Local subgroups and the stable category (Q2781403)

Let \(G\) be a finite group and let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(kG\)-Mod denote the category of left \(kG\)-modules and let \(kG\)-mod denote the full subcategory of finitely generated left \(kG\)-modules. If \(M\) and \(M'\) are objects in \(kG\)-Mod, let \(P\Hom_{kG}(M,M')\) denote the \(k\)-subspace of \(\Hom_{kG}(M,M')\) consisting of those maps that factor through a projective \(kG\)-module. The stable category \(kG\)-\(\underline{\text{Mod}}\) consists of the objects of \(kG\)-mod with morphisms from \(M\) to \(M'\) in \(kG\)-Mod defined by: NEWLINE\[NEWLINE\Hom_{kG}(M,M')=\Hom_{kG}(M,M')/P\Hom_{kG}(M,M').NEWLINE\]NEWLINE The full subcategory of \(kG\)-\(\underline{\text{Mod}}\) consisting of finitely generated \(kG\)-modules is denoted by \(kG\)-\(\underline{\text{mod}}\). It is well-known that \(kG\)-\(\underline{\text{Mod}}\) and \(kG\)-\(\underline{\text{mod}}\) are triangulated with translation functor \(\Omega^{-1}\).NEWLINENEWLINENEWLINELet \({\mathcal P}(G)\) denote the collection of all \(p\)-subgroups of \(G\). In Section 3, a category \({\mathcal L}(G,k)\) is defined in which an object \(L\) in \({\mathcal L}(G,k)\) determines a \(kN_G(P)\)-module \(L(P)\) for each \(P\in{\mathcal P}(G)\). Each module \(L(P)\) must satisfy a condition on its variety and the family \(\{L(P)\mid P\in{\mathcal P}(G)\}\) must be compatible under conjugation by arbitrary elements of \(G\) and restriction.NEWLINENEWLINENEWLINESection 3 of this paper presents a complete definition of \(F{\mathcal L}(G,k)\) and defines a canonical functor \({\mathcal F}\colon kG\)-\(\underline{\text{Mod}}\to{\mathcal L}(G,k)\). Section 4 describes how to define the tensor product of a \(kG\)-module \(M\) and an object \(L\) of \({\mathcal L}(G,k)\) to obtain an element of \({\mathcal L}(G,k)\). Finally, in Section 5, it is shown that \({\mathcal F}\colon kG\)-\(\underline{\text{Mod}}\to{\mathcal L}(G,k)\) is a categorical equivalence and that if \(P\in\text{Syl}_p(G)\) and if \(\ell(G,k)\) denotes the full subcategory of objects \(L\) of \({\mathcal L}(G,k)\) such that \(L(P)\) is stably isomorphic to a finitely generated \(kN_G(P)\)-module, then the restriction of \({\mathcal F}\) to \(kG\)-\(\underline{\text{mod}}\), \(f\colon kG\)-\(\text{mod}\to\ell(G,k)\), is also a categorical equivalence.