Comparison of symbolic and ordinary powers of ideals

Abstract

In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning the behavior of symbolic powers of prime ideals in regular rings finitely generated over a field of characteristic 0) to arbitrary regular rings containing a field. The basic theorem states that in such rings, if P is a prime ideal of height c, then for all n, the symbolic (cn)th power of P is contained in the nth power of P. Results are also given in the non-regular case: one must correct by a power of the Jacobian ideal in rings where the Jacobian ideal is defined.

Cited by in corpus (30)

• Counterexamples to the $I^{(3)}\subset I^2$ containment
• Resurgences for ideals of special point configurations in ${\mathbf{P}}^N$ coming from hyperplane arrangements
• Linear subspaces, symbolic powers and Nagata type conjectures
• Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers
• On the containment problem
• Symbolic powers of sums of ideals
• A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
• Membership criteria and containments of powers of monomial ideals
• Expected resurgences and symbolic powers of ideals
• Computations involving symbolic powers
• The Uniform Symbolic Topology Property for Diagonally $F$-regular Algebras
• The 21 reducible polars of Klein's quartic
• Furstenberg sets and Furstenberg schemes over finite fields
• A Zariski-Nagata theorem for smooth $\mathbb{Z}$-algebras
• A vanishing theorem and symbolic powers of planar point ideals
• Algebraic stories from one and from the other pockets
• Generalizations of Stillman's conjecture via twisted commutative algebras
• Uniform Harbourne-Huneke Bounds via Flat Extensions
• On a class of power ideals
• Uniform Symbolic Topologies via Multinomial Expansions
• On the Sylvester-Gallai and the orchard problem for pseudoline arrangements
• Uniform Symbolic Topologies in Normal Toric Rings
• The Initial Degree of Symbolic Powers of Fermat-like Ideals of Planes and Lines Arrangements
• The Alpha Problem & Line Count Configurations
• Symbolic blowup algebras and invariants associated to certain monomial curves in ${\mathbb{P}}^3$
• Asymptotic Hilbert Polynomial and a bound for Waldschmidt constants
• On the stable Harbourne conjecture for ideals defining space monomial curves
• Saturation bounds for smooth varieties
• Uniform approximation of Abhyankar valuation ideals in function field of prime characteristic
• Asymptotic Behavior of Differential Powers