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<p>We investigate the nearly Gorenstein property of a local ring defined by the maximal minors of a specific <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10508_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(2 \times n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrix with entries in the formal power series ring <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10508_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">\(k[[X_1, X_2, \ldots , X_n]]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> over a field <i>k</i>. Our findings allow us to present numerous concrete examples, such as nearly Gorenstein rings that are not almost Gorenstein and vice versa.</p>

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Nearly Gorenstein local rings defined by maximal minors of a \(2 \times n\) matrix

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月24日

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  • Shinya Kumashiro,
  • Naoyuki Matsuoka,
  • Taiga Nakashima

摘要

We investigate the nearly Gorenstein property of a local ring defined by the maximal minors of a specific \(2 \times n\) 2 × n matrix with entries in the formal power series ring \(k[[X_1, X_2, \ldots , X_n]]\) k [ [ X 1 , X 2 , … , X n ] ] over a field k. Our findings allow us to present numerous concrete examples, such as nearly Gorenstein rings that are not almost Gorenstein and vice versa.