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

 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
Description
This function takes as input a matrix
m with linear entries, which we think of as a presentation matrix for a positively graded
Smodule
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 quickanddirty 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

bgg  the ith differential of the complex R(M)
Ways to use symExt :

"symExt(Matrix,PolynomialRing)"