Fundamental theorems for the K-theory of ${\bf S}$-algebras, I: the connective case

TL;DR

This paper extends fundamental theorems in algebraic K-theory to the context of connective S-algebras, providing new localization results for their relative K-theory.

Contribution

It generalizes the Bass-Quillen Fundamental Theorem to Waldhausen K-theory of connective S-algebras and establishes localization theorems for 1-connected morphisms.

Findings

Extension of the Fundamental Theorem to connective S-algebras

Localization theorems for relative K-theory

New techniques for analyzing K-theory of structured ring spectra

Abstract

We extend the Bass-Quillen Fundamental Theorem of Algebraic K-theory to the Waldhausen K-theory of connective ${\bf S}$-algebras. The same technique used in this extension also yields two localization theorems for the relative K-theory of a 1-connected morphism of connective ${\bf S}$-algebras.