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.

Splittings and Symbolic Powers of Square-free Monomial Ideals

We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

A CATEGORIFICATION OF A QUANTUM FROBENIUS MAP

A quantum Frobenius map a la Lusztig for $\mathfrak{s}\mathfrak{l}_{2}$ is categorified at a prime root of unity.

SEPARABLY CLOSED FIELDS AND CONTRACTIVE ORE MODULES

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.