Finite determination of regular (a,b)-modules

Daniel Barlet (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 IUF - Institut universitaire de France

The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple ''abstract algebraic structure'', but very rich, whose prototype is the formal completion of the Brieskorn-module of an isolated singularity.\\ The aim of this article is to prove a very basic theorem on regular (a,b)-modules showing that a given regular (a,b)-module is completely characterized by some ''finite order jet'' of its structure. Moreover a very simple bound for such a sufficient order is given in term of the rank and of two very simple invariants : the regularity order which count the number of times you need to apply \ $b^{-1}.a \simeq \partial_z.z$ \ in order to reach a simple pole (a,b)-module. The second invariant is the ''width'' which corresponds, in the simple pole case, to the maximal integral difference between to eigenvalues of \ $b^{-1}.a$ \ (the logarithm of the monodromy). \\ In the computation of examples this theorem is quite helpfull because it tells you at which power of \ $b$ \ in the expansions you may stop without loosing any information.

Mots clés: Brieskorn module; (a; b)-module

Domaines: Variables complexes [math.CV]

Daniel Barlet. Finite determination of regular (a,b)-modules. 2007. ⟨hal-00142834v2⟩