  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 in complex analytic geometry.
