On a sequence of Grothendieck groups

TL;DR

This paper demonstrates that a classical exact sequence in K-theory extends to numerical K-groups under specific conditions related to the categories involved, especially concerning compactness preservation or torsion-freeness.

Contribution

It establishes conditions under which a known K-theory exact sequence applies to numerical K-groups, expanding its applicability.

Findings

The sequence descends to numerical K-groups if categories have numerical K-groups.

Preservation of compactness by the quotient functor ensures the sequence's validity.

Torsion-freeness of the K-group of the quotient is sufficient for the sequence to hold.

Abstract

We show that a well-known exact sequence in K-theory for quotients of triangulated categories descends to numerical K-groups provided that the category, the quotient and the category we take the quotient with has a numerical K-group, and if either the quotient functor preserves compactness or the K-group of the quotient is torsion-free.