Algebraic K-theory of finitely projective modules on $\mathbb{E}_{\infty}$-rings

TL;DR

This paper explores the algebraic K-theory of finitely generated projective modules over $ ext{E}_$-rings within spectral algebraic geometry, establishing connections with classical K-theory of ordinary rings.

Contribution

It introduces a framework linking the K-theory of modules over $ ext{E}_$-rings to that of modules over ordinary rings, advancing spectral algebraic geometry.

Findings

Established equivalence between K-theory of spectral modules and classical modules

Extended algebraic K-theory concepts to higher modules in spectral algebraic geometry

Provided new tools for computing K-theory in spectral settings

Abstract

In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain $\mathbb{E}_{\infty}$-rings with the K-theory of an ordinary category of finitely generated projective modules on ordinary rings.