The Frobenius functor and injective modules

Abstract

We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show that it includes all one-dimensional F F -pure rings. We also give a characterization of Gorenstein local rings in terms of T o r i R ( R f , E ) \mathrm {Tor}_i^R(R^{f},E) , where E E is the injective hull of the residue field and R f R^{f} is the ring R R whose right R R -module action is given by the Frobenius map.