Symbolic power of an ideal

In algebra and algebraic geometry, given a commutative Noetherian ring formula_1 and an ideal formula_2 in it, the "n"-th symbolic power of formula_2 is the ideal
where formula_5 is the localization of formula_1 to formula_7 and the intersection runs through all of the associated primes of formula_8
Though this definition does not require formula_2 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 formula_10-primary component of formula_11. Very roughly, it consists of functions with zeros of order "n" along the variety defined by formula_2. We have: formula_13 and if is a maximal ideal, then formula_14.
Symbolic powers induce the following chain of ideals:
Uses.
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 formula_2 defining a curve in a complete local domain, the powers of formula_2 are cofinal with the symbolic powers of formula_2. This important property of being cofinal was further developed by Schenzel in the 1970s.
In algebraic geometry.
Though generators for "ordinary powers" of formula_2 are well understood when formula_2 is given in terms of its generators as formula_21, it is still very difficult in many cases to determine the generators of symbolic powers of formula_2. But in the geometric setting, there is a clear geometric interpretation in the case when formula_2 is a radical ideal over an algebraically closed field of characteristic zero.
If formula_24 is an irreducible variety whose ideal of vanishing is formula_2, then the "differential power" of formula_2 consists of all the functions in formula_1 that vanish
to order ≥ "n" on formula_24, i.e.
Or equivalently, if formula_30 is the maximal ideal for a point formula_31, formula_32.
Theorem (Nagata, Zariski) Let formula_2 be a prime ideal in a polynomial ring formula_34 over an algebraically closed field. Then
This result can be extended to any radical ideal. 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 formula_37
In fact, if formula_24 is a smooth variety over a perfect field, then
Containments.
It is natural to consider whether or not symbolic powers agree with ordinary powers, i.e. does formula_40 hold? In general this is not the case. One example of this is the prime ideal formula_41. Here we have that formula_42. However, formula_43 does hold and the generalization of this inclusion is well understood. Indeed, the containment formula_44follows from the definition. Further, it is known that formula_45 if and only if formula_46. The proof follows from Nakayama's lemma.
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 and was expanded to positive characteristic by Hochster and Huneke. Their papers both build upon the results of Irena Swanson in "Linear Equivalence of Ideal Topologies" (2000).
Theorem (Ein, Lazarfeld, Smith; Hochster, Huneke) Let formula_47 be a homogeneous ideal. Then the inclusion
It was later verified that the bound of formula_50 in the theorem cannot be tightened for general ideals. However, following a question posed by Bocci, Harbourne, and Huneke, it was discovered that a better bound exists in some cases.
Theorem The inclusion formula_51 for all formula_52 holds 