In algebra and algebraic geometry, given a commutative Noetherian ring R {\displaystyle R} and an ideal I {\displaystyle I} in it, the n-th symbolic power of I {\displaystyle I} is the ideal
I ( n ) = ⋂ p ∈ Ass ( R / I ) φ p − 1 ( I n R p ) {\displaystyle I^{(n)}=\bigcap _{{\mathfrak {p}}\in \operatorname {Ass} (R/I)}\varphi _{\mathfrak {p}}^{-1}(I^{n}R_{\mathfrak {p}})}
where R p {\displaystyle R_{\mathfrak {p}}} is the localization of R {\displaystyle R} at p {\displaystyle {\mathfrak {p}}} , we set φ p : R → R p {\displaystyle \varphi _{\mathfrak {p}}\colon R\to R_{\mathfrak {p}}} is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of R / I {\displaystyle R/I} . Though this definition does not require I {\displaystyle I} 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 I {\displaystyle I} -primary component of I n {\displaystyle I^{n}} . Very roughly, it consists of functions with zeros of order n {\displaystyle n} along the variety defined by I {\displaystyle I} . We have: I ( 1 ) = I {\displaystyle I^{(1)}=I} and if I {\displaystyle I} is a maximal ideal, then I ( n ) = I n {\displaystyle I^{(n)}=I^{n}} . Symbolic powers induce the following chain of ideals:
I ( 0 ) = R ⊃ I = I ( 1 ) ⊃ I ( 2 ) ⊃ I ( 3 ) ⊃ I ( 4 ) ⊃ ⋯ {\displaystyle I^{(0)}=R\supset I=I^{(1)}\supset I^{(2)}\supset I^{(3)}\supset I^{(4)}\supset \cdots }
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 I {\displaystyle I} defining a curve in a complete local domain, the powers of I {\displaystyle I} are cofinal with the symbolic powers of I {\displaystyle I} . This important property of being cofinal was further developed by Schenzel in the 1970s.
In algebraic geometry Though generators for ordinary powers of I {\displaystyle I} are well understood when I {\displaystyle I} is given in terms of its generators as I = ( f 1 , … , f k ) {\displaystyle I=(f_{1},\ldots ,f_{k})} , it is still very difficult in many cases to determine the generators of symbolic powers of I {\displaystyle I} . But in the geometric setting, there is a clear geometric interpretation in the case when I {\displaystyle I} is a radical ideal over an algebraically closed field of characteristic zero. If X {\displaystyle X} is an irreducible variety whose ideal of vanishing is I {\displaystyle I} , then the differential power of I {\displaystyle I} consists of all the functions in R {\displaystyle R} that vanish to order ≥ n on X {\displaystyle X} , i.e.
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