Iterated socles and integral dependence in regular rings

Alberto Corso; Shiro Goto; Craig Huneke; Claudia Polini; Bernd Ulrich

doi:10.1090/tran/6926

Iterated socles and integral dependence in regular rings

Alberto Corso, Shiro Goto, Craig Huneke, Claudia Polini, Bernd Ulrich

Mathematics

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3 Scopus citations

Abstract

Let R be a formal power series ring over a field, with maximal ideal m, and let I be an ideal of R. We study iterated socles of I, that is, ideals of the form I:_R m^s for positive integers s. 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 I, with reduction number at most one, provided s ≤ o(I₁ (ϕ_d)) − 1, where o(I₁ (ϕ_d)) is the order of the ideal of entries of the last map in a minimal free R-resolution of R/I. In characteristic zero, we also provide formulas for the generators of iterated socles whenever s ≤ o(I₁ (ϕ_d)). This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal I in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free R-resolution of R/I. 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.

Original language	English
Pages (from-to)	53-72
Number of pages	20
Journal	Transactions of the American Mathematical Society
Volume	370
Issue number	1
DOIs	https://doi.org/10.1090/tran/6926
State	Published - Jan 2018

Bibliographical note

Funding Information:
Received by the editors March 27, 2015, and, in revised form, January 22, 2016. 2010 Mathematics Subject Classification. Primary 13B22, 13D02, 13N15; Secondary 13C40, 13D07. Key words and phrases. Socle of a local ring, Jacobian ideals, integral dependence of ideals, free resolutions, determinantal ideals. The third author was partially supported by NSF grant DMS-1259142. The fourth author was partially supported by NSF grant DMS-1202685 and NSA grant H98230-12-1-0242. The fifth author was partially supported by NSF grant DMS-1205002 and as a Simons Fellow.

Funding Information:
Part of this work was done at the Mathematical Sciences Research Institute (MSRI) in Berkeley, where the authors spent time in connection with the 2012-13 thematic year on Commutative Algebra, supported by NSF grant 0932078000. The authors would like to thank MSRI for its hospitality and partial support. The authors would also like to thank David Eisenbud for helpful discussions about the material of this paper.

Publisher Copyright:
© 2017 American Mathematical Society.

Keywords

Determinantal ideals
Free resolutions
Integral dependence of ideals
Jacobian ideals
Socle of a local ring

ASJC Scopus subject areas

Mathematics (all)
Applied Mathematics

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10.1090/tran/6926

Cite this

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title = "Iterated socles and integral dependence in regular rings",

abstract = "Let R be a formal power series ring over a field, with maximal ideal m, and let I be an ideal of R. We study iterated socles of I, that is, ideals of the form I:R ms for positive integers s. 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 I, with reduction number at most one, provided s ≤ o(I1 (ϕd)) − 1, where o(I1 (ϕd)) is the order of the ideal of entries of the last map in a minimal free R-resolution of R/I. In characteristic zero, we also provide formulas for the generators of iterated socles whenever s ≤ o(I1 (ϕd)). This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal I in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free R-resolution of R/I. 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.",

keywords = "Determinantal ideals, Free resolutions, Integral dependence of ideals, Jacobian ideals, Socle of a local ring",

author = "Alberto Corso and Shiro Goto and Craig Huneke and Claudia Polini and Bernd Ulrich",

note = "Funding Information: Received by the editors March 27, 2015, and, in revised form, January 22, 2016. 2010 Mathematics Subject Classification. Primary 13B22, 13D02, 13N15; Secondary 13C40, 13D07. Key words and phrases. Socle of a local ring, Jacobian ideals, integral dependence of ideals, free resolutions, determinantal ideals. The third author was partially supported by NSF grant DMS-1259142. The fourth author was partially supported by NSF grant DMS-1202685 and NSA grant H98230-12-1-0242. The fifth author was partially supported by NSF grant DMS-1205002 and as a Simons Fellow. Funding Information: Part of this work was done at the Mathematical Sciences Research Institute (MSRI) in Berkeley, where the authors spent time in connection with the 2012-13 thematic year on Commutative Algebra, supported by NSF grant 0932078000. The authors would like to thank MSRI for its hospitality and partial support. The authors would also like to thank David Eisenbud for helpful discussions about the material of this paper. Publisher Copyright: {\textcopyright} 2017 American Mathematical Society.",

year = "2018",

month = jan,

doi = "10.1090/tran/6926",

language = "English",

volume = "370",

pages = "53--72",

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TY - JOUR

T1 - Iterated socles and integral dependence in regular rings

AU - Corso, Alberto

AU - Goto, Shiro

AU - Huneke, Craig

AU - Polini, Claudia

AU - Ulrich, Bernd

N1 - Funding Information: Received by the editors March 27, 2015, and, in revised form, January 22, 2016. 2010 Mathematics Subject Classification. Primary 13B22, 13D02, 13N15; Secondary 13C40, 13D07. Key words and phrases. Socle of a local ring, Jacobian ideals, integral dependence of ideals, free resolutions, determinantal ideals. The third author was partially supported by NSF grant DMS-1259142. The fourth author was partially supported by NSF grant DMS-1202685 and NSA grant H98230-12-1-0242. The fifth author was partially supported by NSF grant DMS-1205002 and as a Simons Fellow. Funding Information: Part of this work was done at the Mathematical Sciences Research Institute (MSRI) in Berkeley, where the authors spent time in connection with the 2012-13 thematic year on Commutative Algebra, supported by NSF grant 0932078000. The authors would like to thank MSRI for its hospitality and partial support. The authors would also like to thank David Eisenbud for helpful discussions about the material of this paper. Publisher Copyright: © 2017 American Mathematical Society.

PY - 2018/1

Y1 - 2018/1

N2 - Let R be a formal power series ring over a field, with maximal ideal m, and let I be an ideal of R. We study iterated socles of I, that is, ideals of the form I:R ms for positive integers s. 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 I, with reduction number at most one, provided s ≤ o(I1 (ϕd)) − 1, where o(I1 (ϕd)) is the order of the ideal of entries of the last map in a minimal free R-resolution of R/I. In characteristic zero, we also provide formulas for the generators of iterated socles whenever s ≤ o(I1 (ϕd)). This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal I in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free R-resolution of R/I. 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.

AB - Let R be a formal power series ring over a field, with maximal ideal m, and let I be an ideal of R. We study iterated socles of I, that is, ideals of the form I:R ms for positive integers s. 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 I, with reduction number at most one, provided s ≤ o(I1 (ϕd)) − 1, where o(I1 (ϕd)) is the order of the ideal of entries of the last map in a minimal free R-resolution of R/I. In characteristic zero, we also provide formulas for the generators of iterated socles whenever s ≤ o(I1 (ϕd)). This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal I in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free R-resolution of R/I. 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.

KW - Determinantal ideals

KW - Free resolutions

KW - Integral dependence of ideals

KW - Jacobian ideals

KW - Socle of a local ring

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Iterated socles and integral dependence in regular rings

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