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

- Usage:
    `isSupportedInZeroLocus_B M`
- Inputs:
  - B, an ideal,
  - M, an ideal, a module, or a graded module,
- 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 second 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Saturation/saturate-doc.m2:139:0.