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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((R,\mathfrak {m})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi mathvariant="fraktur">m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a complete local ring, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(G=\textrm{gr}_{\mathfrak {m}}(R)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mtext>gr</mtext> <mi mathvariant="fraktur">m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be its associated graded ring. We introduce a homogenization technique which allows to relate <i>G</i> to the special fiber and <i>R</i> to the generic fiber of a “Gröbner-like” deformation. Using this technique we prove sharp results concerning the connectedness of <i>R</i> and <i>G</i>. 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 <i>R</i> is connected in codimension one.</p>

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From a local ring to its associated graded algebra

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月7日

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  • Alessandro De Stefani,
  • Maria Evelina Rossi,
  • Matteo Varbaro

摘要

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