Higher Algebraic K-theory of Ring Epimorphisms

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 C be an additive category, and let Y → X be a covariant morphism of objects inEnd C ,Y (X) is the quotient ring of the endomorphism ring End 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) ≃ K n (R/J) ⊕ K n (eRe) for all n ∈ 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 typeÃ.

Recollements of derived categories III: finitistic dimensions

“…For covariant morphisms, we have the following result which follows from Corollary and [, Lemma 3.2]. Corollary Let be a covariant morphism in an additive category .…”

“…Recall from that a morphism of objects in an additive category is said to be covariant if the induced map is injective, and the induced map is a split epimorphism of ‐modules. Covariant morphisms capture traces of modules, which guarantee the ubiquity of covariant morphisms (see ).…”

“…Note that D(λ * ) always has a left adjoint functor S ⊗ L R − : D(R) → D(S). For some new advances on homological ring epimorphisms phrased in terms of recollements of derived categories, we refer the reader to [8,9,10]. Applying Corollary 3.6(1) to homological ring epimorphisms, we have the following simple result.…”

“…Applying Corollary 3.12 to triangular matrix rings, we re-obtain the following well-known result (for example, see [14,Corollary 4.21]). Recall from [10] that a morphism λ : Y → X of objects in an additive category C is said to be co-…”

Derived equivalences of algebras

“…For more details on the proof of Theorem and further information on computation formulas for algebraic ‐groups of other type of matrix subrings, we refer the reader to . For applications of recollements to computation of algebraic ‐groups of rings, we refer to .…”