Bigraded structures and the depth of blow-up algebras

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 ${\mathrm{\Sigma }}^{I,J}$. The study of the bigraded Hilbert function of $\text{SIJ}$ 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 $g{r}_I(R)$. The module $\text{SIJ}$ can be considered as a refinement of the Sally’s module previously introduced by W. Vasconcelos.