Comparison of abelian categories recollements
type: Item
instance of: scholarly article
title: Comparison of abelian categories recollements
zbMATH Open document ID: 1060.18008
author: Vincent Franjou
author: Teimuraz Pirashvili
published in: Documenta Mathematica
publication date: 2004-08-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0403081
full work available at URL: https://eudml.org/doc/124449
full work available at URL: 03.html
review text: A triple of abelian categories \({\mathcal A}'\), \({\mathcal A}\), \({\mathcal A}''\) is a recollement situation if there is a collection of functors between them which show that \({\mathcal A}'\) is (equivalent to) a localizing and colocalizing subcategory of \({\mathcal A}\), with quotient category \({\mathcal A}''\). If \({\mathcal A}'\), \({\mathcal B}\), \({\mathcal A}''\) is also a recollement situation, a comparison functor from \({\mathcal A}\) to \({\mathcal B}\) is one that respects the functors implicit in the definitions of the recollement situations. The question then arises as to whether a comparison functor is necessarily an equivalence of categories. For abelian categories in general, an example is given to show that a comparison functor need not be an equivalence; on the other hand, it is known that, in the case of triangulated categories, a comparison functor must be an equivalence. In this paper, the authors give a precise criterion for a comparison functor to be an equivalence, under the condition that \({\mathcal A}\) and \({\mathcal B}\) both have enough projectives and injectives. Given two functors \(F,G: {\mathcal A}''\to{\mathcal A}'\) between abelian categories and a natural transformation \(\xi: F\to G\), MacPherson and Vilonen have constructed a canonical recollement \({\mathcal A}(\xi)\). Given that the categories have enough projectives, the authors determine when a given recollement is equivalent to \({\mathcal A}(\xi)\).
Mathematics Subject Classification ID: 18E40
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 18G10
zbMATH DE Number: 2095716
zbMATH Keywords: abelian category
zbMATH Keywords: recollement situation
zbMATH Keywords: localizing subcategory
MaRDI profile type: Publication
arXiv ID: math/0403081
description: scientific article