saturate -- saturation of ideal or submodule

saturate -- saturation of ideal or submodule

saturate(I,J,options) -- computes the saturation (I : J^*) of I with respect to J. If J is not given, the ideal J is taken to be the ideal generated by the variables of the ring R of I.

If I is either an ideal or a submodule of a module M, the saturation (I : J^*) is defined to be the set of elements f in the ring (first case) or in M (second case) such that J^N * f is contained in I, for some N large enough.

For example, one way to homogenize an ideal is to homogenize the generators and then saturate with respect to the homogenizing variable.

i1 : R = ZZ/32003[a..d];

i2 : I = ideal(a^3-b, a^4-c)



             3       4

o2 = ideal (a  - b, a  - c)



o2 : Ideal of R

i3 : Ih = homogenize(I,d)



             3      2   4      3

o3 = ideal (a  - b*d , a  - c*d )



o3 : Ideal of R

i4 : saturate(Ih,d)



                        2     2    3      2   3      2

o4 = ideal (a*b - c*d, a c - b d, b  - a*c , a  - b*d )



o4 : Ideal of R

We can use this command to remove graded submodules of finite length.

i5 : m = ideal vars R



o5 = ideal (a, b, c, d)



o5 : Ideal of R

i6 : M = R^1 / (a * m^2)



o6 = cokernel | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 |



                            1

o6 : R-module, quotient of R

i7 : M / saturate 0_M



o7 = cokernel | a a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 |



                            1

o7 : R-module, quotient of R

The computation is currently not stored anywhere: this means that the computation cannot be continued after an interrupt. This will be changed in a later version.

Ways to use saturate :

saturate Ideal

saturate Module

saturate Vector

saturate(Ideal,Ideal)

saturate(Ideal,RingElement)

saturate(Module,Ideal)

saturate(Module,RingElement)

saturate(MonomialIdeal,MonomialIdeal)

Optional arguments :

saturate(..., DegreeLimit => ...) -- compute up to a certain degree

saturate(..., MinimalGenerators => ...)

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