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 |