localize(Ideal,Ideal) -- localize an ideal at a prime ideal

localize(Ideal,Ideal) -- localize an ideal at a prime ideal

Synopsis:

Usage: J = localize(I,P)

Function: localize

Input:

I, an instance of class Ideal: an ideal in a (quotient of a) polynomial ring R.

P, an instance of class Ideal: a prime ideal in the same ring.

Output:

J, an instance of class Ideal: the extension contraction ideal I R_P intersect R.

Optional arguments :

localize(..., Strategy) -- specify a computational strategy

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

Caveat:

The ideal P is not checked to be prime.

Author and maintainer: C. Yackel, cyackel@math.indiana.edu. Last modified June 2000.

See also:

primaryDecomposition Ideal -- find a primary decomposition of an ideal

radical -- compute the radical of an ideal

decompose -- irreducible components of an ideal

top -- compute the top dimensional components

removeLowestDimension -- remove components of lower dimension