Remarks on a triangulated version of Auslander-Kleiner's Green correspondence

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.