Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Some finiteness properties of Lyubeznik's F-modules by Melvin HOCHSTER, University of Michigan Abstract: Lyubeznik proved that for F-modules over a regular ring that is a finitely generated algebra over a regular local ring of characteristic p > 0, F-finite implies DCC, i.e., finite length as an F-module. We discuss Lyubeznik's results, and then strengthen them by showing that whenever DCC holds for an F-module over a regular ring, there are only finitely many F-submodules. We also discuss consequences of this result for Artinian modules M over an arbitrary local ring with an action of Frobenius F such that F(rm) = F(r)F(m) for all r in the ring and and m in the module. Part of this work is joint with Florian Enescu.