Bernstein-Sato polynomials and test modules in positive characteristic

2014/02/06 by Manuel Blickle, Blickle, Manuel, Axel Stäbler +1
Mathematics · #13A35 (Primary) #14F10(Secondary) #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13A35

paper · pdf · doi:10.48550/arxiv.1402.1333

13 pages; v2: Corrected a mistake in Lemma 4.1, a comparison to Stadnik's theory of $b$-functions (arXiv:1206.4039) was added; final version

arxiv created 2015/04/21 · arxiv updated 2015/04/22

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 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 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.

Citations

• Bernstein-Sato polynomials in positive characteristic
• The Lemma on b-functions in Positive Characteristic

Related

• The associated graded module of the test module filtration
• Bernstein-Sato polynomials in positive characteristic
• Multiplier modules, V-filtrations and Bernstein-Sato polynomials on singular ambient varieties
• Bernstein-Sato theory for arbitrary ideals in positive characteristic
• Bernstein-Sato polynomials of arbitrary varieties