Idempotents in nilpotent quotients and triangulated categories
type: Item
instance of: scholarly article
title: Idempotents in nilpotent quotients and triangulated categories
author: Teimuraz Pirashvili
published in: Journal of Pure and Applied Algebra
publication date: 2024-09-03T00:00:00Z
review text: Recall that an ideal \(\mathbb I\) of an additive category \(\mathbb A\) is a subbifunctor of the bifunctor \(\Hom_{\mathbb A}(-, -) \colon\mathbb A^{op} \times\mathbb A \to Ab\), where \(Ab\) is the category of abelian groups. The quotient category \(\mathbb A/\mathbb I\), which has the same objects as \(\mathbb A\), while morphisms in \(\mathbb A/\mathbb I\) are given by \(\Hom_{\mathbb A/\mathbb I}(A, B) := \Hom_{\mathbb A}(A, B)/\mathbb I(A, B)\).\N\NLet \(\mathbb I\) and \(\mathbb J\) be ideals of \(\mathbb A\), the product \(\mathbb I\mathbb J\) is defined by: For all objects \(A\) and \(B\), \( \mathbb I\mathbb J(A, B)\) is the set of all products \(fg\), where \(f \in \mathbb I(C, B)\) and \(g \in \mathbb J(A, C)\), for some \(C\). Am ideal \(\mathbb I\) is called nilpotent if \(\mathbb I^n=0\) for some \(n\in\mathbb N\).\N\NFor a nilpotent ideal \(\mathbb I\) of an additive category \(\mathbb A\) and \(e \colon A \to A\) an idempotent in \(\mathbb A\), the author proves that \(e\) splits iff \(F(e)\) splits, where \(F \colon \mathbb A \to \mathbb A/\mathbb I\) is the canonical functor. As an application to triangulated categories, the author gives a short proof of [\textit{J. Le} and \textit{X.-W. Chen}, J. Algebra 310, No. 1, 452--457 (2007; Zbl 1112.18009), Proposition 2.3], which is a key observation to prove Karoubianness of a triangulated category with bounded t-structure.
reviewed by: Zhe Han
Mathematics Subject Classification ID: 18G80
zbMATH DE Number: 7905688
zbMATH Keywords: nilpotent ideal
zbMATH Keywords: additive categories
zbMATH Keywords: triangulated categories
MaRDI profile type: Publication
cites work: Karoubianness of a triangulated category
cites work: On splitting idempotents in pre-triangulated categories
cites work: Simplicial degrees of functors
cites work: Q5515129
cites work: Q4342453
DOI: 10.1016/J.JPAA.2024.107755
description: scientific article; zbMATH DE number 7905688