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From a local ring to its associated graded algebra

Alessandro De Stefani, Maria Evelina Rossi, Matteo Varbaro

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 "Gröbner-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.

From a local ring to its associated graded algebra

Abstract

Let be a complete local ring, and be its associated graded ring. We introduce a homogenization technique which allows to relate to the special fiber and to the generic fiber of a "Gröbner-like" deformation. Using this technique we prove sharp results concerning the connectedness of and . 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 is connected in codimension one.

Paper Structure

This paper contains 7 sections, 16 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Gröbner deformations for complete local rings
  3. Rings of equal characteristic
  4. Rings of mixed characteristic
  5. Subdimensions and connectedness
  6. Connectedness of the associated graded ring
  7. Bounds on numerical invariants of local domains

Key Result

Theorem A

Let $(R,\mathfrak{m},k)$ be a complete local ring. There exists a complete local ring $\operatorname{hom} (R)$, and non-zero divisor $t$ on $\operatorname{hom} (R)$, such that where $V=k$ if $R$ has equal characteristic, while it is a complete unramified discrete valuation ring with residue field $k$ if $R$ has mixed characteristic.

Theorems & Definitions (54)

  • Theorem A: see Theorems \ref{['t:deform equichar']} and \ref{['t:deform mixed']}
  • Theorem B: see Theorem \ref{['t:connectedness']}
  • Example C: Example \ref{['Ex']}
  • Theorem D: Theorem \ref{['thm h2 and multiplicity linear forms']}
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 44 more