Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: September 19, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Bernard Angéniol and Monique Lejeune-Jalabert, Calcul différentiel et classes caractéristiques en géométrie algébrique, Travaux en Cours [Works in Progress], vol. 38, Hermann, Paris, 1989 (French). With an English summary. MR 1001363
  • [2] Luchezar L. Avramov, Ragnar-Olaf Buchweitz, Srikanth B. Iyengar, and Claudia Miller, Homology of perfect complexes, Adv. Math. 223 (2010), no. 5, 1731-1781. MR 2592508,
  • [3] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
  • [4] Lindsay Burch, On ideals of finite homological dimension in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 941-948. MR 0229634
  • [5] Alberto Corso, Craig Huneke, Claudia Polini, and Bernd Ulrich, Distance in resolutions with applications to integral dependence, Loewy length, and blowup algebras, in preparation.
  • [6] Alberto Corso, Craig Huneke, and Wolmer V. Vasconcelos, On the integral closure of ideals, Manuscripta Math. 95 (1998), no. 3, 331-347. MR 1612078,
  • [7] Alberto Corso and Claudia Polini, Links of prime ideals and their Rees algebras, J. Algebra 178 (1995), no. 1, 224-238. MR 1358263,
  • [8] Alberto Corso and Claudia Polini, Reduction number of links of irreducible varieties, J. Pure Appl. Algebra 121 (1997), no. 1, 29-43. MR 1471122,
  • [9] Alberto Corso, Claudia Polini, and Wolmer V. Vasconcelos, Links of prime ideals, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 431-436. MR 1269930,
  • [10] Shiro Goto, Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), no. 1, 151-160. MR 887198,
  • [11] Jürgen Herzog, Canonical Koszul cycles, International Seminar on Algebra and its Applications (Spanish) (México City, 1991) Aportaciones Mat. Notas Investigación, vol. 6, Soc. Mat. Mexicana, México, 1992, pp. 33-41. MR 1310371
  • [12] Jürgen Herzog and Craig Huneke, Ordinary and symbolic powers are Golod, Adv. Math. 246 (2013), 89-99. MR 3091800,
  • [13] Le Tuan Hoa, Jürgen Stückrad, and Wolfgang Vogel, Towards a structure theory for projective varieties of $ {\rm degree}={\rm codimension}+2$, J. Pure Appl. Algebra 71 (1991), no. 2-3, 203-231. MR 1117635,
  • [14] Melvin Hochster and John A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058. MR 0302643
  • [15] Anthony A. Iarrobino, Tangent cone of a Gorenstein singularity, Proceedings of the conference on algebraic geometry (Berlin, 1985), Teubner-Texte Math., vol. 92, Teubner, Leipzig, 1986, pp. 163-176. MR 922910
  • [16] Anthony A. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra, Mem. Amer. Math. Soc. 107 (1994), no. 514, viii+115. MR 1184062,
  • [17] Anthony A. Iarrobino and Jacques Emsalem, Some zero-dimensional generic singularities; finite algebras having small tangent space, Compositio Math. 36 (1978), no. 2, 145-188. MR 515043
  • [18] Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975
  • [19] Joseph Lipman, Adjoints of ideals in regular local rings, Math. Res. Lett. 1 (1994), no. 6, 739-755. With an appendix by Steven Dale Cutkosky. MR 1306018,
  • [20] Claudia Polini and Bernd Ulrich, Linkage and reduction numbers, Math. Ann. 310 (1998), no. 4, 631-651. MR 1619911,
  • [21] Günter Scheja and Uwe Storch, Über Spurfunktionen bei vollständigen Durchschnitten, J. Reine Angew. Math. 278/279 (1975), 174-190 (German). MR 0393056
  • [22] Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
  • [23] John Tate, The different and the discriminant. Appendix to: Barry Mazur and Lawrence Roberts, Local Euler characteristics, Invent. Math. 9 (1970), 201-234. MR 0258844
  • [24] Wolmer Vasconcelos, Integral closure: Rees algebras, multiplicities, algorithms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2153889
  • [25] Hsin-Ju Wang, Links of symbolic powers of prime ideals, Math. Z. 256 (2007), no. 4, 749-756. MR 2308888,
  • [26] Kei-ichi Watanabe and Ken-ichi Yoshida, A variant of Wang's theorem, J. Algebra 369 (2012), 129-145. MR 2959790,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13B22, 13D02, 13N15, 13C40, 13D07

Retrieve articles in all journals with MSC (2010): 13B22, 13D02, 13N15, 13C40, 13D07

Additional Information

Alberto Corso
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan

Craig Huneke
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Claudia Polini
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Bernd Ulrich
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

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.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society