View source for Symbolic power of an ideal

In [[algebra]] and [[algebraic geometry]], given a [[commutative]] [[Noetherian ring]] <math>R</math> and an [[Ideal (ring theory)|ideal]] <math>I</math> in it, the '''''n''-th symbolic power''' of <math>I</math> is the ideal

:<math> I^{(n)} = \bigcap_{P\in \operatorname{Ass}(R/I)} \varphi_P^{-1}(I^n R_P) </math>

where <math>R_P </math> is the [[Localization of a ring|localization]] of <math>R</math> at <math>P</math>, we set <math>\varphi_P : R \to R_P</math> is the canonical map from a ring to its localization, and the intersection runs through all of the [[associated prime]]s of <math>R/I</math>.

Though this definition does not require <math>I</math> to be [[prime ideal|prime]], this assumption is often worked with because in the case of a [[prime ideal]], the symbolic power can be equivalently defined as the <math>I </math>-[[Primary decomposition|primary component]] of  <math>I^n</math>. Very roughly, it consists of functions with zeros of order ''n'' along the variety defined by <math>I</math>.  We have: <math>I^{(1)} = I</math> and if <math>I</math> is a [[maximal ideal]], then <math>I^{(n)} = I^n</math>.

Symbolic powers induce the following chain of ideals:

:<math>  I^{(0)}=R\supset I=I^{(1)}\supset I^{(2)}\supset I^{(3)}\supset I^{(4)}\supset \cdots </math>

== Uses ==
The study and use of symbolic powers has a long history in [[commutative algebra]]. [[Wolfgang Krull|Krull’s]] famous proof of his [[Krull's principal ideal theorem|principal ideal theorem]] uses them in an essential way. They first arose after [[primary decomposition]]s were proved for [[Noetherian rings]]. [[Oscar Zariski|Zariski]] used symbolic powers in his study of the analytic [[Normal space|normality]] of [[algebraic varieties]]. [[Claude Chevalley|Chevalley's]] famous lemma comparing [[topology|topologies]] states that in a [[local ring|complete local domain]] the symbolic powers [[topology]] of any [[prime number|prime]] is [[Comparison of topologies|finer]] than the [[completion (algebra)|''m''-adic topology]]. A crucial step in the vanishing theorem on [[local cohomology]] of Hartshorne and Lichtenbaum uses that for a prime <math>I</math> defining a [[curve]] in a [[local ring|complete local domain]], the powers of <math>I</math> are [[cofinal (mathematics)|cofinal]] with the symbolic powers of <math>I</math>. This important property of being [[cofinality|cofinal]] was further developed by Schenzel in the 1970s.<ref name=":0" />

== In algebraic geometry ==
Though [[generator (mathematics)|generators]] for ''ordinary powers'' of <math>I</math> are well understood when <math>I</math> is given in terms of its generators as <math> I = (f_1, \ldots, f_k)</math>, it is still very difficult in many cases to determine the generators of symbolic powers of <math>I</math>. But in the [[Geometry|geometric]] setting, there is a clear geometric interpretation in the case when <math>I</math> is a [[radical ideal]] over an [[algebraically closed field]] of [[characteristic zero]].

If <math>X</math> is an [[Irreducible component|irreducible]] [[algebraic variety|variety]] whose ideal of vanishing is <math>I</math>, then the ''[[Differential ideal|differential power]]'' of <math>I</math>  consists of all the [[Function (mathematics)|functions]] in <math>R</math> that vanish
to order ≥ ''n'' on <math>X</math>, i.e.

:<math>I^{\langle n\rangle} := \{ f \in R \mid f \text{ vanishes to order} \geq n \text{ on all of }X \}.</math>

Or equivalently, if <math>\mathbf{m}_p</math> is the [[maximal ideal]] for a point <math>p\in X</math>, <math>I^{\langle n\rangle}=\bigcap _{p \in X} \mathbf{m}_p^n</math>.

'''Theorem (Nagata, Zariski)'''<ref>David Eisenbud. Commutative Algebra: with a view toward algebraic geometry, volume 150. Springer Science & Business Media, 2013.</ref> Let <math>I</math> be a prime ideal in a [[polynomial ring]] <math>K[x_1, \ldots , x_N ]</math> over an algebraically closed field. Then
:<math>I^{(m)}=I^{\langle m\rangle}</math>
This result can be extended to any [[radical ideal]].<ref>{{cite arXiv |eprint=math/0611696|last1=Sidman|first1=Jessica|title=Prolongations and computational algebra|last2=Sullivant|first2=Seth|year=2006}}</ref> This formulation is very useful because, in [[characteristic zero]], we can compute the differential powers in terms of generators as:

:<math>I^{\langle m\rangle}=\left\langle f \mid \frac{\partial^\mathbf{a}f}{\partial x^\mathbf{a}}\in I \text{ for all }\mathbf{a}\in \mathbb{N}
^N \text{ where }|\mathbf{a}|=\sum_{i=1}^N a_i\leq m-1 \right\rangle </math>

For another formulation, we can consider the case when the base [[Ring (mathematics)|ring]] is a [[polynomial ring]] over a [[field (mathematics)|field]]. In this case, we can interpret the ''n''-th symbolic power as the [[sheaf (mathematics)|sheaf]] of all function [[germ (mathematics)|germs]] over <math> X = \operatorname{Spec}(R) \text{ vanishing to order} \geq n \text{ at } Z=V(I)</math>
In fact, if <math>X</math> is a [[smooth scheme|smooth variety]] over a [[perfect field]], then

:<math> I^ {(n)} = \{f \in R \mid f \in \mathbf{m}^ n \text{ for every closed point }\mathbf{m} \in  Z\}</math><ref name=":0">{{cite arXiv|last=Dao|first=Hailong|last2=De Stefani|first2=Alessandro|last3=Grifo|first3=Eloísa|last4=Huneke|first4=Craig|last5=Núñez-Betancourt|first5=Luis|date=2017-08-09|title=Symbolic powers of ideals|eprint=1708.03010|class=math.AC}}</ref>

== Containments ==
It is natural to consider whether or not symbolic powers agree with ordinary powers, i.e. does <math>I^n=I^{(n)}</math> hold? In general this is not the case. One example of this is the prime ideal <math>P =(x^4-yz,\, y^2-xz, \,x^3y-z^2) \subseteq K[x,y,z]</math>. Here we have that <math>P^2\neq P^{(2)}</math>.<ref name=":0" />  However, <math>P^2\subset P^{(2)}</math> does hold and the generalization of this [[inclusion order|inclusion]] is well understood. Indeed, the containment <math>I^n\subseteq I^{(n)}</math>follows from the definition. Further, it is known that <math>I^r\subseteq I^{(m)}</math> if and only if <math>m\leq r</math>. The proof follows from [[Nakayama's lemma]].<ref name=":2">{{cite conference | last1 = Bauer | first1 = Thomas | last2 = Di Rocco | first2 = Sandra | author2-link = Sandra Di Rocco | last3 = Harbourne | first3 = Brian | last4 = Kapustka | first4 = Michał | last5 = Knutsen | first5 = Andreas | last6 = Syzdek | first6 = Wioletta | last7 = Szemberg | first7 = Tomasz | editor1-last = Bates | editor1-first = Daniel J. | editor2-last = Besana | editor2-first = GianMario | editor3-last = Di Rocco | editor3-first = Sandra | editor4-last = Wampler | editor4-first = Charles W. | contribution = A primer on Seshadri constants | doi = 10.1090/conm/496/09718 | location = Providence, Rhode Island | mr = 2555949 | pages = 33–70 | publisher = American Mathematical Society | series = Contemporary Mathematics | title = Interactions of classical and numerical algebraic geometry: Papers from the conference in honor of Andrew Sommese held at the University of Notre Dame, Notre Dame, IN, May 22–24, 2008 | volume = 496 | year = 2009| arxiv = 0810.0728 }}</ref>

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 <ref>Lawrence Ein, Robert Lazarsfeld, and Karen E Smith. Uniform bounds and symbolic powers on smooth varieties. Inventiones mathematicae, 144(2):241–252, 2001</ref> and was expanded to [[Characteristic (algebra)|positive characteristic]] by Hochster and Huneke.<ref>Melvin Hochster and Craig Huneke. Comparison of symbolic and ordinary powers of ideals. Inventiones mathematicae, 147(2):349–369, 2002.</ref> Their papers both build upon the results of [[Irena Swanson]] in ''Linear Equivalence of Ideal Topologies'' (2000).<ref>[[Irena Swanson]]. Linear equivalence of ideal topologies. Mathematische Zeitschrift, 234(4):755–775, 2000</ref>

'''Theorem  (Ein, Lazarfeld, Smith; Hochster, Huneke)''' Let <math>I\subset K[x_1,x_2,\ldots,x_N]</math> be a [[graded ring|homogeneous ideal]]. Then the inclusion
:<math>I^{(m)}\subset I^r</math>  holds for all <math>m\geq Nr.</math>
It was later verified that the [[upper and lower bounds|bound]] of <math> N</math> in the theorem cannot be tightened for general ideals.<ref name=":1">{{cite arXiv |eprint=0706.3707|last1=Bocci|first1=Cristiano|title=Comparing powers and symbolic powers of ideals|last2=Harbourne|first2=Brian|class=math.AG|year=2007}}</ref> However, following a question posed<ref name=":1" /> by Bocci, Harbourne, and Huneke, it was discovered that a better bound exists in some cases.

'''Theorem''' The inclusion <math>I^{(m)}\subseteq I^r</math> for all <math>m\geq Nr-N+1</math> holds 
# for arbitrary ideals in characteristic 2;<ref>Tomasz Szemberg and Justyna Szpond. On the containment problem. Rendiconti del Circolo Matematico di Palermo Series 2, pages 1–13, 2016.</ref>
# for [[monomial ideal]]s in arbitrary characteristic<ref name=":2" /> 
# for ideals of [[star polygon|d-stars]]<ref name=":1" /> 
# for ideals of general points in [[Projective space|<math>\mathbb{P}^ 2 \text{ and }\mathbb{P}^ 3</math>]]<ref>Marcin Dumnicki. Containments of symbolic powers of ideals of generic points in P 3 . Proceedings of the American Mathematical Society, 143(2):513–530, 2015.</ref><ref>{{cite arXiv |eprint=1103.5809|last1=Harbourne|first1=Brian|title=Are symbolic powers highly evolved?|last2=Huneke|first2=Craig|class=math.AC|year=2011}}</ref>

== References ==
[[File:Sz2010a.jpg|thumb|''From left:'' Brian Harbourne, [[Sandra Di Rocco]], {{ill|Tomasz Szemberg|pl}}, and Thomas Bauer at the [[Mathematical Research Institute of Oberwolfach|MFO]] mini-workshop ''Linear Series on Algebraic Varieties'', 2010]]
{{reflist}}

==External links==
* [[Melvin Hochster]], [http://www.math.lsa.umich.edu/~hochster/711F07/L09.07.pdf Math 711: Lecture of September 7, 2007]

[[Category:Algebra]]