The associated graded module of the test module filtration

Abstract

We show that each direct summand of the associated graded module of the test module filtration $\tau (M,f^{\lambda }{)}_{\lambda \ge 0}$ admits a natural Cartier structure. If $\lambda$ is an $F$-jumping number, then this Cartier structure is nilpotent on $\tau (M,f^{\lambda -\epsilon })/\tau (M,f^{\lambda })$ if and only if the denominator of $\lambda$ is divisible by $p$. We also show that these Cartier structures coincide with certain Cartier structures that are obtained by considering certain $\mathcal{D}$-modules associated to $M$ that were used to construct Bernstein-Sato polynomials. Moreover, we point out that the zeros of the Bernstein-Sato polynomial $b_{M,f}$ attached to an \emph{$F$-regular} Cartier module correspond to its $F$-jumping numbers. This generalizes Theorem 5.4 of arXiv:1402.1333 where a stronger version of $F$-regularity was used. Finally, we develop a basic theory of \emph{non-$F$-pure modules} and prove a weaker connection between Bernstein-Sato polynomials $b_{M,f}$ and Cartier modules $(M,\kappa )$ for which $M_f$ is $F$-regular and certain jumping numbers attached to $M$.

arXiv comment15 pages; v2: extended results, new section on non-F-pure modules and on the connection with Bernstein-Sato polynomials, 22 pages v3: fixed a mistake, improved exposition