saturate
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.
saturate(Ideal)
saturate(Ideal,Ideal)
saturate(Ideal,RingElement)
saturate(Module)
saturate(Module,Ideal)
saturate(Module,RingElement)
saturate(Vector)
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 {0} | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 |
1
o6 : R - module, quotient of R |
i7 : M / saturate 0_M
o7 = cokernel {0} | a a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 |
1
o7 : R - module, quotient of R |
Allowable options include:
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.
