ass Ideal -- find the associated primes of an ideal

ass Ideal -- find the associated primes of an ideal

Synopsis:

Usage: L = ass(I)

Function: ass

Input:

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

Output:

L, an instance of class List: a list of prime ideals in R.

Optional arguments :

ass(..., PrintLevel => ...)

ass(..., Strategy => ...) -- specify a computational strategy

Computes the set of associated primes for the ideal I. The resulting list is stashed in I under the key Assasinator.

i1 : R = ZZ/101[a..d];

i2 : I = intersect(ideal(a^2,b),ideal(a,b,c^5),ideal(b^4,c^4))



             4     4   2 4

o2 = ideal (b , b*c , a c )



o2 : Ideal of R

i3 : ass(I)



o3 = {ideal (c, b), ideal (b, a)}



o3 : List

i4 : R = ZZ/7[x,y,z]/(x^2,x*y);

i5 : I=ideal(0_R);



o5 : Ideal of R

i6 : ass I



o6 = {ideal x, ideal (y, x)}



o6 : List

The associated primes are found using the Ext modules: The associated primes of codimension i of I and Ext^i(R^1/I,R) are identical, as shown in Eisenbud-Huneke-Vasconcelos, Invent math, 110, 207-235 (1992).

Caveat:

This function uses decompose, which currently only works over finite ground fields, not the rationals or integers.

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