A representation for algebraic K-theory of quasi-coherent modules over affine spectral schemes

TL;DR

This paper develops a new representation for algebraic K-theory of quasi-coherent modules over affine spectral schemes, using sheafification and classifying spaces to better understand its structure.

Contribution

It introduces a novel sheafification-based representation of K-theory for spectral schemes, connecting it with classifying spaces of general linear groups.

Findings

The K-theory functor is represented by a specific group completion.

Sheafification of K-theory aligns with classifying spaces of colimits of affine spectral schemes.

The approach applies to both Zariski and Nisnevich topologies.

Abstract

In this paper, we study K-theory of spectral schemes by using locally free sheaves. Let us regard the K-theory as a functor K on affine spectral schemes. Then, we prove that the group completion $\Omega B^{\mathcal{G}}(B^{\mathcal{G}}GL)$ represents the sheafification of K with respect to Zariski (resp. Nisnevich) topology $\mathcal{G}$, where $B^{\mathcal{G}}GL$ is a classifying space of a colimit of affine spectral schemes $GL_n$.