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# firstOrderDeformation -- Makes a first order deformation.

- Usage:
    `firstOrderDeformation(mg,v)`
    `firstOrderDeformation(I,v)`
- Inputs:
  - I, an ideal, monomial, reduced, in a PolynomialRing, or
  - I, a monomial ideal, reduced
  - mg, a matrix, row, with generators of a reduced monomial ideal.
- Outputs:
  - a first order deformation,

## Description

Generates a first order deformation ideal mg. If it is given an ideal I then generators of I are stored to have a reference for parameters. You can also give non-minimal generators.

    i1 : R=QQ[x_0..x_4];

    i2 : addCokerGrading(R)

    o2 = | -1 -1 -1 -1 |
         | 1  0  0  0  |
         | 0  1  0  0  |
         | 0  0  1  0  |
         | 0  0  0  1  |

                  5       4
    o2 : Matrix ZZ  <-- ZZ

    i3 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0)

    o3 = ideal (x x , x x , x x , x x , x x )
                 0 1   1 2   2 3   3 4   0 4

    o3 : Ideal of R

    i4 : mg=mingens I;

                 1      5
    o4 : Matrix R  <-- R

    i5 : f=firstOrderDeformation(mg, vector {-1,1,0,0,0})

         x
          1
    o5 = --
         x
          0

    o5 : first order deformation space of dimension 1

## See also

- FirstOrderDeformation -- The class of all first order deformations of monomial ideals.
- dim(FirstOrderDeformation) -- Compute the dimension of a deformation.
- addCokerGrading -- Stores a cokernel grading in a polynomial ring.

## Ways to use firstOrderDeformation:

- `firstOrderDeformation(Ideal,Vector)`
- `firstOrderDeformation(Matrix,Vector)`
- `firstOrderDeformation(MonomialIdeal,Vector)`

## For the programmer

The object firstOrderDeformation is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SRdeformations.m2:5029:0.