The Grothendieck group of an \(n\)-angulated category
type: Item
instance of: scholarly article
title: The Grothendieck group of an \(n\)-angulated category
zbMATH Open document ID: 1291.18015
author: Petter Andreas Bergh
author: Marius Thaule
published in: Journal of Pure and Applied Algebra
publication date: 2014-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1205.5697
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 6244918
zbMATH Keywords: \(n\)-angulated category
zbMATH Keywords: triangulated category
zbMATH Keywords: Grothendieck group
zbMATH Keywords: cluster tilting category
zbMATH Keywords: tensor ideal
MaRDI profile type: Publication
OpenAlex ID: W2963167842
arXiv ID: 1205.5697
cites work: Q3096591
cites work: The axioms for \(n\)-angulated categories
cites work: Refining thick subcategory theorems
cites work: n-angulated categories
cites work: Q4132708
cites work: Axiomatic stable homotopy theory
cites work: The connection between May's axioms for a triangulated tensor product and Happel's description of the derived category of the quiver \(D_4\)
cites work: Cluster-tilted algebras are Gorenstein and stably Calabi-Yau
cites work: \(K\)-theory and patching for categories of complexes
cites work: Q4215784
cites work: The additivity of traces in triangulated categories
cites work: Q3128212
DOI: 10.1016/J.JPAA.2013.06.007
review text: An \(n\)-angulated category is a ``higher-dimensional'' analogue of a triangulated category. Where the latter is an additive category with a class of so-called triangles which are in some sense the equivalent of short exact sequences in an abelian category, in the former we are given sequences of length \(n\) which have a similar purpose. In fact, a \(3\)-angulated category is precisely a triangulated category. In the paper under review the authors define and study the Grothendieck group of an \(n\)-angulated category. Roughly speaking, one takes, similar to the triangulated case, the free abelian group on the set of isomorphism clases of objects modulo the Euler relations corresponding to the \(n\)-angles.NEWLINENEWLINEIn Section 2 the authors recall the definition of an \(n\)-angulated category, define its Grothendieck group and establish some of its basic properties. In the following section the following example is studied: If \(\mathcal{T}\) is a triangulated category and \(\mathcal{C}\) a so-called \((n-2)\)-cluster tilting subcategory satisfying an additional technical condition, then \(\mathcal{C}\) can be made into an \(n\)-angulated category. It is then shown that its Grothendieck group \(K_0(\mathcal{C})\) surjects onto the Grothendieck group of \(\mathcal{T}\).NEWLINENEWLINEIn Section 4 the main result of the paper is proved. Namely, if \(\mathcal{C}\) is an \(n\)-angulated category and \(n\) is odd, there is a one-to-one correspondence between subgroups of \(K_0(\mathcal{C})\) and complete and dense \(n\)-angulated subcategories of \(\mathcal{C}\). Here, a subcategory \(\mathcal{A}\) is dense if every object in \(\mathcal{C}\) is a direct summand of an object of \(\mathcal{C}\) and it is complete if the following holds: in an \(n\)-angle in \(\mathcal{C}\) in which \(n-1\) vertices are in \(\mathcal{A}\), the last vertex is also in \(\mathcal{A}\). This result generalizes the corresponding statement for a triangulated category proved by Thomason.NEWLINE…
description: scientific article; zbMATH DE number 6244918