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NumericalGodeaux :: standardResolution

standardResolution -- compute a standard resolution of an S-module R obtained from the given input

Synopsis

Description

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

See also

Ways to use standardResolution :