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• W2092458402 abstract "The core of an ideal is the intersection of all its reductions. In 2005, Polini and Ulrich explicitly described the core as a colon ideal of a power of a single reduction and a power of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a broader class of ideals, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an ideal in a local Cohen-Macaulay ring. In this paper, we show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an ideal of analytic spread <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a Noetherian local ring with infinite residue field, then with some mild conditions on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we have <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c o r e left-parenthesis upper I right-parenthesis superset-of-or-equal-to upper J left-parenthesis upper J Superscript n Baseline colon upper I Superscript n Baseline right-parenthesis equals upper I left-parenthesis upper J Superscript n Baseline colon upper I Superscript n Baseline right-parenthesis equals left-parenthesis upper J Superscript n plus 1 Baseline colon upper I Superscript n Baseline right-parenthesis intersection upper I> <mml:semantics> <mml:mrow> <mml:mi>core</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>⊇<!-- ⊇ --></mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {core} (I)supseteq J(J^n: I^n)=I(J^n: I^n)=(J^{n+1}: I^n)cap I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any minimal reduction <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding=application/x-tex>J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n much-greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≫<!-- ≫ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ngg 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2092458402 date "2007-11-23" @default.
• W2092458402 modified "2023-09-24" @default.
• W2092458402 title "Core of ideals of Noetherian local rings" @default.
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