Vol. 308, No. 2, 2020

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Green correspondence and relative projectivity for pairs of adjoint functors between triangulated categories

Alexander Zimmermann

Vol. 308 (2020), No. 2, 473–509
Abstract

Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produced additive quotients of certain subcategories giving the classical Green correspondence in the special setting of modular representation theory. Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the classical setting of modular representation theory is actually an equivalence between triangulated categories with respect to a nonstandard triangulated structure. We first define and study versions of relative projectivity and relative injectivity with respect to pairs of adjoint functors. We then modify Auslander and Kleiner’s construction such that the correspondence holds in the setting of triangulated categories.

Keywords
Green correspondence, relative projectivity, Verdier localisation, triangulated category, adjoint functors, vertex of modules
Mathematical Subject Classification 2010
Primary: 16E35
Secondary: 16S34, 18D10, 18E30, 20C05
Milestones
Received: 3 May 2019
Revised: 6 May 2020
Accepted: 6 June 2020
Published: 9 December 2020
Authors
Alexander Zimmermann
Département de mathématiques et LAMFA (UMR 7352 du CNRS)
Université de Picardie
Amiens
France