Homology, mixed multiplicities and Hilbert coefficients of the fiber cone

In this paper, we compare the depth of the fiber cone and the associated graded ring. To achieve this, we construct a bi-graded complex corresponding to a bi-graded, Noetherian, Hilbert filtration. The vanishing of the homology modules of this complex helps us to compare the depth of the fiber cone of the filtration and the depth of the corresponding associated graded ring. We also give a formula for the fiber coefficients in terms of the lengths of certain homology modules. We give an upper bound for the first fiber coefficient and show that when this bound is attained, the fiber cone has good depth.