Stanley decompositions of multigraded modules $M$ over polynomials rings have
been discussed intensively in recent years. There is a natural notion of depth
that goes with a Stanley decomposition, called the Stanley depth. Stanley
conjectured that the Stanley depth of a module $M$ is always at least the
(classical) depth of $M$. In this paper we introduce a weaker type of
decomposition, which we call Hilbert decomposition, since it only depends on
the Hilbert function of $M$, and an analogous notion of depth, called Hilbert
depth.