A Fundamental Theorem for the $K$-theory of connective $\mathbb{S}$-algebras

TL;DR

This paper extends the Fundamental Theorem of $K$-theory from simplicial rings to $S$-algebras, broadening its applicability in algebraic topology and higher algebra.

Contribution

It generalizes the Fundamental Theorem of $K$-theory to the setting of $S$-algebras using the density argument, building on recent advances and prior results.

Findings

Extended $K$-theory Fundamental Theorem to $S$-algebras

Proved the theorem for simplicial rings using recent results

Recovered the theorem for the $K$-theory of spaces as a special case

Abstract

Invoking the density argument of Dundas-Goodwillie-McCarthy, we extend the Fundamental Theorem of $K$-theory from the category of simplicial rings to the category of $\mathbb{S}$-algebras. As an intermediate step, we prove the Fundamental Theorem for simplicial rings appealing to recent results from the first author's thesis. This recovers as a special case the Fundamental Theorem for the $K$-theory of spaces appearing in H\"uttemann-Klein-Vogell-Waldhausen-Williams.