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Auslander and Kleiner proved in 1994 an abstract version of Greencorrespondence for pairs of adjoint functors between three categories. Theyproduce additive quotients of certain subcategories giving the classical Greencorrespondence in the special setting of modular representation theory.Carlson, Peng and Wheeler showed in 1998 that Green correspondence in theclassical setting of modular representation theory is actually an equivalencebetween triangulated categories with respect to a non standard triangulatedstructure. In the present note we first define and study a version of relativeprojectivity, respectively relative injectivity with respect to pairs ofadjoint functors. We then modify Auslander Kleiner's construction such that thecorrespondence holds in the setting of triangulated categories.

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