  Let R be a local ring of prime characteristic. We study the ring of Frobenius
operators F(E), where E is the injective hull of the residue field of R. In
particular, we examine the finite generation of F(E) over its degree zero
component, and show that F(E) need not be finitely generated when R is a
determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in
good generality that we use, for example, to prove the discreteness of
F-jumping numbers for arbitrary ideals in determinantal rings.
