From a local ring to its associated graded algebra

Let (R,m) be a complete local ring, and G= grₘ(R) be its associated graded ring. We introduce a homogenization technique which allows to relate G to the special fiber and R to the generic fiber of a "Gr\"obner-like" deformation. Using this technique we prove sharp results concerning the connectedness of R and G. We also construct a family of local domains which fail to satisfy Abhyankar's inequality for the Hilbert-Samuel multiplicity. However, we prove a version of the inequality which holds when R is connected in codimension one.