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

## Synopsis

• 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