Abstract
Let S be a Cohen-Macaulay ring which is local or standard graded over a field, and let I be an unmixed ideal that is generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal J⊆I(m) equals the m-th symbolic power I(m) of I. Second, we provide a saturation-type formula to compute I(m) and employ it to deduce a theoretical criterion for when I(m)=Im. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of I(m). Along the way, we prove a generalized version of a conjecture raised by Eisenbud and Mazur about annS(I(m)/Im), and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals.
Original language | English |
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Pages (from-to) | 1-21 |
Number of pages | 21 |
Journal | Journal of Algebra |
Volume | 658 |
DOIs | |
State | Published - Nov 15 2024 |
Bibliographical note
Publisher Copyright:© 2024 Elsevier Inc.
Keywords
- Jacobian ideal
- Powers of ideals
- Symbolic defect
- Symbolic power
ASJC Scopus subject areas
- Algebra and Number Theory