Bernstein-Sato polynomials and test modules in positive characteristic

In analogy with the complex analytic case, Mustaţă constructed (a family of) Bernstein-Sato polynomials for the structure sheaf $\mathcal{O}_X$ and a hypersurface $(f=0)$ in $X$, where $X$ is a regular variety over an $F$-finite field of positive characteristic (see arxiv:0711.3794). He shows that the suitably interpreted zeros of his Bernstein-Sato polynomials correspond to the jumping numbers of the test ideal filtration $τ(X,f^t)$. In the present paper we generalize Mustaţă's construction replacing $\mathcal{O}_X$ by an arbitrary $F$-regular Cartier module $M$ on $X$ and show an analogous correspondence of the zeros of our Bernstein-Sato polynomials with the jumping numbers of the associated filtration of test modules $τ(M,f^t)$ provided that $f$ is a non-zero divisor on $M$.

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