Algebraic K-theory of endomorphism rings

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 $K_n{(}_{\mathcal{C}}(X\oplus Y))\simeq K_n{(}_{\mathcal{C},Y}(X))\oplus K_n{(}_{\mathcal{C}}(Y))$ 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 ${}_{R}J$ 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 $\tilde{A}$.

21 pages. Representation-theoretic methods are used to study the algebraic K-theory of rings

References in corpus (2)

• Higher algebraic $K$-groups and $\mathcal{D}$-split sequences
• Stratifications of derived categories from tilting modules over tame hereditary algebras

Cited by in corpus (3)

• Recollements of derived categories II: Algebraic K-theory
• Ladders and simplicity of derived module categories
• Cellularity of centrosymmetric matrix algebras and Frobenius extensions