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• W2964227910 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a formal power series ring over a field, with maximal ideal <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German m> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>m</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <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> be an ideal of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We study iterated socles 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>, that is, ideals of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I colon Subscript upper R Baseline German m Superscript s> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> </mml:msub> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>m</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>I :_R {mathfrak m}^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for positive integers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We are interested in iterated socles in connection with the notion of integral dependence of ideals. In this article we show that iterated socles are integral over <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>, with reduction number at most one, provided <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s less-than-or-equal-to o left-parenthesis upper I 1 left-parenthesis phi Subscript d Baseline right-parenthesis right-parenthesis minus 1> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mtext>o</mml:mtext> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>φ<!-- φ --></mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>s leq text {o}(I_1(varphi _d))-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=o left-parenthesis upper I 1 left-parenthesis phi Subscript d Baseline right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:mtext>o</mml:mtext> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>φ<!-- φ --></mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>text {o}(I_1(varphi _d))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the order of the ideal of entries of the last map in a minimal free <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-resolution of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R slash upper I> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>R/I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In characteristic zero, we also provide formulas for the generators of iterated socles whenever <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s less-than-or-equal-to o left-parenthesis upper I 1 left-parenthesis phi Subscript d Baseline right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mtext>o</mml:mtext> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>φ<!-- φ --></mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sleq text {o}(I_1(varphi _d))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal <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> in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-resolution of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R slash upper I> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>R/I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Applications are given to iterated socles of determinantal ideals with generic height. In particular, we give surprisingly simple formulas for iterated socles of height two ideals in a power series ring in two variables. The generators of these socles are suitable determinants obtained from the Hilbert-Burch matrix." @default.
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• W2964227910 date "2017-09-19" @default.
• W2964227910 modified "2023-10-18" @default.
• W2964227910 title "Iterated socles and integral dependence in regular rings" @default.
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