Quantum mechanics

Triangulated category

Pure mathematics

Converse

Abelian group

Unification

Power (physics)

Computer science

Physics

Organic chemistry

Cluster (spacecraft)

Mathematics

Subcategory

Geometry

Category of groups

Combinatorics

Functor

Generator (circuit theory)

Chemistry

Programming language

Grothendieck group

Group (periodic table)

Derived category

Journal of Algebra

Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. We consider in this article the Grothendieck group K0(C) of an extriangulated category C. We show that a locally finite extriangulated category C always has almost split extensions and in this case the relations of the Grothendieck group K0(C) are generated by the Auslander-Rieten E-triangles. We give a partial converse result when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category with a cluster tilting subcategory, the relations of the Grothendieck group are generated by AR-triangles if and only if the triangulated category is locally finite. We also show that there is a one-to-one correspondence between subgroups of K0(C) containing the image of G and dense G−(co)resolving subcategories of C where G a generator of C, which generalizes results about classifying subcategories of a triangulated or exact category C by subgroups of K0(C).

Grothendieck groups in extriangulated categories