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D.15.18.4 symExt

Procedure from library tateProdCplxNegGrad.lib (see tateProdCplxNegGrad_lib).

Usage:
symExt(m); m matrix

Purpose:
computes differential R(M_0) -> R(M_1) for the module M over S corresponding to the linear presentation matrix m, however, in order to get the result, m has to be fetched to the exterior algebra E

Assume:
m a matrix, linear presentation matrix over S; Note: also works for nonlinear matrices, but makes no sense to use it in this case

Return:
matrix B representing R(M_0) -> R(M_1)

Note:
output lives in S (not as in Macaulay2 in the ring E, to get the same result, just fetch the matrix to E)

Example:
 
LIB "tateProdCplxNegGrad.lib";
intvec c = 1,2;
def (S,E) = productOfProjectiveSpaces(c);
setring(S);
matrix m[4][2] = x(0)(0), x(1)(0),x(0)(1),0,0,x(1)(1), 0,x(1)(2);
matrix A = symExt(m);
print(A);
==> 0,       x(0)(0),0,       0,      
==> 0,       0,      x(0)(0), 0,      
==> 0,       0,      0,       x(0)(0),
==> x(0)(1), 0,      0,       0,      
==> -x(0)(0),x(0)(1),0,       0,      
==> 0,       0,      x(0)(1), 0,      
==> 0,       0,      0,       x(0)(1),
==> 0,       x(1)(0),0,       0,      
==> 0,       0,      x(1)(0), 0,      
==> 0,       0,      0,       x(1)(0),
==> x(1)(1), 0,      0,       0,      
==> 0,       x(1)(1),0,       0,      
==> -x(1)(0),0,      x(1)(1), 0,      
==> 0,       0,      0,       x(1)(1),
==> x(1)(2), 0,      0,       0,      
==> 0,       x(1)(2),0,       0,      
==> 0,       0,      x(1)(2), 0,      
==> 0,       0,      -x(1)(1),x(1)(2) 
setring(E);
print(fetch(S,A));
==> 0,       e(0)(0),0,       0,      
==> 0,       0,      e(0)(0), 0,      
==> 0,       0,      0,       e(0)(0),
==> e(0)(1), 0,      0,       0,      
==> -e(0)(0),e(0)(1),0,       0,      
==> 0,       0,      e(0)(1), 0,      
==> 0,       0,      0,       e(0)(1),
==> 0,       e(1)(0),0,       0,      
==> 0,       0,      e(1)(0), 0,      
==> 0,       0,      0,       e(1)(0),
==> e(1)(1), 0,      0,       0,      
==> 0,       e(1)(1),0,       0,      
==> -e(1)(0),0,      e(1)(1), 0,      
==> 0,       0,      0,       e(1)(1),
==> e(1)(2), 0,      0,       0,      
==> 0,       e(1)(2),0,       0,      
==> 0,       0,      e(1)(2), 0,      
==> 0,       0,      -e(1)(1),e(1)(2)