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# regularityVar -- computes the Castelnuovo-Mumford regularity of homogeneous ideals in terms of Betti numbers, with respect to some of the variables of the ring

- Usage:
    `regularityVar(l,I)`
- Inputs:
  - l, a list, list of variables of the polynomial ring R to take into account for computing the Castelnuovo-Mumford regularity
  - I, an ideal, ideal of a polynomial ring

## Description

regularityVar computes the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring by computing the shifts and degrees of generators in a minimal free resolution of the homogeneous ideal.

The list of variables l contains the variables of the ring having degree 1. Those variables on the ring not in l have automatically degree 0, as well as the elements on the coefficient ring

    i1 : R=QQ[a..i,x,y,z]

    o1 = R

    o1 : PolynomialRing

    i2 : f1 = a*x+b*y+c*z

    o2 = a*x + b*y + c*z

    o2 : R

    i3 : f2 = d*x+e*y+f*z

    o3 = d*x + e*y + f*z

    o3 : R

    i4 : f3 = g*x+h*y+i*z

    o4 = g*x + h*y + i*z

    o4 : R

    i5 : I = ideal(f1,f2,f3)

    o5 = ideal (a*x + b*y + c*z, d*x + e*y + f*z, g*x + h*y + i*z)

    o5 : Ideal of R

    i6 : l = {x,y,z}

    o6 = {x, y, z}

    o6 : List

    i7 : regularityVar (l,I)

    o7 = 1

## See also

- coefficientRing -- get the coefficient ring
- freeResolution -- compute a free resolution of a module or ideal

## Ways to use regularityVar:

- `regularityVar(List,Ideal)`

## For the programmer

The object regularityVar is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/EliminationMatrices.m2:1642:0.