On relative projective functors (Q1336474)

Let \(G\) be a finite group, \(k\) a field of characteristic \(p>0\) and \(\text{mmod }kG\) the category of all contravariant functors \(F:\text{mod }kG\to\text{mod }k\). If \(V\) is a \(kG\)-module, consider the functor \((-,V)=\Hom_{kG}(-,V)\) and its subfunctor \(P(-,V)\), where \(P(X,V)=\{f\in\text{Hom}(X,V)\mid f\) factors through a projective \(kG\)-module\}. The main result of the paper states that if \(M\) is an indecomposable \(kG\)-module with vertex \(H\) and source \(Y\) then \(P(-,M)\) is indecomposable in \(\text{mmod }kG\), has vertex \(H\) and source \(P(-,Y)\). The determination of the vertices and sources of the simple functor \(SV=(-,V)/r(-,V)\), with \(V\) a projective indecomposable \(kG\)-module is then deduced as a direct consequence of this theorem.