  Let $R$ be a Cohen-Macaulay local ring, and let $I\subset R$ be an ideal with
minimal reduction $J$. In this paper we attach to the pair $I$, $J$ a
non-standard bigraded module $\Sigma^{I,J}$.
  The study of the bigraded Hilbert function of $\SIJ$ allows us to prove a
improved version of Wang's conjecture and a weak version of Sally's conjecture,
both on the depth of the associated graded ring $gr_I(R)$. The module $\SIJ$
can be considered as a refinement of the Sally's module previously introduced
by W. Vasconcelos.
