Grothendieck groups in extriangulated categories

Abstract

The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulated category C has Auslander-Reiten E−triangles and the relations of the Grothendieck group K0(C ) are generated by the Auslander-Rieten E−triangles. A partial converse result is given when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category C with a cluster tilting subcategory, the relations of the Grothendieck group K0(C ) are generated by Auslander-Reiten triangles if and only if the triangulated category C is locally finite. It is also shown 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 is a generator of C , which generalizes results about classifying subcategories of a triangulated [T] or an exact category C [M] by subgroups of K0(C ).

The Grothendieck Group of an $n$-exangulated Category

“…Inspired by the classification results for triangulated, (n + 2)-angulated and exact categories mentioned above, a natural question to ask is whether there is a similar connection between subcategories and subgroups of the Grothendieck group for n-exangulated categories. Independently of our work, Zhu-Zhuang recently gave a partial answer to this question in the case n = 1 [16,Theorem 5.7]. In this paper we prove a more general classification result for n-exangulated categories with n odd.…”

“…A 1-(co)generator is often just called a (co)generator. Our notion of a (co)generator essentially coincides with what is used in [11] and [16]. There, however, it is not assumed that the subcategory G is additive.…”

Exact structures and degeneration of Hall algebras

“…Here by supp Hom(−, M) we mean the set of indecomposable objects X such that Hom(X, M) = 0. When A is locally finite, thanks to the theorem of Zhu and Zhuang [71] mentioned above, Theorem E applies to all pairs of Ext 1 −finite exact structures.…”

“…Extriangulated categories. Let us note that Zhu and Zhuang [71] actually work in the setting of extriangulated categories. This notion was recently introduced by Nakaoka and Palu [55] as a unification of exact and of triangulated categories.…”

“…Every Krull-Schmidt category A admits a unique maximal admissible exact structure, namely the structure E whose category of defects eff E is the full subcategory of modules of finite length in eff(A, E max ). [71] that the class of Krull-Schmidt categories A whose maximal exact structures are admissible is much larger than that of additively finite categories. In particular, it includes all Hom −finite, locally finite categories.…”