A May-type spectral sequence for higher topological Hochschild homology

TL;DR

This paper develops a spectral sequence for higher topological Hochschild homology (THH) based on filtrations of commutative monoids, providing new computational tools and bounds for THH groups of certain ring spectra.

Contribution

It introduces a May-type spectral sequence for higher THH derived from filtrations of commutative monoids in a symmetric monoidal stable model category, extending computational methods.

Findings

Constructed a spectral sequence linking graded and ungraded higher THH.

Derived bounds on THH groups for $E_{}$-ring spectra with given homotopy rings.

Provided examples of filtrations and their implications for THH computations.

Abstract

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 $A$ such that $\pi_*(A) \cong R$.