Bernstein–sato Polynomials and Test Modules in Positive Characteristic

Abstract

In analogy with the complex analytic case, Mustaţă constructed (a family of) Bernstein-Sato polynomials for the structure sheaf O X and a hypersurface (f = 0) in X, where X is a regular variety over an F -finite field of positive characteristic (see [23]). He shows that the suitably interpreted zeros of his Bernstein-Sato polynomials correspond to the F -jumping numbers of the test ideal filtration τ (X, f t ). In the present paper we generalize Mustaţă's construction replacing 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 .

Functorial test modules

“…Note that it was previously observed in [Stä15b, Proposition 4.2] that τ for F -regular Cartier modules in the sense of [Bli13, Definition 3.4] is functorial. We also remark that many results of [Stä15b] and the main result of [BS16] were proved using the test module theory as developed in [Bli13] under the assumption that (M, κ) is F -regular and that the Cartier algebra is of the form C e = κf tp e R.…”

“…There is, by now, ample evidence that the associated graded of the test module filtration is connected to the étale p-torsion nearby cycles functor (cf. [Stä15b], [Sta14], [Stä15a], [BS16]). However, it also seems to avoid pathologies that only occur for -torsion nearby cycles with = p. The connection is similar to that of the multiplier ideal filtration with nearby cycles in the complex case.…”

Bernstein-Sato Polynomials in Commutative Algebra

“…Relative versions of the above result, for an element in a unit F -module, were considered by Stadnik [Sta14] and Blickle and Stäbler [BS16]. 6.3.…”

Test module filtrations for unit F-modules

“…Fix one such a. By the first part of the proof of [5,Lemma 4.3] (which only requires F -purity and no assumptions on t) we have…”