From a local ring to its associated graded algebra

TL;DR

This paper introduces a homogenization technique linking a complete local ring and its associated graded ring to their fibers, proving connectedness results and exploring Hilbert-Samuel multiplicity inequalities.

Contribution

It develops a homogenization method relating local rings to their associated graded rings and proves new connectedness and multiplicity inequalities.

Findings

Established sharp connectedness results for $R$ and $G$.

Constructed local domains violating Abhyankar's inequality.

Proved a version of the inequality for rings connected in codimension one.

Abstract

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.