Higher K-theory of polynomial categories

TL;DR

This paper proves that the K-theory of an abelian category remains invariant under polynomial extension, generalizing known results about schemes and providing a new perspective on algebraic K-theory.

Contribution

It establishes that the base change functor induces an isomorphism on K-theory for noetherian abelian categories with enough projectives, extending $bA^1$-homotopy invariance to a categorical setting.

Findings

K-theory is invariant under polynomial extension for certain categories

The main theorem generalizes $bA^1$-homotopy invariance to abelian categories

Provides a new framework connecting polynomial categories and algebraic K-theory

Abstract

The main theorem in this paper is that the base change functor from an abelian category $\cA$ to its polynomial category in the sense of Schlichting $-\otimes_{\cA}\bbZ[t]:\cA \to \cA[t]$ induces an isomorphism on their $K$-theories if $\cA$ is noetherian and has enough projective objects. The main theorem implies the well-known fact that $\mathbb{A}^1$-homotopy invariance of $K'$-theory for noetherian schemes.