Symbolic power of an ideal

in it, the n-th symbolic power of

is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of

to be prime, this assumption is often worked with because in the case of a prime ideal, the symbolic power can be equivalently defined as the

Very roughly, it consists of functions with zeros of order n along the variety defined by

Symbolic powers induce the following chain of ideals: The study and use of symbolic powers has a long history in commutative algebra.

Krull’s famous proof of his principal ideal theorem uses them in an essential way.

They first arose after primary decompositions were proved for Noetherian rings.

Zariski used symbolic powers in his study of the analytic normality of algebraic varieties.

Chevalley's famous lemma comparing topologies states that in a complete local domain the symbolic powers topology of any prime is finer than the m-adic topology.

A crucial step in the vanishing theorem on local cohomology of Hartshorne and Lichtenbaum uses that for a prime

defining a curve in a complete local domain, the powers of

This important property of being cofinal was further developed by Schenzel in the 1970s.

, it is still very difficult in many cases to determine the generators of symbolic powers of

is a radical ideal over an algebraically closed field of characteristic zero.

is an irreducible variety whose ideal of vanishing is

be a prime ideal in a polynomial ring

over an algebraically closed field.

Then This result can be extended to any radical ideal.

[3] This formulation is very useful because, in characteristic zero, we can compute the differential powers in terms of generators as: For another formulation, we can consider the case when the base ring is a polynomial ring over a field.

In this case, we can interpret the n-th symbolic power as the sheaf of all function germs over

is a smooth variety over a perfect field, then It is natural to consider whether or not symbolic powers agree with ordinary powers, i.e. does

does hold and the generalization of this inclusion is well understood.

The proof follows from Nakayama's lemma.

[4] There has been extensive study into the other containment, when symbolic powers are contained in ordinary powers of ideals, referred to as the Containment Problem.

Once again this has an easily stated answer summarized in the following theorem.

It was developed by Ein, Lazarfeld, and Smith in characteristic zero [5] and was expanded to positive characteristic by Hochster and Huneke.

[6] Their papers both build upon the results of Irena Swanson in Linear Equivalence of Ideal Topologies (2000).

[7] Theorem (Ein, Lazarfeld, Smith; Hochster, Huneke) Let

in the theorem cannot be tightened for general ideals.

[8] However, following a question posed[8] by Bocci, Harbourne, and Huneke, it was discovered that a better bound exists in some cases.