On the Discreteness and Rationality of Jumping Coefficients

In [HY] Hara and Yoshida defined the notion of generalized test ideals τ(a) ⊆ R for ideals a of regular local rings R and non-negative parameters c ∈ R. Since then several papers studied the dependence of these ideals on the real parameter c. Notably, in [BMS1] the authors studied the jumping coefficients originating of these generalized test ideals: these are the non-negative c ∈ R for which τ(a) 6= τ(a) for all ǫ > 0. In [BMS1] it was shown that, when R is of essentially finite type over a field and F -finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In [BMS2] these results have been extended to principal ideals in F -finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily F -finite) complete regular local rings containing fields of positive characteristic. To establish these results we need to understand the jumping coefficients of ideals of power series rings and we henceforth fix R to be the power series ring K[[x1, . . . , xn]] where K is a field of prime characteristic p. We shall denote the Frobenius map of R with f and we let E be the injective hull of the residue field. One way to think about E is as the module of inverse polynomials K[x1 , . . . , x − n ] where we can define an R p linear map sending each x1 1 ·. . . x −αn n (where α1, . . . , αn > 0) to x −pα1 1 ·. . . x −pαn n . We shall think of the R[T ; f ]-leftmodule structure on E arising from this map as its standard R[T ; f ]-left-module structure and use it as the basis for the construction of other non-standard structures. Given a fixed g ∈ R we can define for all a ∈ N and β ∈ N an R[Θa,β; f]-left-module structure (cf. [K] for notation and properties of these skew-polynomial rings) given by Θa,βm = g T m. We shall investigate the R[Θa,β; f ]-submodule of nilpotent elements defined as