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SYMBOLIC ANALYTIC SPREAD: UPPER BOUNDS AND APPLICATIONS

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

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)
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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
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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