This function takes a graded module M over a polynomial ring S that is a second syzygy, and returns a three-generator ideal j whose second syzygy is M, so that the resolution of S/j, from the third step, is isomorphic to the resolution of M. Alternately bruns takes a matrix whose cokernel is a second syzygy, and finds a 3-generator ideal whose second syzygy is the image of that matrix.
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 : f=F.dd_3
o6 = {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 4
o6 : Matrix S <--- S
|
i7 : j=bruns M; o7 : Ideal of S |
i8 : betti res j -- the ideal has 3 generators
0 1 2 3 4
o8 = total: 1 3 5 4 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . 3 . . .
4: . . . . .
5: . . . . .
6: . . 5 . .
7: . . . 4 .
8: . . . . 1
o8 : BettiTally
|
Here is a more complicated example, also involving a complete intersection. You can see that columns three and four in the two Betti diagrams are the same.
i9 : kk=ZZ/32003 o9 = kk o9 : QuotientRing |
i10 : S=kk[a..d] o10 = S o10 : PolynomialRing |
i11 : i=ideal(a^2,b^3,c^4, d^5)
2 3 4 5
o11 = ideal (a , b , c , d )
o11 : Ideal of S
|
i12 : betti (F=res i)
0 1 2 3 4
o12 = total: 1 4 6 4 1
0: 1 . . . .
1: . 1 . . .
2: . 1 . . .
3: . 1 1 . .
4: . 1 1 . .
5: . . 2 . .
6: . . 1 1 .
7: . . 1 1 .
8: . . . 1 .
9: . . . 1 .
10: . . . . 1
o12 : BettiTally
|
i13 : M = image F.dd_3
o13 = image {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | a2 0 0 d5 |
{7} | 0 -b3 -c4 0 |
{8} | 0 a2 0 -c4 |
{9} | 0 0 a2 b3 |
6
o13 : S-module, submodule of S
|
i14 : f=F.dd_3
o14 = {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | a2 0 0 d5 |
{7} | 0 -b3 -c4 0 |
{8} | 0 a2 0 -c4 |
{9} | 0 0 a2 b3 |
6 4
o14 : Matrix S <--- S
|
i15 : j1=bruns f; o15 : Ideal of S |
i16 : betti res j1
0 1 2 3 4
o16 = total: 1 3 5 4 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . . . . .
4: . . . . .
5: . . . . .
6: . . . . .
7: . . . . .
8: . 1 . . .
9: . 2 . . .
10: . . . . .
11: . . . . .
12: . . . . .
13: . . . . .
14: . . . . .
15: . . 1 . .
16: . . 1 . .
17: . . 2 . .
18: . . 1 1 .
19: . . . 1 .
20: . . . 1 .
21: . . . 1 .
22: . . . . 1
o16 : BettiTally
|
i17 : j=bruns M; o17 : Ideal of S |
i18 : betti res j
0 1 2 3 4
o18 = total: 1 3 5 4 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . . . . .
4: . . . . .
5: . . . . .
6: . . . . .
7: . . . . .
8: . 1 . . .
9: . 2 . . .
10: . . . . .
11: . . . . .
12: . . . . .
13: . . . . .
14: . . . . .
15: . . 1 . .
16: . . 1 . .
17: . . 2 . .
18: . . 1 1 .
19: . . . 1 .
20: . . . 1 .
21: . . . 1 .
22: . . . . 1
o18 : BettiTally
|
In the next example, we perform the "Brunsification" of a rational curve.
i19 : kk=ZZ/32003 o19 = kk o19 : QuotientRing |
i20 : S=kk[a..e] o20 = S o20 : PolynomialRing |
i21 : i=monomialCurveIdeal(S, {1,3,4,5})
2 2 2 3 2
o21 = ideal (d - c*e, b*d - a*e, c - b*e, b*c - a*d, a*c*d - b e, b - a c)
o21 : Ideal of S
|
i22 : betti (F=res i)
0 1 2 3 4
o22 = total: 1 5 8 5 1
0: 1 . . . .
1: . 4 2 . .
2: . 1 6 5 1
o22 : BettiTally
|
i23 : time j=bruns F.dd_3;
-- used 0.357412 seconds
o23 : Ideal of S
|
i24 : betti res j
0 1 2 3 4
o24 = total: 1 3 6 5 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . . . . .
4: . 3 . . .
5: . . . . .
6: . . . . .
7: . . 2 . .
8: . . 4 5 1
o24 : BettiTally
|
The object bruns is a method function.