Frobenius and Cartier algebras of Stanley-Reisner rings

摘要

We study the generation of the Frobenius algebra of the injective\udhull of a complete Stanley–Reisner ring over a field with positive\udcharacteristic. In particular, by extending the ideas used by\udM. Katzman to give a counterexample to a question raised by\udG. Lyubeznik and K.E. Smith about the finite generation of Frobenius\udalgebras, we prove that the Frobenius algebra of the injective\udhull of a complete Stanley–Reisner ring can be only principally\udgenerated or infinitely generated. Also, by using our explicit description\udof the generators of such algebra and applying the recent\udwork by M. Blickle about Cartier algebras and generalized test ideals,\udwe are able to show that the set of F -jumping numbers of\udgeneralized test ideals associated to complete Stanley–Reisner rings\udform a discrete subset inside the non-negative real numbers.