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
