Theory of (a, b)-Modules. I

Abstract

The aim of this chapter is to discuss a very simple algebraic structure that gives a systematic approach to a point of view that has appeared in Kyoji Saito [3] and Morihiko Saito [4, 5] in their study of isolated singularities of complex hypersurfaces. The idea is that the basic operation on asymptotic expansions at 0 with one variable (say s) is termwise integration (without constant). This operation is denoted by b. A second operation, denoted by a, is multiplication by s. The commutation relation ab — ba = b ² shows that it is interesting to have a complete b-adic topology to work with. This leads us to a finiteness hypothesis over the ring ℂ[[b]] that is satisfied by the formal completion of the Brieskorn lattice of an isolated hypersurface singularity.