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<p>In this paper, we prove that the canonical trace ideal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\text {trace}}_A(\omega _A)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>trace</mtext> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mi>A</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an Ulrich ideal for any two-dimensional rational triple point <i>A</i>. Using this, we classify all Ulrich ideals on rational triple points. Moreover, we show that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((A,{\mathfrak m})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi mathvariant="fraktur">m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a two-dimensional quotient singularity with the multiplicity <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(e \ge 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\mathfrak m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">m</mi> </math></EquationSource> </InlineEquation> is the unique Ulrich ideal of <i>A</i>. As a result, we can classify all Ulrich ideals of <i>A</i> if <i>A</i> is either a rational triple point or a quotient singularity.</p>

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Ulrich Ideals on Rational Triple Points of Dimension Two

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Kyosuke Maeda,
  • Ken-ichi Yoshida

摘要

In this paper, we prove that the canonical trace ideal \({\text {trace}}_A(\omega _A)\) trace A ( ω A ) is an Ulrich ideal for any two-dimensional rational triple point A. Using this, we classify all Ulrich ideals on rational triple points. Moreover, we show that if \((A,{\mathfrak m})\) ( A , m ) is a two-dimensional quotient singularity with the multiplicity \(e \ge 4\) e ≥ 4 , then \({\mathfrak m}\) m is the unique Ulrich ideal of A. As a result, we can classify all Ulrich ideals of A if A is either a rational triple point or a quotient singularity.