Iterated socles and integral dependence in regular rings
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- by Alberto Corso, Shiro Goto, Craig Huneke, Claudia Polini and Bernd Ulrich PDF
- Trans. Amer. Math. Soc. 370 (2018), 53-72 Request permission
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.References
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Additional Information
- Alberto Corso
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- MR Author ID: 348795
- 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
- MR Author ID: 192104
- Email: goto@math.meiji.ac.jp
- Craig Huneke
- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 89875
- Email: huneke@virginia.edu
- Claudia Polini
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 340709
- Email: cpolini@nd.edu
- Bernd Ulrich
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 175910
- Email: ulrich@math.purdue.edu
- 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.
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3717974