symExt -- the first differential of the complex R(M)

• Usage:

  symExt(m,E)

• Inputs:
  • m, a matrix, a presentation matrix for a positively graded module M over a polynomial ring
  • E, a polynomial ring, exterior algebra
• Outputs:
  • a matrix, a matrix representing the map M_1 ** omega_E <-- M_0 ** omega_E

i1 : S = ZZ/32003[x_0..x_2]; 

i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];

i3 : M = coker matrix {{x_0^2, x_1^2}};

i4 : m = presentation truncate(regularity M,M);

             4      8
o4 : Matrix S  <-- S

i5 : symExt(m,E)

o5 = {-1} | e_2 0   0   0   |
     {-1} | e_1 e_2 0   0   |
     {-1} | e_0 0   e_2 0   |
     {-1} | 0   e_0 e_1 e_2 |

             4      4
o5 : Matrix E  <-- E