Algebraic K-theory of endomorphism rings

TL;DR

This paper provides formulas for computing higher algebraic K-groups of endomorphism rings in additive categories, simplifying calculations by relating them to quotient and corner rings, with applications to various algebraic structures.

Contribution

It introduces new formulas for calculating higher algebraic K-groups of endomorphism rings, linking them to quotient and corner rings, expanding computational tools in algebraic K-theory.

Findings

K-groups of endomorphism rings decompose via quotient rings.

K-groups of a ring relate to those of quotient and corner rings under certain conditions.

Applicable to stratified rings, hereditary orders, and affine Hecke algebras.

Abstract

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism of objects in ${\mathcal C}$. Then $K_n\big(_{\mathcal C}(X\oplus Y)\big)\simeq K_n\big(_{{\mathcal C},Y}(X)\big)\oplus K_n\big(_{\mathcal C}(Y)\big)$ for all $1\le n\in \mathbb{N}$, where $_{{\mathcal C},Y}(X)$ is the quotient ring of the endomorphism ring $_{\mathcal C}(X)$ of $X$ modulo the ideal generated by all those endomorphisms of $X$ which factorize through $Y$. Moreover, let $R$ be a ring with identity, and let $e$ be an idempotent element in $R$. If $J:=ReR$ is homological and $_RJ$ has a finite projective resolution by finitely generated projective $R$-modules, then $K_n(R)\simeq K_n(R/J)\oplus K_n(eRe)$ for all $n\in \mathbb{N}$. This reduces calculations of the higher algebraic $K$-groups of $R$ to those of the quotient ring $R/J$ and the corner ring $eRe$, and can be applied to a large variety of rings: Standardly stratified rings, hereditary orders, affine cellular algebras and extended affine Hecke algebras of type $\tilde{A}$.