  Let $(R,\mathfrak{m})$ be a complete local ring, and $G={\rm
gr}_{\mathfrak{m}}(R)$ 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.
