On the interplay between the Frobenius functor and its dual

For a commutative Noetherian ring [Formula: see text] of prime characteristic, denote by [Formula: see text] the ring [Formula: see text] with the left structure given by the Frobenius map. We develop Thomas Marley’s work on the property of the Frobenius functor [Formula: see text] and show some interplays between F and its dual [Formula: see text], which is introduced by Jürgen Herzog.