We first substitute the unknown entries of the matrices d1’ and d2 by the corresponding entries of the matrix subsPoint. These matrices are defined over the polynomial ring S = k[x0,x1,y0,...,y3]. From d2 we compute the missing first row of the first matrix, and hence the complete syzygy matrix d1. The S-module R := coker d1 has then the prescribed Betti numbers. As a final step, we check that the syzygy matrices are modulo x0,x1 of the form fixed in the procedure complexModuloRegularSequence and that the second syzygy matrix is skew-symmetric. Such a minimal free resolution is called a standard resolution.
i1 : kk = ZZ/23; |
i2 : s = "1111"; |
i3 : (relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s); |
i4 : Sa = getP11(relPfaf); |
i5 : point1 = randomPoint(ideal relPfaf,Sa);
o5 : Ideal of kk[a , a , a , a , a , a , a , a , a , a , a , a , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , o , o , o , o , o , o , o , o , o , o , o , o , x , x , y , y , y , y ]
3,2,3 3,1,3 3,0,3 2,2,3 2,1,2 2,0,2 1,1,3 1,1,2 1,0,1 0,0,3 0,0,2 0,0,1 0,0 0,2 0,4 0,6 0,7 1,1 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,5 2,7 3,0 3,1 3,3 3,5 3,7 1,0,0 2,0,1 2,1,2 3,0,0 3,1,0 4,0,1 4,2,1 4,3,3 5,1,2 5,2,2 5,3,3 5,4,3 0 1 0 1 2 3
|
i6 : (randLine,subsLine) = randomLine(point1,relPfaf,Sa); |
i7 : (solutionMat,restVars) = solutionMatrix(relLin); |
i8 : (randPoint,subsPoint) = randomSection(solutionMat,restVars,subsLine); |
i9 : F = standardResolution(d1',d2,subsPoint,s); |
i10 : betti F
0 1 2 3
o10 = total: 8 26 26 8
0: 1 . . .
1: . . . .
2: . . . .
3: . . . .
4: 4 . . .
5: 3 6 . .
6: . 12 . .
7: . 8 8 .
8: . . 12 .
9: . . 6 3
10: . . . 4
11: . . . .
12: . . . .
13: . . . .
14: . . . 1
o10 : BettiTally
|
i11 : F.dd_2 + transpose F.dd_2 == 0 o11 = true |
i12 : F.dd_1 - transpose F.dd_3 == 0 o12 = true |