Adjoint functors and an extension of the Green correspondence for group representations
The results of this paper are inspired by the observation that the proofs of the Green correspondence and the Burry-Carlson-Puig theorem rely on the Mackey decomposition and on the fact that induction and restriction are adjoint functors. The main results here are then a Green correspondence and Burry-Carlson-Puig theorem for adjoint functors. The general setting is as follows: consider additive categories \(\mathcal D\), \(\mathcal H\), \(\mathcal G\) and functors \(S : {\mathcal H} \to {\mathcal G}\), \(T : {\mathcal G} \to {\mathcal H}\), \(S' : {\mathcal D} \to {\mathcal H}\), \(T' : {\mathcal H} \to {\mathcal D}\), such that \(S\) (\(S'\)) is a left adjoint of \(T\) (\(T'\)), and \(TS = I_{\mathcal H} \oplus U\), and let finally \(\mathcal Y\) be a suitable subcategory of \(\mathcal H\). The Green correspondence is an equivalence between certain quotient categories defined with the aid of the above data. For the Burry-Carlson-Puig type theorem, \(S\) is assumed to be both a left and a right adjoint of \(T\), and \(\mathcal H\), \(\mathcal G\) to be Krull- Schmidt categories. Two statements of uniqueness concerning the Green correspondence are proved, one of them being new even for group representations. The applications to group algebras are obtained by taking the groups \(D \leq H \leq G\), the categories \({\mathcal D} =\text{mod-}RD\), \({\mathcal H} =\text{mod-}RH\), \({\mathcal G} =\text{mod-}RG\) (where \(R\) is a noetherian commutative ring), the restriction functors \(S\), \(S'\) and the induction functors \(T\), \(T'\), and also two different choices for \(\mathcal Y\) are considered. It is to be mentioned that the general results not only throw a new light on known facts, but also provide new results in the particular cases.
scientific article; zbMATH DE number 611205