Let $R = S/I$, and $w_R = \operatorname{Ext}^c(R, S^{-n-1})$, where $c$ is the codimension of $I$ and $n+1$ is the number of variables of the polynomial ring $S$.
If there exists an injective homomorphism $f \colon w_R \to R$ of degree $d$, this function returns the ideal defining the cokernel of a random such map. If none exist, null is returned. If after trying the number of trials given by the optional argument Count, none that are injective can be found (this is very unlikely), null is also returned. Setting the global variable debugLevel to a positive value will let you know how many times it took to find one (if it didn't find it right away).
If $S/I$ is arithmetically Cohen-Macaulay of codimension $c$, then the cokernel of $f$ will be arithmetically Gorenstein of codimension $c+1$.
See section 2.5 of [QQ] for more details and references.
i1 : S = ZZ/101[a..d];
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i2 : I = pointsIdeal randomPoints(S, 6)
2 2 2
o2 = ideal (a*c - 14b*c + 43c + 31a*d - 28b*d - 17c*d + 28d , b + 11b*c +
------------------------------------------------------------------------
2 2 2
37c - 40a*d - 45b*d - 10c*d - 23d , a*b + 35b*c - 18c - 31a*d - 38b*d
------------------------------------------------------------------------
2 2 2 2
+ 7c*d - 35d , a + 49b*c + 40c - 4a*d - 22b*d - 45c*d - 5d )
o2 : Ideal of S
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i3 : betti res I
0 1 2 3
o3 = total: 1 4 5 2
0: 1 . . .
1: . 4 2 .
2: . . 3 2
o3 : BettiTally
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i4 : doubling(5, I)
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i5 : J = doubling(8, I)
2 2 2
o5 = ideal (a*c - 14b*c + 43c + 31a*d - 28b*d - 17c*d + 28d , b + 11b*c +
------------------------------------------------------------------------
2 2 2
37c - 40a*d - 45b*d - 10c*d - 23d , a*b + 35b*c - 18c - 31a*d - 38b*d
------------------------------------------------------------------------
2 2 2 2 2
+ 7c*d - 35d , a + 49b*c + 40c - 4a*d - 22b*d - 45c*d - 5d , c d -
------------------------------------------------------------------------
2 2 2 3 2 2 2 3
28a*d + 9b*d + 50c*d + 50d , b*c*d - 15a*d + 39b*d - 26c*d - d )
o5 : Ideal of S
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i6 : betti res J
0 1 2 3 4
o6 = total: 1 6 10 6 1
0: 1 . . . .
1: . 4 2 . .
2: . 2 6 2 .
3: . . 2 4 .
4: . . . . 1
o6 : BettiTally
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i7 : S = ZZ/101[x_0..x_5];
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i8 : M = genericSymmetricMatrix(S, 3)
o8 = | x_0 x_1 x_2 |
| x_1 x_3 x_4 |
| x_2 x_4 x_5 |
3 3
o8 : Matrix S <-- S
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i9 : I = trim minors(2, M)
2 2 2
o9 = ideal (x - x x , x x - x x , x x - x x , x - x x , x x - x x , x -
4 3 5 2 4 1 5 2 3 1 4 2 0 5 1 2 0 4 1
------------------------------------------------------------------------
x x )
0 3
o9 : Ideal of S
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i10 : doubling(4, I) -- no such map exists
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i11 : betti res doubling(6,I)
0 1 2 3 4
o11 = total: 1 9 16 9 1
0: 1 . . . .
1: . 9 16 9 .
2: . . . . 1
o11 : BettiTally
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i12 : betti res doubling(7,I)
0 1 2 3 4
o12 = total: 1 9 16 9 1
0: 1 . . . .
1: . 6 8 3 .
2: . 3 8 6 .
3: . . . . 1
o12 : BettiTally
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i13 : betti res doubling(8,I)
0 1 2 3 4
o13 = total: 1 9 16 9 1
0: 1 . . . .
1: . 6 8 3 .
2: . . . . .
3: . 3 8 6 .
4: . . . . 1
o13 : BettiTally
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i14 : J = doubling(8, I);
o14 : Ideal of S
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i15 : (dim J, degree J)
o15 = (2, 14)
o15 : Sequence
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i16 : (dim I, degree I)
o16 = (3, 4)
o16 : Sequence
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i17 : S = ZZ/101[x_0..x_8];
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i18 : M = genericMatrix(S, 3, 3)
o18 = | x_0 x_3 x_6 |
| x_1 x_4 x_7 |
| x_2 x_5 x_8 |
3 3
o18 : Matrix S <-- S
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i19 : I = trim minors(2, M)
o19 = ideal (x x - x x , x x - x x , x x - x x , x x - x x , x x - x x ,
5 7 4 8 2 7 1 8 5 6 3 8 4 6 3 7 2 6 0 8
-----------------------------------------------------------------------
x x - x x , x x - x x , x x - x x , x x - x x )
1 6 0 7 2 4 1 5 2 3 0 5 1 3 0 4
o19 : Ideal of S
|
i20 : betti res doubling(8,I)
0 1 2 3 4 5
o20 = total: 1 10 25 25 10 1
0: 1 . . . . .
1: . 10 16 9 . .
2: . . 9 16 10 .
3: . . . . . 1
o20 : BettiTally
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i21 : J = doubling(8, I);
o21 : Ideal of S
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i22 : (dim J, degree J)
o22 = (4, 12)
o22 : Sequence
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i23 : (dim I, degree I)
o23 = (5, 6)
o23 : Sequence
|