Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <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> be an analytically unramified Cohen-Macaulay local ring of dimension <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$d \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq3.gif" Format="GIF" Height="11" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathfrak {a}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">a</mi> </math></EquationSource> </InlineEquation> be an <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq4.gif" Format="GIF" Height="11" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathfrak {m}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">m</mi> </math></EquationSource> </InlineEquation>-primary ideal in A. If I is an ideal in A, then let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$I^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>I</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> be the integral closure of I in A. Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq6.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="210" /> </InlineMediaObject> <EquationSource Format="TEX">$G_{{\mathfrak {a}}}(A)^ = \bigoplus _{n\ge 0 }({\mathfrak {a}}^n)^/({\mathfrak {a}}^{n+1})^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi mathvariant="fraktur">a</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> <mo>=</mo> <msub> <mo>⨁</mo> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="fraktur">a</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="fraktur">a</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> be the associated graded ring of the integral closure filtration of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq3.gif" Format="GIF" Height="11" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathfrak {a}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">a</mi> </math></EquationSource> </InlineEquation>. Itoh conjectured in 1992 that if the third Hilbert coefficient of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$G_{{\mathfrak {a}}}(A)^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi mathvariant="fraktur">a</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, i.e., <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq9.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$e_3^{{\mathfrak {a}}^}(A) = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>e</mi> <mn>3</mn> <msup> <mrow> <mi mathvariant="fraktur">a</mi> </mrow> <mo>∗</mo> </msup> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and A is Gorenstein then <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3852_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$G_{{\mathfrak {a}}}(A)^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi mathvariant="fraktur">a</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> is Cohen-Macaulay. In this paper we prove Itoh’s conjecture (more generally for analytically unramified Cohen-Macaulay local rings).

A solution to Itoh’s conjecture for integral closure filtration

期刊论文
发表日期：

2025年9月20日

查看全文

Tony J. Puthenpurakal

摘要

Let $(A,{\mathfrak {m}})$ $(A, m)$ be an analytically unramified Cohen-Macaulay local ring of dimension $d \ge 3$ $d \geq 3$ and let ${\mathfrak {a}}$ $a$ be an ${\mathfrak {m}}$ $m$ -primary ideal in A. If I is an ideal in A, then let $I^*$ $I^{*}$ be the integral closure of I in A. Let $G_{{\mathfrak {a}}}(A)^* = \bigoplus _{n\ge 0 }({\mathfrak {a}}^n)^*/({\mathfrak {a}}^{n+1})^*$ $G_{a} {(A)}^{*} = ⨁_{n \geq 0} {(a^{n})}^{*} / {(a^{n + 1})}^{*}$ be the associated graded ring of the integral closure filtration of ${\mathfrak {a}}$ $a$ . Itoh conjectured in 1992 that if the third Hilbert coefficient of $G_{{\mathfrak {a}}}(A)^*$ $G_{a} {(A)}^{*}$ , i.e., $e_3^{{\mathfrak {a}}^*}(A) = 0$ $e_{3}^{a^{*}} (A) = 0$ and A is Gorenstein then $G_{{\mathfrak {a}}}(A)^*$ $G_{a} {(A)}^{*}$ is Cohen-Macaulay. In this paper we prove Itoh’s conjecture (more generally for analytically unramified Cohen-Macaulay local rings).