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K3Carpets :: canonicalHomotopies

canonicalHomotopies -- Homotopies on the resolution of a K3 carpet

Synopsis

Description

By default the option FineGrading is set to false. With FineGrading=>true the script returns the $\ZZ^4$-graded resolution, and the function h returns the homotopies one graded component at a time as a HashTable.

Note that the homotopies are 0 except in the middle part of the resolution, where there is a generator degree common to two consecutive free modules.

i1 : (F,h0) = canonicalHomotopies(7,3)

         ZZ                  1        ZZ                  10    
o1 = ((-----[x ..x , y ..y ])  <-- (-----[x ..x , y ..y ])   <--
       32003  0   3   0   3         32003  0   3   0   3        
                                                                
      0                            1                            
     ------------------------------------------------------------------------
        ZZ                  16        ZZ                  16    
     (-----[x ..x , y ..y ])   <-- (-----[x ..x , y ..y ])   <--
      32003  0   3   0   3          32003  0   3   0   3        
                                                                
     2                             3                            
     ------------------------------------------------------------------------
        ZZ                  10        ZZ                  1
     (-----[x ..x , y ..y ])   <-- (-----[x ..x , y ..y ])  <-- 0, h0)
      32003  0   3   0   3          32003  0   3   0   3         
                                                                6
     4                             5

o1 : Sequence
i2 : betti F

            0  1  2  3  4 5
o2 = total: 1 10 16 16 10 1
         0: 1  .  .  .  . .
         1: . 10 16  .  . .
         2: .  .  . 16 10 .
         3: .  .  .  .  . 1

o2 : BettiTally
i3 : netList apply(length F, j-> sum(rank F_1, i->h0(i,j)))

     +-------------------------------------------------------+
o3 = |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     |{2} | 1 |                                              |
     +-------------------------------------------------------+
     |0                                                      |
     +-------------------------------------------------------+
     |{5} | -1 -1 1  -1 -1 0  0  0  0  0  0  0  0  0  0  0  ||
     |{5} | -1 -1 0  0  0  1  -1 -1 0  0  0  0  0  0  0  0  ||
     |{5} | -1 0  -2 0  0  2  0  0  -2 -2 -1 0  0  0  0  0  ||
     |{5} | 0  1  -2 0  0  2  0  0  0  -2 0  -1 0  0  0  0  ||
     |{5} | -1 0  0  -1 0  0  1  0  -1 0  0  0  -1 0  0  0  ||
     |{5} | 0  1  0  -1 0  0  1  0  0  -1 0  0  0  -1 0  0  ||
     |{5} | 0  0  2  -1 0  0  0  0  2  -2 0  0  0  0  -1 0  ||
     |{5} | 1  0  0  0  -2 0  0  2  0  -2 -1 0  2  0  0  0  ||
     |{5} | 0  1  0  0  2  0  0  -2 0  0  0  1  0  -2 0  0  ||
     |{5} | 0  0  1  0  1  0  0  0  0  -2 -1 1  0  0  -1 0  ||
     |{5} | 0  0  0  1  2  0  0  0  0  0  0  0  -2 2  -1 0  ||
     |{5} | 0  0  0  0  0  -2 1  0  -2 2  0  0  0  0  0  -1 ||
     |{5} | 0  0  0  0  0  -1 0  -1 0  2  1  -1 0  0  0  -1 ||
     |{5} | 0  0  0  0  0  0  -1 -2 0  0  0  0  2  -2 0  -1 ||
     |{5} | 0  0  0  0  0  0  0  0  -1 -2 -1 0  1  0  -1 -1 ||
     |{5} | 0  0  0  0  0  0  0  0  0  -1 0  -1 0  1  -1 -1 ||
     +-------------------------------------------------------+
     |0                                                      |
     +-------------------------------------------------------+
     |{8} | -1 -1 1 -1 1 1 1 -1 1 1 |                        |
     +-------------------------------------------------------+
i4 : H = makeHomotopies1(F.dd_1, F);
i5 : (F,h0) = canonicalHomotopies(7,3, FineGrading=>true);
i6 : h0(0,2)

o6 = HashTable{{0, 5, 6, 9} => 0                    }
               {0, 5, 7, 8} => 0
               {1, 4, 5, 10} => 0
               {1, 4, 6, 9} => 0
               {1, 4, 7, 8} => {1, 4, 7, 8} | 0 1 |
               {1, 4, 8, 7} => {1, 4, 8, 7} | 1 |
               {2, 3, 5, 10} => 0
               {2, 3, 6, 9} => {2, 3, 6, 9} | -1 0 |
               {2, 3, 7, 8} => {2, 3, 7, 8} | 1  0 |
                               {2, 3, 7, 8} | -2 1 |
               {2, 3, 8, 7} => {2, 3, 8, 7} | 0 |
                               {2, 3, 8, 7} | 1 |
               {2, 3, 9, 6} => 0
               {3, 2, 6, 9} => {3, 2, 6, 9} | -1 |
               {3, 2, 7, 8} => {3, 2, 7, 8} | 1 |
                               {3, 2, 7, 8} | 0 |
               {3, 2, 8, 7} => 0
               {3, 2, 9, 6} => 0
               {4, 1, 7, 8} => 0
               {4, 1, 8, 7} => 0

o6 : HashTable
i7 : homotopyRanks(7,3)
       0  1  2  3  4 5
total: 1 10 16 16 10 1
    0: 1  .  .  .  . .
    1: . 10 16  .  . .
    2: .  .  . 16 10 .
    3: .  .  .  .  . 1

     +-------------------------+---------------+
o7 = || x_1^2-x_0x_2 |         |{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || x_1x_2-x_0x_3 |        |{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || x_2^2-x_1x_3 |         |{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || x_2y_0-2x_1y_1+x_0y_2 ||{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || x_3y_0-2x_2y_1+x_1y_2 ||{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || x_2y_1-2x_1y_2+x_0y_3 ||{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || x_3y_1-2x_2y_2+x_1y_3 ||{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || y_1^2-y_0y_2 |         |{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || y_1y_2-y_0y_3 |        |{1, 0, 8, 0, 1}|
     +-------------------------+---------------+
     || y_2^2-y_1y_3 |         |{1, 0, 8, 0, 1}|
     +-------------------------+---------------+

See also

Ways to use canonicalHomotopies :

For the programmer

The object canonicalHomotopies is a method function with options.