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D.4.8.3 cupproduct

Procedure from library homolog.lib (see homolog_lib).

Usage:
cupproduct(M,N,P,p,q[,any]); M,N,P modules, p,q integers

Compute:
cup-product Ext^p(M',N') x Ext^q(N',P') ---> Ext^(p+q)(M',P'), where M':=R^m/M, if M in R^m, R basering (i.e. M':=coker(matrix(M)))

Assume:
all Ext's are of finite dimension

Return:
- if called with 5 arguments: matrix of the associated linear map Ext^p (tensor) Ext^q --> Ext^(p+q), i.e. the columns of <matrix> present the coordinates of the cup products (b_i & c_j) with respect to a kbase of Ext^p+q (b_i resp. c_j are the chosen bases of Ext^p, resp. Ext^q).
- if called with 6 arguments: list L,
 
      L[1] = matrix (see above)
      L[2] = matrix of kbase of Ext^p(M',N')
      L[3] = matrix of kbase of Ext^q(N',P')
      L[4] = matrix of kbase of Ext^p+q(N',P')

Note:
printlevel >=1; shows what is going on.
printlevel >=2; shows the result in another representation.
For computing the cupproduct of M,N itself, apply proc to syz(M), syz(N)!

Example:
 
LIB "homolog.lib";
int p      = printlevel;
ring  rr   = 32003,(x,y,z),(dp,C);
ideal  I   = x4+y3+z2;
qring  o   = std(I);
module M   = [x,y,0,z],[y2,-x3,z,0],[z,0,-y,-x3],[0,z,x,-y2];
print(cupproduct(M,M,M,1,3));
==> 1,0, 0, 0,0,0,0,0,0,0, 0,0,0,0,0,0,
==> 0,-1,0, 0,1,0,0,0,0,0, 0,0,0,0,0,0,
==> 0,0, -1,0,0,0,0,0,1,0, 0,0,0,0,0,0,
==> 0,0, 0, 1,0,0,1,0,0,-1,0,0,1,0,0,0 
printlevel = 3;
list l     = (cupproduct(M,M,M,1,3,"any"));
==> // vdim Ext(M,N) = 4
==> // kbase of Ext^p(M,N)
==> //  - the columns present the kbase elements in Hom(F(p),G(0))
==> //  - F(*),G(*) are free resolutions of M and N
==> 0, 0, 1, 0,  
==> 0, y, 0, 0,  
==> 1, 0, 0, 0,  
==> 0, 0, 0, y,  
==> 0, -1,0, 0,  
==> 0, 0, x2,0,  
==> 0, 0, 0, -x2,
==> 1, 0, 0, 0,  
==> 0, 0, 0, -1, 
==> -1,0, 0, 0,  
==> 0, 1, 0, 0,  
==> 0, 0, 1, 0,  
==> -1,0, 0, 0,  
==> 0, 0, 0, x2y,
==> 0, 0, x2,0,  
==> 0, -y,0, 0   
==> // vdim Ext(N,P) = 4
==> // kbase of Ext(N,P):
==> 0, 0, 1,  0,  
==> 0, 0, 0,  y,  
==> 1, 0, 0,  0,  
==> 0, -y,0,  0,  
==> 0, -1,0,  0,  
==> 1, 0, 0,  0,  
==> 0, 0, 0,  -x2,
==> 0, 0, -x2,0,  
==> 0, 0, 0,  -1, 
==> 0, 0, 1,  0,  
==> 0, 1, 0,  0,  
==> 1, 0, 0,  0,  
==> -1,0, 0,  0,  
==> 0, -y,0,  0,  
==> 0, 0, x2, 0,  
==> 0, 0, 0,  -x2y
==> // kbase of Ext^q(N,P)
==> //  - the columns present the kbase elements in Hom(G(q),H(0))
==> //  - G(*),H(*) are free resolutions of N and P
==> 0, 0, 1,  0,  
==> 0, 0, 0,  y,  
==> 1, 0, 0,  0,  
==> 0, -y,0,  0,  
==> 0, -1,0,  0,  
==> 1, 0, 0,  0,  
==> 0, 0, 0,  -x2,
==> 0, 0, -x2,0,  
==> 0, 0, 0,  -1, 
==> 0, 0, 1,  0,  
==> 0, 1, 0,  0,  
==> 1, 0, 0,  0,  
==> -1,0, 0,  0,  
==> 0, -y,0,  0,  
==> 0, 0, x2, 0,  
==> 0, 0, 0,  -x2y
==> // vdim Ext(M,P) = 4
==> // kbase of Ext^p+q(M,P)
==> //  - the columns present the kbase elements in Hom(F(p+q),H(0))
==> //  - F(*),H(*) are free resolutions of M and P
==> 0, 0, 1,  0,  
==> 0, 0, 0,  y,  
==> 1, 0, 0,  0,  
==> 0, -y,0,  0,  
==> 0, -1,0,  0,  
==> 1, 0, 0,  0,  
==> 0, 0, 0,  -x2,
==> 0, 0, -x2,0,  
==> 0, 0, 0,  -1, 
==> 0, 0, 1,  0,  
==> 0, 1, 0,  0,  
==> 1, 0, 0,  0,  
==> -1,0, 0,  0,  
==> 0, -y,0,  0,  
==> 0, 0, x2, 0,  
==> 0, 0, 0,  -x2y
==> // lifting of kbase of Ext^p(M,N)
==> //  - the columns present liftings of kbase elements in Hom(F(p+q),G(q))
==> 1,0, 0, 0,  
==> 0,-y,0, 0,  
==> 0,0, x2,0,  
==> 0,0, 0, x2y,
==> 0,1, 0, 0,  
==> 1,0, 0, 0,  
==> 0,0, 0, -x2,
==> 0,0, x2,0,  
==> 0,0, -1,0,  
==> 0,0, 0, y,  
==> 1,0, 0, 0,  
==> 0,y, 0, 0,  
==> 0,0, 0, -1, 
==> 0,0, -1,0,  
==> 0,-1,0, 0,  
==> 1,0, 0, 0   
==> // matrix of cup-products (in Ext^p+q)
==> 0, 0, -1, 0,   0, 0, 0,   y,   1,  0,  0,  0,  0,   y,   0,  0,   
==> 0, 0, 0,  y,   0, 0, y,   0,   0,  -y, 0,  0,  y,   0,   0,  0,   
==> 1, 0, 0,  0,   0, y, 0,   0,   0,  0,  x2, 0,  0,   0,   0,  -x2y,
==> 0, y, 0,  0,   -y,0, 0,   0,   0,  0,  0,  x2y,0,   0,   x2y,0,   
==> 0, 1, 0,  0,   -1,0, 0,   0,   0,  0,  0,  x2, 0,   0,   x2, 0,   
==> 1, 0, 0,  0,   0, y, 0,   0,   0,  0,  x2, 0,  0,   0,   0,  -x2y,
==> 0, 0, 0,  -x2, 0, 0, -x2, 0,   0,  x2, 0,  0,  -x2, 0,   0,  0,   
==> 0, 0, x2, 0,   0, 0, 0,   -x2y,-x2,0,  0,  0,  0,   -x2y,0,  0,   
==> 0, 0, 0,  -1,  0, 0, -1,  0,   0,  1,  0,  0,  -1,  0,   0,  0,   
==> 0, 0, -1, 0,   0, 0, 0,   y,   1,  0,  0,  0,  0,   y,   0,  0,   
==> 0, -1,0,  0,   1, 0, 0,   0,   0,  0,  0,  -x2,0,   0,   -x2,0,   
==> 1, 0, 0,  0,   0, y, 0,   0,   0,  0,  x2, 0,  0,   0,   0,  -x2y,
==> -1,0, 0,  0,   0, -y,0,   0,   0,  0,  -x2,0,  0,   0,   0,  x2y, 
==> 0, y, 0,  0,   -y,0, 0,   0,   0,  0,  0,  x2y,0,   0,   x2y,0,   
==> 0, 0, -x2,0,   0, 0, 0,   x2y, x2, 0,  0,  0,  0,   x2y, 0,  0,   
==> 0, 0, 0,  -x2y,0, 0, -x2y,0,   0,  x2y,0,  0,  -x2y,0,   0,  0    
==> ////// end level 2 //////
==> // the associated matrices of the bilinear mapping 'cup' 
==> // corresponding to the kbase elements of Ext^p+q(M,P) are shown,
==> //  i.e. the rows of the final matrix are written as matrix of
==> //  a bilinear form on Ext^p x Ext^q
==> //----component 1:
==> 1,0,0,0,
==> 0,0,0,0,
==> 0,0,0,0,
==> 0,0,0,0 
==> //----component 2:
==> 0,-1,0,0,
==> 1,0, 0,0,
==> 0,0, 0,0,
==> 0,0, 0,0 
==> //----component 3:
==> 0,0,-1,0,
==> 0,0,0, 0,
==> 1,0,0, 0,
==> 0,0,0, 0 
==> //----component 4:
==> 0,0, 0,1,
==> 0,0, 1,0,
==> 0,-1,0,0,
==> 1,0, 0,0 
==> ////// end level 3 //////
show(l[1]);show(l[2]);
==> // matrix, 4x16
==> 1,0, 0, 0,0,0,0,0,0,0, 0,0,0,0,0,0,
==> 0,-1,0, 0,1,0,0,0,0,0, 0,0,0,0,0,0,
==> 0,0, -1,0,0,0,0,0,1,0, 0,0,0,0,0,0,
==> 0,0, 0, 1,0,0,1,0,0,-1,0,0,1,0,0,0 
==> // matrix, 16x4
==> 0, 0, 1, 0,  
==> 0, y, 0, 0,  
==> 1, 0, 0, 0,  
==> 0, 0, 0, y,  
==> 0, -1,0, 0,  
==> 0, 0, x2,0,  
==> 0, 0, 0, -x2,
==> 1, 0, 0, 0,  
==> 0, 0, 0, -1, 
==> -1,0, 0, 0,  
==> 0, 1, 0, 0,  
==> 0, 0, 1, 0,  
==> -1,0, 0, 0,  
==> 0, 0, 0, x2y,
==> 0, 0, x2,0,  
==> 0, -y,0, 0   
printlevel = p;


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