Hostname: page-component-5b777bbd6c-5mwv9 Total loading time: 0 Render date: 2025-06-20T17:53:52.721Z Has data issue: false hasContentIssue false
  • English
  • Français

SYMBOLIC ANALYTIC SPREAD: UPPER BOUNDS AND APPLICATIONS

Part of: Special varieties General commutative ring theory Theory of modules and ideals Local rings and semilocal rings

Published online by Cambridge University Press:  07 May 2020

Hailong Dao  and
Jonathan Montaño
Hailong Dao
Affiliation:
Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Blvd., Lawrence, KS66045, USA (hdao@ku.edu)
Jonathan Montaño
Affiliation:
Department of Mathematical Sciences, New Mexico State University, PO Box 30001, Las Cruces, NM88003-8001, USA (jmon@nmsu.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

The symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article, we find upper bounds for the symbolic analytic spread under certain conditions in terms of other invariants of $I$. Our methods also work for more general systems of ideals. As applications, we provide bounds for the (local) Kodaira dimension of divisors, the arithmetic rank, and the Frobenius complexity. We also show sufficient conditions for an ideal to be a set-theoretic complete intersection.

Keywords

analytic spreadsymbolic powersarithmetic rankKodaira dimensionset-theoretic complete intersectionFrobenius complexity

MSC classification

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Secondary: 13A15: Ideals; multiplicative ideal theory 14M10: Complete intersections 13C40: Linkage, complete intersections and determinantal ideals 13C15: Dimension theory, depth, related rings (catenary, etc.) 13H15: Multiplicity theory and related topics
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 6 , November 2021 , pp. 1969 - 1981
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aberbach, I. and Polstra, T., Local cohomology bounds and test ideals, Preprint.Google Scholar
Brodmann, M., Asymptotic stability of Ass (M/I n M), Proc. Amer. Math. Soc. 74 (1979), 16–18.Google Scholar
Bruns, W. and Herzog, J., Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, Volume 39, (Cambridge University Press, Cambridge, 1993).Google Scholar
Bruns, W. and Schwänzl, R., The number of equations defining a determinantal variety, Bull. Lond. Math. Soc. 22 (1990), 439–445.CrossRefGoogle Scholar
Bruns, W. and Vetter, U., Determinantal Rings, Lecture Notes in Mathematics, Volume 1327, (Springer, Berlin, 1988).10.1007/BFb0080378CrossRefGoogle Scholar
Burch, L., Codimension and analytic spread, Proc. Cambridge Philos. Soc. 72 (1972), 369–373.CrossRefGoogle Scholar
Cutkosky, S. D., Herzog, J. and Srinivasan, H., Asymptotic growth of algebras associated to powers of ideals, Math. Proc. Cambridge Philos. Soc. 148 (2010), 55–72.CrossRefGoogle Scholar
Dao, H., De Stefani, A., Grifo, E., Huneke, C. and Núñez-Betancourt, L., Symbolic powers of ideals, in Singularities and Foliations. Geometry, Topology and Applications, Springer Proceedings in Mathematics and Statistics, Volume 222, pp. 387–432 (Springer, Cham, 2018).CrossRefGoogle Scholar
De Stefani, A., Grifo, E. and Jeffries, J., A Zariski–Nagata theorem for smooth ℤ-algebras, J. Reine Angew. Math. 761 (2020), 123–140.CrossRefGoogle Scholar
Dutta, S. P., Symbolic powers, intersection multiplicity, and asymptotic behaviour of Tor, J. Lond. Math. Soc. (2) 28 (1983), 261–281.CrossRefGoogle Scholar
Enescu, F. and Pérez, F., The Frobenius exponent of Cartier subalgebras, J. Pure Appl. Algebra 223 (2019), 3879–3888.CrossRefGoogle Scholar
Enescu, F. and Yao, Y., The Frobenius complexity of a local ring of prime characteristic, J. Algebra 459 (2016), 133–156.CrossRefGoogle Scholar
Enescu, F. and Yao, Y., On the Frobenius complexity of determinantal rings, J. Pure Appl. Algebra 222 (2018), 414–432.CrossRefGoogle Scholar
Herzog, J., Hibi, T. and Trung, N. V., Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 210 (2007), 304–322.CrossRefGoogle Scholar
Hoa, L. T., Kimura, K., Terai, N. and Trung, T. N., Stability of depths of symbolic powers of Stanley–Reisner ideal, J. Algebra 473 (2017), 307–323.CrossRefGoogle Scholar
Huneke, C. and Swanson, I., Integral Closures of Ideals, Rings and Modules, London Mathematical Society Lecture Note Series, Volume 336, (Cambridge University Press, 2006).Google Scholar
Giorgi, E., On the irreducible components of the form ring and an application to intersection cycles, Comm. Algebra 34 (2006), 2755–2767.10.1080/00927870600636365CrossRefGoogle Scholar
Goto, S., Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), 151–160.CrossRefGoogle Scholar
Grifo, E. and Huneke, C., Symbolic powers of ideals defining F-pure and strongly F-regular rings, Int. Math. Res. Not. IMRN 2019 (2019), 2999–3014.10.1093/imrn/rnx213CrossRefGoogle Scholar
Katz, D. and Validashti, J., Multiplicity and Rees valuations, Collect. Math. 61 (2010), 1–24.CrossRefGoogle Scholar
Katzman, M., Schwede, K., Singh, A. K. and Zhang, W., Rings of Frobenius operators, Math. Proc. Camb. Phil. Soc. 157 (2014), 151–167.CrossRefGoogle Scholar
Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 195–279.CrossRefGoogle Scholar
Lyubeznik, G., On set-theoretic intersections, J. Algebra 87 (1984), 105–112.CrossRefGoogle Scholar
Lyubeznik, G., On the arithmetical rank of monomial ideals, J. Algebra 112 (1988), 86–89.CrossRefGoogle Scholar
Ma, L. and Schwede, K., Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers, Invent. Math. 214 (2018), 913–955.CrossRefGoogle Scholar
Montaño, J. and Núñez-Betancourt, L., Splittings and symbolic powers of square-free monomial Ideals, Int. Math. Res. Not. IMRN. to appear, rnz138, doi: 10.1093/imrn/rnz138.Google Scholar
Nguyen, H. D. and Trung, N. V., Depth functions of symbolic powers of homogeneous ideals, Invent. Math. 218 (2019), 779–827.10.1007/s00222-019-00897-yCrossRefGoogle Scholar
Page, J., The Frobenius complexity of Hibi rings, J. Pure Appl. Algebra 223 (2019), 580–604.10.1016/j.jpaa.2018.04.008CrossRefGoogle Scholar
Page, J., Smolkin, D. and Tucker, K., Symbolic and ordinary powers of ideals in Hibi rings, Preprint, 2018, arXiv:1810.00149.Google Scholar
Ratliff, L. J., Notes on essentially powers filtrations, Michigan Math. J. 26 (1979), 313–324.CrossRefGoogle Scholar
Roberts, P., A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94 (1985), 589–592.CrossRefGoogle Scholar
Rossi, M. and Valla, G., Hilbert Functions of Filtered Modules, Lect. Notes Unione Mat. Ital., Volume 9, (Springer, Berlin; UMI, Bologna, 2010).Google Scholar
Varbaro, M., Symbolic powers and matroids, Proc. Amer. Math. Soc. 139 (2011), 2357–2366.CrossRefGoogle Scholar