isSupportedInZeroLocus -- whether support of a module is contained in the zero locus of the (irrelevant) ideal

Synopsis

Usage:

isSupportedInZeroLocus_B M
Inputs:
- M, a module,
- B, an ideal,
Outputs:
- a Boolean value, true if $M$ (or $R^1/I$, if an ideal of $R$ is given) is supported only on the zero locus of $B$; that is, whether $\mathrm{supp}(M) \subset \mathrm{V}(B)$

Description

Given an module $M$ and an ideal $B$, isSupportedInZeroLocus checks whether $\mathrm{ann}(M):B^\infty=R$. If it is, isSupportedInZeroLocus returns true otherwise it returns false. If the first argument is an ideal, $M = R^1/I$ is taken as the module.

i1 : S = ZZ/32003[x_0..x_4, Degrees=>{2:{1,0}, 3:{0,1}}];

i2 : irr = intersect(ideal(x_0,x_1), ideal(x_2,x_3,x_4));

o2 : Ideal of S

i3 : M = S^1/(irr^2);

i4 : isSupportedInZeroLocus_irr M

o4 = true

This is done without computing saturation of $M$. Instead, we check whether for each generator of $B$ some power of it annihilates the module $M$.

Ways to use isSupportedInZeroLocus :

isSupportedInZeroLocus(Ideal,GradedModule)
isSupportedInZeroLocus(Ideal,Ideal)
isSupportedInZeroLocus(Ideal,Module)

For the programmer

The object isSupportedInZeroLocus is a method function.