REMARKS ON A TRIANGULATED VERSION OF AUSLANDER-KLEINER'S GREEN CORRESPONDENCE
Résumé
For a finite group G and an algebraically closed field k of characteristic p > 0 for any indecomposable finite dimensional kG-module M with vertex D and a subgroup H of G containing N G (D) there is a unique indecomposable kH-module N of vertex D being a direct summand of the restriction of M to H. This correspondence, called Green correspondence, was generalised by Auslander-Kleiner to the situation of pairs of adjoint functors between additive categories. In the original situation of group rings Carlson-Peng-Wheeler proved that this correspondence is actually restriction of triangle functors between triangulated quotient categories of the corresponding module categories. We review this theory and show how we got a common generalisation of the approaches of Auslander-Kleiner and Carlson-Peng-Wheeler, using Verdier localisations.
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