Uniqueness of the multiplicative cyclotomic trace

TL;DR

This paper characterizes the multiplicative structures of algebraic K-theory and related trace maps using noncommutative motives, proving their uniqueness and establishing a multiplicative Morita theory.

Contribution

It introduces a multiplicative Morita theory and proves the uniqueness of the multiplicative Dennis and cyclotomic trace maps in algebraic K-theory.

Findings

The space of all multiplicative structures on algebraic K-theory is contractible.

Algebraic K-theory functor is lax symmetric monoidal.

E_n ring spectra induce E_{n-1} ring algebraic K-theory spectra.

Abstract

Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology (TC). Moreover, we prove that the space of all multiplicative structures on algebraic K-theory is contractible. We also show that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, which in particular implies that E_n ring spectra give rise to E_{n-1} ring algebraic K-theory spectra. Along the way, we develop a "multiplicative Morita theory", establishing a symmetric monoidal equivalence between the infinity category of small idempotent-complete stable infinity categories and the Morita localization of the infinity category of spectral categories.