LAUSR

Abstract We show that each direct summand of the associated graded module of the test module filtration admits a natural Cartier structure. If λ is an F-jumping number, then this Cartier structure is nilpotent on if and only if the denominator of λ is divisible by p. We also show that these Cartier structures coincide with certain Cartier structures that are obtained by considering certain -modules associated to M that were used to construct Bernstein-Sato polynomials. Moreover, we point out that the zeros of the Bernstein-Sato polynomial attached to an F-regular Cartier module correspond to its F-jumping numbers. This generalizes [6, Theorem 5.4] where a stronger version of F-regularity was used. Finally, we develop a basic theory of non-F-pure modules and prove a weaker connection between Bernstein-Sato polynomials and Cartier modules for which Mf is F-regular and certain jumping numbers attached to M.