Bernstein-Sato polynomials and test modules in positive characteristic (Q2828013)

From now on, \(k\) will always denote a field of \textit{prime characteristic} \(p\), \(R\) will denote a commutative Noetherian regular ring, essentially of finite type over \(k\), let \(f\in R\), and let \(M\) be an \(R\)-module.NEWLINENEWLINEFor the convenience of the reader, we review the following notions; first of all, we say that an additive map \(\phi : M\to M\) is \textit{\(p^{-1}\)-linear} if, for any \(r\in R\) and any \(m\in M\), NEWLINE\[NEWLINE \phi (r^p m)=r \phi (m). NEWLINE\]NEWLINE Moreover, a \textit{Cartier module} is just an \(R\)-module \(M\) equipped with a \(p^{-1}\)-linear map \(\phi : M\to M\); in this case, one says that \(M\) is \textit{\(F\)-regular} provided \(\phi\) is surjective and, given any other \(R\)-submodule \(N\subseteq M\) such that \(\phi (N)\subseteq N\) and \(N_{\mathfrak{p}}=M_{\mathfrak{p}}\) for any generic point \(\mathfrak{p}\in\text{Spec}(R)\), then \(N=M\).NEWLINENEWLINEIn the paper under review, the authors define a family of Bernstein-Sato polynomials \(b_{M,f}^e (s)\in\mathbb{Q}[s]\) (see Definition 3.7) for an arbitrary \(F\)-regular Cartier module \(M\); it is worth noting that their definition agree with the one obtained by \textit{M. Mustaţă} [J. Algebra 321, No. 1, 128--151 (2009; Zbl 1157.32012)] where \(M=R\) and \(e\gg 0\). Moreover, their main result about this family of polynomials (see Theorem 5.4) is that, if \(f\) is a non-zero divisor on \(M\), then the roots of \(b_{M,f}^e (s)\) are given for \(e\gg 0\) by NEWLINE\[NEWLINE \frac{\lceil\lambda p^e\rceil-1}{p^e}, NEWLINE\]NEWLINE where \(\lambda\) runs over all the rational numbers \(t\) between \(0\) and \(1\) which are \(F\)-jumping numbers of the test modules \(\tau (M,f^t)\). This can be regarded as the prime characteristic analog of what happens in characteristic zero for the classical Bernstein-Sato polynomial of \(f\); indeed, as a consequence of work by \textit{N. Budur} and \textit{M. Saito} [J. Algebr. Geom. 14, No. 2, 269--282 (2005; Zbl 1086.14013)], it is known that, when \(R=\mathbb{C}[x_1,\ldots ,x_d]\), the jumping numbers of the multiplier ideal \(\mathcal{J} (R,f^t)\) (here, \(t\) runs again through \(0\) and \(1\)) are roots of the classical Bernstein-Sato polynomial \(b_f (s)\); notice that, in this setting, it is well-known that jumping numbers of multiplier ideals are discrete and rational, as a consequence of the existence of resolution of singularities of algebraic varieties over \(\mathbb{C}\).NEWLINENEWLINEFinally, it is also worth noting that in Theorem 6.9 the authors obtain a comparison result between their Bernstein-Sato polynomials and the ones defined by \textit{T. J. Stadnik} in [``The lemma of \(b\)-functions in positive characteristic'', preprint, \url{arXiv:1206.4039}].