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

Synopsis

Description

This function takes as input a matrix m with linear entries, which we think of as a presentation matrix for a positively graded S-module M matrix representing the map M_1 ** omega_E <-- M_0 ** omega_E which is the first differential of the complex R(M).
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

Caveat

This function is a quick-and-dirty tool which requires little computation. However if it is called on two successive truncations of a module, then the maps it produces may NOT compose to zero because the choice of bases is not consistent.

See also

Ways to use symExt :

For the programmer

The object symExt is a method function.