On a theory of the b-function in positive characteristic

Abstract

We present a theory of the b-function (or Bernstein-Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic p > 0. Its b-function b f is defined to be an ideal of the algebra of continuous k-valued functions on Z p . The zero-locus of the b-function is thus naturally interpreted as a subset of Z p , which we call the set of roots of b f . We prove that b f has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Mustaţȃ and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of b f and relate it to the test ideals of f.

Length of Local Cohomology in Positive Characteristic and Ordinarity

“…In this note, we compute the positive characteristic D-module length of the first local cohomology module of the structure sheaf with support in a hypersurface, in a large class of examples. Our main result can also be seen as part of our study of the b-function in positive characteristic, see [2]. On the one hand, in [2] using D-module (or unit Fmodule) techniques, for D the ring of Grothendieck differential operators, we associate to a non-constant polynomial f with coefficients in a perfect field of positive characteristic a set of p-adic integers, called the roots of the b-function of f. On the other, one may consider the F -jumping exponents of the generalised test ideals of f, see [12].…”

“…Our main result can also be seen as part of our study of the b-function in positive characteristic, see [2]. On the one hand, in [2] using D-module (or unit Fmodule) techniques, for D the ring of Grothendieck differential operators, we associate to a non-constant polynomial f with coefficients in a perfect field of positive characteristic a set of p-adic integers, called the roots of the b-function of f. On the other, one may consider the F -jumping exponents of the generalised test ideals of f, see [12]. These are positive real numbers which are characterised by their intersection with the unit interval p0, 1s and have been shown to be rational numbers in [7].…”

“…These are positive real numbers which are characterised by their intersection with the unit interval p0, 1s and have been shown to be rational numbers in [7]. In [2] we prove that the roots of the b-function of f are exactly the opposites of the F -jumping exponents of f which are in Q X Z p . It would thus seem that the information provided by the Fjumping exponents of f which are not in Q X Z p , i.e.…”

“…We claim that at the very least the D-module (or unit F -module) length of the module N f used to define the b-function in [2] distinguishes ordinary primes from supersingular ones, for large enough primes. More precisely, using the terminology of [2], one can see that the joint eigenspace of the action of the higher Euler operators on N f corresponding to the root -1 of the b-function of f is isomorphic to the first local cohomology module…”

“…Since this assertion does not appear in the literature, let us give a proof. By [2,Proposition 1], this joint eigenspace is the limit of the direct system: For a d-dimensional proper variety Z over a field of characteristic p ą 0, we let the p-genus g p pZq of Z be the dimension of the stable part [14]…”