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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 $d_1'$ and $d_2$ by the corresponding entries of the matrix subsPoint. These matrices are defined over the polynomial ring $S = k[x_0,x_1,y_0,\ldots,y_3]$. From $d_2$ we compute the missing first row of the first matrix, and hence the complete syzygy matrix $d_1$. The $S$-module $R := coker d_1$ has then the prescribed Betti numbers. As a final step, we check that the syzygy matrices are modulo $x_0,x_1$ 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   , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , x ..x , 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   1,6   2,1   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   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 :

For the programmer

The object standardResolution is a method function.