On the Frobenius twist in the category of unstable modules

The Frobenius twist of strict polynomial functors induces natural morphisms between Ext-groups, which are injective according to Friedlander and Suslin [10]. This fact plays an important role in homological computations of strict polynomial functors. In algebraic topology, the category P-d of homogeneous strict polynomial functors of degree d is connected to the category U of unstable modules over the Steenrod algebra via Hai's functor [11], which is proved to be fully faithful in [4]. The Frobenius twist is extended to the category U, under the form of the double functor Phi, but remains mysterious there. This article aims to study the Frobenius twist Phi and its effects on Ext-groups of unstable modules. We compute explicitly many Ext-groups in U and show that in these cases, the morphisms induced by the Frobenius twist are injective. These results are obtained by constructing the minimal injective resolution of the free unstable module F(1).