The associated graded module of the test module filtration

We show that each direct summand of the associated graded module test filtration τ(M,fλ)λ≥0 admits a natural Cartier structure. If λ is an F-jumping number, then this structure nilpotent on τ(M,fλ−ε)/τ(M,fλ) if and only denominator divisible by p. also these structures coincide with certain are obtained considering D-modules to M were used construct Bernstein-Sato polynomials. Moreover, we point out zeros polynomial bM,f attached F-regular correspond its numbers. This generalizes [6 Blickle, M., Stäbler, A. (2016). polynomials modules in positive characteristic. Nagoya Math. J. 222(1):74–99. DOI: 10.1017/nmj.2016.11.[Crossref], [Web Science ®] , [Google Scholar], Theorem 5.4] where stronger version F-regularity was used. Finally, develop basic theory non-F-pure prove weaker connection between (M,κ) for which Mf jumping numbers M.