it is concentrated in even degrees, generated by powers of the Bökstedt generator $\sigma$, generalizing classical Bökstedt periodicity for $R={\mathbb{F}}_p$. We study an equivariant generalization of the Postnikov filtration, the *regular slice filtration*, on $\mathrm{THH}(R;{\mathbb{Z}}_p)$. The slice filtration is again concentrated in even degrees, generated by $RO(\mathbb{T})$-graded classes which can loosely be thought of as the *norms* of $\sigma$. The slices are expressible as $RO(\mathbb{T})$-graded suspensions of Mackey functors obtained from the Witt Mackey functor. We obtain a sort of filtration by $q$-factorials. A key ingredient, which may be of independent interest, is a close connection between the Hill-Yarnall characterization of the slice filtration and Anschütz-le Bras' $q$-deformation of Legendre's formula.

48 pages, 10 figures

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• Motivic structures in non-commutative geometry
• A naive approach to genuine $G$-spectra and cyclotomic spectra
• On categories of slices