Algebra

Symbolic power of an ideal

In algebra and algebraic geometry, given a commutative Noetherian ring and an ideal in it, the n-th symbolic power of is the ideal where is the localization of at , we set is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of . Though this definition does not require 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 -primary component of . Very roughly, it consists of functions with zeros of order n along the variety defined by . We have: and if is a maximal ideal, then . Symbolic powers induce the following chain of ideals: (Wikipedia).

Symbolic power of an ideal
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