Algebraic $K$-theory of coherent spaces

TL;DR

This paper characterizes the algebraic K-theory of sheaf categories on locally coherent spaces, linking it to scissors congruence K-theory, Topological Hochschild Homology, and measure space K-theory.

Contribution

It provides a new description of the algebraic K-theory of sheaves on locally coherent spaces, including spectra of rings, and explores their connections to other invariants.

Findings

K-theory of sheaves on locally coherent spaces is described explicitly.

Links between scissors congruence K-theory and Topological Hochschild Homology are established.

Algebraic K-theory of measure spaces is analyzed.

Abstract

We give a description of the value of a finitary localizing invariant, such as algebraic $K$-theory, on the category of sheaves on a locally coherent space $X$. This in particular includes all spaces that arise as spectra of commutative rings. As applications we discuss the connection between scissors congruence $K$-theory and Topological Hochschild Homology of certain locally coherent spaces, as well as the algebraic $K$-theory of a measure space.