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resDM -- uses a "killing cycles"-style construction to find a free resolution of a differential module

Synopsis

Description

Given a differential module D, this method creates a free flag resolution of D, using Algorithm 2.7 from the accompanying paper "The multigraded BGG correspondence in Macaulay2". The default resDM(D) runs the algorithm for the number of steps determined by the dimension of the ambient ring. Alternatively, resDM(D,k) iterates k steps of the algorithm.

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

The default algorithm runs for dim R + 1 steps. Adding the number of steps as a second argument is like adding a LengthLimit.

i7 : R = QQ[x]/(x^3);
i8 : phi = map(R^1,R^1,x^2, Degree=>2);

             1      1
o8 : Matrix R  <-- R
i9 : D = differentialModule phi;
i10 : r = resDM(D)

       1      1      1
o10 = R  <-- R  <-- R
                     
      -1     0      1

o10 : DifferentialModule
i11 : r.dd_1

o11 = 0

              1      1
o11 : Matrix R  <-- R
i12 : r = resDM(D,6)

       6      6      6
o12 = R  <-- R  <-- R
                     
      -1     0      1

o12 : DifferentialModule
i13 : r.dd_1

o13 = {1}  | 0 x 0  -1 1  0  |
      {0}  | 0 0 x2 0  x  -1 |
      {0}  | 0 0 0  x  0  -1 |
      {-1} | 0 0 0  0  x2 0  |
      {-1} | 0 0 0  0  0  x  |
      {-2} | 0 0 0  0  0  0  |

              6      6
o13 : Matrix R  <-- R

See also

Ways to use resDM :

For the programmer

The object resDM is a method function.