# localize -- localize an ideal at a prime ideal

## Synopsis

• Usage:
localize(I, P)
• Inputs:
• I, an ideal, an ideal in a (quotient of a) polynomial ring R
• P, an ideal, a prime ideal in the same ring
• Optional inputs:
• Strategy => ..., default value 1
• Outputs:
• an ideal, the extension contraction ideal $I R_P \cap R$.

## Description

The result is the ideal obtained by first extending to the localized ring and then contracting back to the original ring.

 i1 : R = ZZ/(101)[x,y]; i2 : I = ideal (x^2,x*y); o2 : Ideal of R i3 : P1 = ideal (x); o3 : Ideal of R i4 : localize(I,P1) o4 = ideal x o4 : Ideal of R i5 : P2 = ideal (x,y); o5 : Ideal of R i6 : localize(I,P2) 2 o6 = ideal (x , x*y) o6 : Ideal of R
 i7 : R = ZZ/31991[x,y,z]; i8 : I = ideal(x^2,x*z,y*z); o8 : Ideal of R i9 : P1 = ideal(x,y); o9 : Ideal of R i10 : localize(I,P1) o10 = ideal (y, x) o10 : Ideal of R i11 : P2 = ideal(x,z); o11 : Ideal of R i12 : localize(I,P2) 2 o12 = ideal (z, x ) o12 : Ideal of R

The strategy option value should be one of the following, with default value 1.

• Strategy => 0 -- Uses the algorithm of Eisenbud-Huneke-Vasconcelos

This strategy does not require the calculation of the assassinator, but can require the computation of high powers of ideals. The method appears in Eisenbud-Huneke-Vasconcelos, Invent. Math. 110 (1992) 207-235.

• Strategy => 1 -- Uses a separator to find the localization

This strategy uses a separator polynomial - a polynomial in all of the associated primes of { t I} but { t P} and those contained in { t P}. In this strategy, the assassinator of the ideal will be recalled, or recomputed using Strategy => 1 if unknown. The separator polynomial method is described in Shimoyama-Yokoyama, J. Symbolic computation, 22(3) 247-277 (1996). This is the same as Strategy => 1 except that, if unknown, the assassinator is computed using Strategy => 2.

• Strategy => 2 -- Uses a separator to find the localization

Authored by C. Yackel. Last modified June, 2000.

## Caveat

The ideal P is not checked to be prime.