Bökstedt周期性的切片精化
A slice refinement of Bökstedt periodicity
摘要：
设$R$是完全类环. Hesselholt和Bhatt-Morrow-Scholze在$\mathrm{THH}(R;\mathbb Z_p)$上发现了Postnikov过滤：它集中在偶数度，由Bökstedt生成器$σ$的幂生成，推广了$R=\mathbb F_p$的经典Bökstedt周期性。我们研究了$\mathrm{THH}(R;\mathbb Z_p)$上的Postnikov过滤，* 规则切片过滤 * 的等变推广。切片过滤再次集中在偶数度，由$RO(\mathbb T)$分级类生成，这些分级类可以松散地被认为是$σ$的 * 范数 *。切片可表示为从Witt Mackey函子得到的Mackey函子的$RO(\mathbb T)$ -分级悬置。利用$q$ -因子得到了一种滤波。一个关键的成分，这可能是独立的利益，是一个密切的联系之间的Hill-Yarnall表征的切片过滤和Anschütz-le Bras的$q$ -变形勒让德的公式。
英文摘要：
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 $σ$, 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 $σ$. 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.
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