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Transactions of the American Mathematical Society

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Iterated socles and integral dependence in regular rings


Authors: Alberto Corso, Shiro Goto, Craig Huneke, Claudia Polini and Bernd Ulrich
Journal: Trans. Amer. Math. Soc. 370 (2018), 53-72
MSC (2010): Primary 13B22, 13D02, 13N15; Secondary 13C40, 13D07
DOI: https://doi.org/10.1090/tran/6926
Published electronically: September 19, 2017
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Abstract: Let $ R$ be a formal power series ring over a field, with maximal ideal $ \mathfrak{m}$, and let $ I$ be an ideal of $ R$. We study iterated socles of $ I$, that is, ideals of the form $ I :_R {\mathfrak{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 \leq$$ \text {o}(I_1(\varphi _d))-1$, where $ \text {o}(I_1(\varphi _d))$ is the order of the ideal of entries of the last map in a minimal free $ R$-resolution of $ R/I$. In characteristic zero, we also provide formulas for the generators of iterated socles whenever $ s\leq$$ \text {o}(I_1(\varphi _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.


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Additional Information

Alberto Corso
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: alberto.corso@uky.edu

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: goto@math.meiji.ac.jp

Craig Huneke
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: huneke@virginia.edu

Claudia Polini
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: cpolini@nd.edu

Bernd Ulrich
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: ulrich@math.purdue.edu

DOI: https://doi.org/10.1090/tran/6926
Keywords: Socle of a local ring, Jacobian ideals, integral dependence of ideals, free resolutions, determinantal ideals
Received by editor(s): March 27, 2015
Received by editor(s) in revised form: January 22, 2016
Published electronically: September 19, 2017
Additional Notes: 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.
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