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
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i2 : S=kk[a..d]
o2 = S
o2 : PolynomialRing
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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
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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
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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
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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
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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
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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
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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
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i10 : (betti res j1) == (betti res j2)
o10 = true
|