A cohomology attached to a function

In this paper, we study a certain cohomology attached to a smooth function, which arose naturally in Poisson geometry. We explain how this cohomology depends on the function, and we prove that it satisfies both the excision and the Mayer-Vietoris axioms. For a regular function we show that the cohomology is related to the de Rham cohomology. Finally, we use it to give a new proof of a well-known result of A. Dimca [Compositio Math. 76 (1990) 19-47] in complex analytic geometry.