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# brunsIdeal -- Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal

- Usage:
    `j = brunsIdeal i`
- Inputs:
  - i, an ideal, a homogeneous ideal
- Outputs:
  - j, an ideal, a homogeneous ideal generated by three elements whose second syzygy module is isomorphic the second syzygy module of the ideal i.

## Description

This function is a special case of the function bruns. Given an ideal, the user can find another ideal which is 3-generated, and furthermore, the second syzygy modules of both ideals are isomorphic. Although one can use bruns to do this procedure, this function cuts out some of the steps.

    i1 : kk=ZZ/32003

    o1 = kk

    o1 : QuotientRing

    i2 : S=kk[a..d]

    o2 = S

    o2 : PolynomialRing

    i3 : i=ideal(a^2,b^2,c^2, d^2)

                 2   2   2   2
    o3 = ideal (a , b , c , d )

    o3 : Ideal of S

    i4 : betti (F=res i)

                0 1 2 3 4
    o4 = total: 1 4 6 4 1
             0: 1 . . . .
             1: . 4 . . .
             2: . . 6 . .
             3: . . . 4 .
             4: . . . . 1

    o4 : BettiTally

    i5 : M = image F.dd_3

    o5 = image {4} | c2  d2  0   0   |
               {4} | -b2 0   d2  0   |
               {4} | a2  0   0   d2  |
               {4} | 0   -b2 -c2 0   |
               {4} | 0   a2  0   -c2 |
               {4} | 0   0   a2  b2  |

                                 6
    o5 : S-module, submodule of S

    i6 : j1 = bruns M

                      4          2 2    2 2        2 2        4         2 2
    o6 = ideal (-9831d , - 15925b c  - a d  + 6174b d , 15925b  - 12753b d )

    o6 : Ideal of S

    i7 : betti res j1

                0 1 2 3 4
    o7 = total: 1 3 5 4 1
             0: 1 . . . .
             1: . . . . .
             2: . . . . .
             3: . 3 . . .
             4: . . . . .
             5: . . . . .
             6: . . 5 . .
             7: . . . 4 .
             8: . . . . 1

    o7 : BettiTally

    i8 : j2=brunsIdeal i

                      4        2 2    2 2        2 2          4        2 2
    o8 = ideal (-3490d , 11765b c  - a d  + 8771b d , - 11765b  - 5457b d )

    o8 : Ideal of S

    i9 : betti res j2

                0 1 2 3 4
    o9 = total: 1 3 5 4 1
             0: 1 . . . .
             1: . . . . .
             2: . . . . .
             3: . 3 . . .
             4: . . . . .
             5: . . . . .
             6: . . 5 . .
             7: . . . 4 .
             8: . . . . 1

    o9 : BettiTally

    i10 : (betti res j1) == (betti res j2)

    o10 = true

## See also

- bruns -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module

## Ways to use brunsIdeal:

- `brunsIdeal(Ideal)`

## For the programmer

The object brunsIdeal is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Bruns.m2:328:0.