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minimizeDM -- minimizes a square matrix or a differential module

Description

Given a differential module D, this code breaks off trivial blocks, producing a quasi-isomorphic differential module D' with a minimal differential.

i1 : R = QQ[x,y];
 
i2 : M = R^1/ideal(x^2,y^2);
 
i3 : phi = map(M,M,x*y, Degree=>2);

o3 : Matrix M <-- M
 
i4 : D = differentialModule phi;
 
i5 : r = resDM(D)

      8      8      8
o5 = R  <-- R  <-- R
                    
     -1     0      1

o5 : DifferentialModule
 
i6 : r.dd_1

o6 = {1}  | 0 0 y x 0 0 -1 0  |
     {1}  | 0 0 0 0 y x 0  1  |
     {0}  | 0 0 0 0 0 0 x  0  |
     {0}  | 0 0 0 0 0 0 -y 0  |
     {0}  | 0 0 0 0 0 0 0  x  |
     {0}  | 0 0 0 0 0 0 0  -y |
     {-1} | 0 0 0 0 0 0 0  0  |
     {-1} | 0 0 0 0 0 0 0  0  |

             8      8
o6 : Matrix R  <-- R
 
i7 : mr = minimizeDM(r)

      4      4      4
o7 = R  <-- R  <-- R
                    
     -1     0      1

o7 : DifferentialModule
 
i8 : mr.dd_1

o8 = | xy  x2  0   0   |
     | -y2 -xy 0   0   |
     | 0   0   -xy -x2 |
     | 0   0   y2  xy  |

             4      4
o8 : Matrix R  <-- R

See also

Ways to use minimizeDM:

  • minimizeDM(DifferentialModule)

For the programmer

The object minimizeDM is a method function.

The source of this document is in MultigradedBGG.m2:823:0.