The algebraic structure of twisted topological Hochschild homology

TL;DR

This paper investigates the algebraic structure of twisted topological Hochschild homology (THH), performs specific computations for certain spectra, and extends the algebraic framework to twisted B"okstedt spectral sequences.

Contribution

It provides new insights into the algebraic structure of twisted THH and extends the twisted B"okstedt spectral sequence to a broader algebraic context.

Findings

Computed $C_2$-twisted THH of the Real bordism spectrum.

Reduced $C_p$-twisted THH of geometric ring $C_p$-spectra to classical THH.

Extended the algebraic structure to the twisted B"okstedt spectral sequence.

Abstract

Topological Hochschild homology (THH) is an invariant of ring spectra developed by B\"okstedt. In recent years many equivariant analogues to THH have emerged. One example is twisted THH which is an invariant of $C_n$-equivariant ring spectra developed by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell. In this paper, we study the algebraic structure of twisted THH, and perform some computations. Specifically, we compute $C_2$-twisted THH of the Real bordism spectrum and show that the $C_p$-twisted THH of geometric ring $C_p$-spectra reduces to a computation of classical THH. We extend the algebraic structure of twisted THH to the twisted B\"okstedt spectral sequence of Adamyk, Gerhardt, Hess, Klang, and Kong. We show that, under appropriate flatness conditions and for $R$ a commutative ring $C_p$-spectrum, the $C_p$-twisted B\"okstedt spectral sequence is a spectral sequence of commutative $\underline{E}_\star(R)$-algebras.