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NumericalGodeaux :: getChainComplexes

getChainComplexes -- resolve the two linear submatrices of the solution matrices over the coordinate ring of the Pfaffians

Synopsis

Description

In this procedure we work over the coordinate ring $S_Q$ where $Q \subset \ \mathbb{P}^{11}$ is the complete intersection of the four quadratic relations. For $i=1,2$, we resolve the matrices $l_i$ and $l_i^{tr}$ in both directions and put this together to a chain complex $C_i$. Very surprisingly, in the case "1111" the resolution of $l_1$ and $l_2$ over the non-regular ring S_Q is finite in both directions. So far, we have studied these chain complexes mainly for the case "1111". The resulting complexes $C_1$ and $C_2$ are exact outside a codimension 3 respectively 2 subscheme of $Q$. These loci are determined in the procedure homologyLocus.

i1 : kk = ZZ/197;
 
i2 : s = "1111";
 
i3 : (relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s);
 
i4 : (C1,C2) = getChainComplexes(relLin,relPfaf);
 
i5 : betti C1, betti C2

             0  1  2 3          0  1  2 3
o5 = (total: 4 12 12 4, total: 12 30 20 2)
         -3: 4 12 12 4     -4: 12 30 20 .
                           -3:  .  .  . .
                           -2:  .  .  . 2

o5 : Sequence
 
i6 : prune HH C1

o6 = 0 : cokernel {-3} | a_(3,0,3) a_(3,1,3) a_(3,2,3) 0         0        
                  {-3} | 0         0         0         a_(2,0,2) a_(2,1,2)
                  {-3} | 0         0         0         0         0        
                  {-3} | 0         0         0         0         0        
         0         0         0         0         0         0        
         a_(2,2,3) 0         0         0         0         0        
         0         a_(1,0,1) a_(1,1,2) a_(1,1,3) 0         0        
         0         0         0         0         a_(0,0,1) a_(0,0,2)
         0         |
         0         |
         0         |
         a_(0,0,3) |

     1 : 0                                                                

     2 : 0                                                                

     3 : 0                                                                

o6 : GradedModule
 
i7 : apply(4,i->  (pH = prune HH_i(C1); (dim pH,degree pH)))

o7 = {(5, 64), (-1, 0), (-1, 0), (-1, 0)}

o7 : List
 
i8 : dim ring C1

o8 = 8
 
i9 : apply(4,i->  (pH = prune HH_i(C2); (dim pH,degree pH)))

o9 = {(6, 72), (6, 72), (-1, 0), (-1, 0)}

o9 : List

See also

Ways to use getChainComplexes :

For the programmer

The object getChainComplexes is a method function.