D-modules, Bernstein-Sato polynomials and F-invariants of direct summands

We study the structure of D-modules over a ring R which is a direct summand of a polynomial or a power series ring S with coefficients over a field. We relate properties of D-modules over R to D-modules over S. We show that the localization Rf and the local cohomology module HIi(R) have finite length as D-modules over R. Furthermore, we show the existence of the Bernstein-Sato polynomial for elements in R. In positive characteristic, we use this relation between D-modules over R and S to show that the set of F-jumping numbers of an ideal I⊆R is contained in the set of F-jumping numbers of its extension in S. As a consequence, the F-jumping numbers of I in R form a discrete set of rational numbers. We also relate the Bernstein-Sato polynomial in R with the F-thresholds and the F-jumping numbers in R.