The May filtration on THH and faithfully flat descent

用于拓扑Hochschild同调计算的Whitehead滤过

Whitehead Filtrations for Computations in Topological Hochschild Homology

本研究利用Whitehead滤过导出的谱序列结合循环不变性，简化连通环谱的拓扑Hochschild同调计算，给出多个具体算例并推广至一般余纤维情形，还构建了2-局部自共轭K-理论对应计算的通用框架。

A slice refinement of Bökstedt periodicity

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…

A spectrum-level Hodge filtration on topological Hochschild homology

We define a functorial spectrum-level filtration on the topological Hochschild homology of any commutative ring spectrum $R$, and more generally the factorization homology $R \otimes X$ for any space $X$, echoing algebraic constructions of Loday and Pirashvili. We investigate the properties of this filtration and show that it breaks THH up into common eigenspectra of the Adams operations.

对数棱镜上同调：通过对数THH实现

Logarithmic Prismatic Cohomology via Logarithmic THH

受Bhatt-Morrow-Scholze启发，利用对数THH滤过定义Nygaard完备化对数棱镜上同调，并将拟正则半完美体的对数THH描述为普通Hochschild同调的1参数形变。

A May-type spectral sequence for higher topological Hochschild homology

Given a filtration of a commutative monoid $A$ in a symmetric monoidal stable model category $\mathcal{C}$, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of $A$, and whose output is the higher order topological Hochschild homology of $A$. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring $R$, we get an upper bound on the size of the $THH$-groups of $E_{\infty}$-ring spectra $A$ su…

Trace theories, Bokstedt periodicity and Bott periodicity

We flesh out the theory of "trace theories" and "trace functors" sketched in arXiv:1308.3743, extend it to a homotopical setting, and prove a reconstruction theorem claiming that a trace theory is completely determined by the associated trace functor. As an application, we consider Topological Hoshschild Homology $THH(A,M)$ of a algebra $A$ over a perfect field of positive characteristic, with coefficients in a bimodule $M$, and prove two comparison results. Firstly, we give a very simple algebraic model for THH in terms of Hochschild-Witt Homology WHH of arXiv:1604.01588 (and we also identify…