Higher algebraic K-groups and $${\mathcal D}$$ -split sequences

Abstract

In this paper, we use D-split sequences and derived equivalences to provide formulas for calculation of higher algebraic K-groups (or mod-p K-groups) of certain matrix subrings which cover tiled orders, rings related to chains of Glaz-Vasconcelos ideals, and some other classes of rings. In our results, we do not assume any homological requirements on rings and ideals under investigation, and therefore extend sharply many existing results of this type in the algebraic K-theory literature to a more general context.

Higher Algebraic K-theory of Ring Epimorphisms

“…A submodule Y of X is a trace in X if Hom R (Y, X/Y ) = 0, and a weak trace in X if the inclusion from Y to X induces an isomorphism Hom R (Y,Y ) → Hom R (Y, X) of abelian groups. For example, every idempotent ideal of R is a trace of the regular R-module R R, and every GV-ideal J of R is a weak trace of R R (see [28,Section 7] for definition). Also, the socle of any finite dimensional algebra A over a field is a weak trace of A A.…”

“…Note that rings of the form in Corollary 1.3 not only cover some of tiled orders, Hecke orders, and minimal model program for orders over surfaces (see [20,21,4]), but also occur in commutative rings (see [28,Section 7]) and stratification of derived module categories arising from infinitely generated tilting modules over tame hereditary algebras (see [5]). Theorem 1.1 can also be applied to affine cellular algebras (see [13]) and reduces their algebraic K-theory to the one of affine commutative rings.…”

“…Hopefully, this might lead to some knowledge on comprehensive understanding of higher algebraic K-groups K n for rings. This paper is a continuation of the project in this direction started in [28] where the techniques of derived equivalences developed in the representation theory of algebras were used to calculate the algebraic K-groups of rings. More precisely, we employed D-split sequences introduced in [12] to give formulas for computation of the higher algebraic K-groups of a class of rings including many maximal orders in noncommutative algebraic geometry and in arithmetical representations (see [4,20]).…”

“…More precisely, we employed D-split sequences introduced in [12] to give formulas for computation of the higher algebraic K-groups of a class of rings including many maximal orders in noncommutative algebraic geometry and in arithmetical representations (see [4,20]). One of the key techniques used in [28] is to embed a given ring into a big ring which is projective as a module over the given ring. This method is powerful for the rings considered there.…”

“…Thus, it would be interesting to know the K-theory of this kind of rings. This motivates us to consider the following general question (see [28]):…”