A splitting result for the algebraic K-theory of projective toric schemes

TL;DR

This paper proves a splitting result for the algebraic K-theory of projective toric schemes, showing it decomposes into multiple copies of K(R) based on the acyclicity of a twisted line bundle.

Contribution

It introduces a new splitting theorem for the K-theory of projective toric schemes using combinatorial methods applicable to a broad class of rings.

Findings

K-theory decomposes into k+1 summands K(R)

The splitting depends on the minimal k for which L(-k-1) is not acyclic

Applicable to both commutative and left noetherian rings

Abstract

Suppose X is a projective toric scheme defined over a commutative ring R equipped with an ample line bundle L. We prove that its K-theory has k+1 direct summands K(R) where k is minimal among non-negative integers such that the twisted line bundle L(-k-1) is not acyclic. In fact, using a combinatorial description of quasi-coherent sheaves throughout we prove the result for a ring R which is either commutative, or else left noetherian.