Relative cyclotomic structures and equivariant complex cobordism

TL;DR

This paper introduces a new structure on cyclotomic spectra that enables the construction of a relative topological cyclic homology ($TC^{R}$) and explores its applications to equivariant complex cobordism spectra.

Contribution

It develops a framework for relative cyclotomic structures on commutative ring spectra, allowing the construction of $TC^{R}$ and establishing descent results, with applications to equivariant cobordism spectra.

Findings

Constructed $TC^{R}$ for cyclotomic spectra.

Proved descent results relating $TC^{R}$ and $TC$.

Applied framework to equivariant complex cobordism spectra.

Abstract

We describe a structure on a commutative ring (pre)cyclotomic spectrum $R$ that gives rise to a (pre)cyclotomic structure on topological Hochschild homology ($THH$) relative to its underlying commutative ring spectrum. This lets us construct $TC$ relative to $R$, denoted $TC^{R}$, and we prove some descent results relating $TC^{R}$ and $TC$. We explore several examples of this structure on familiar $\mathbb{T}$-equivariant commutative ring spectra including the periodic $\mathbb{T}$-equivariant complex cobordism spectrum $MUP_{\mathbb{T}}$ and a new (connective) equivariant version of the complex cobordism spectrum $MU$.