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D.4.3.2 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 choosen 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)); → 0,1,0, 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, → 0,0,0, -1,0,0,0,0,0,1,0, 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, 0, -1, 0, 0, 0, 0, y, 1, 0, 0, 0, 0, y, 0, 0, → 0,0, 0, 0, y, 0, 0, y, 0, 0, -y, 0, 0, y, 0, 0, 0, → 0,1, 0, 0, 0, 0, y, 0, 0, 0, 0, x2, 0, 0, 0, 0, -x2y, → 0,0, y, 0, 0, -y,0, 0, 0, 0, 0, 0, x2y,0, 0, x2y,0, → 0,0, 1, 0, 0, -1,0, 0, 0, 0, 0, 0, x2, 0, 0, x2, 0, → 0,1, 0, 0, 0, 0, y, 0, 0, 0, 0, x2, 0, 0, 0, 0, -x2y, → 0,0, 0, 0, -x2, 0, 0, -x2, 0, 0, x2, 0, 0, -x2, 0, 0, 0, → 0,0, 0, x2, 0, 0, 0, 0, -x2y,-x2,0, 0, 0, 0, -x2y,0, 0, → 0,0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, → 0,0, 0, -1, 0, 0, 0, 0, y, 1, 0, 0, 0, 0, y, 0, 0, → 0,0, -1,0, 0, 1, 0, 0, 0, 0, 0, 0, -x2,0, 0, -x2,0, → 0,1, 0, 0, 0, 0, y, 0, 0, 0, 0, x2, 0, 0, 0, 0, -x2y, → 0,-1,0, 0, 0, 0, -y,0, 0, 0, 0, -x2,0, 0, 0, 0, x2y, → 0,0, y, 0, 0, -y,0, 0, 0, 0, 0, 0, x2y,0, 0, x2y,0, → 0,0, 0, -x2,0, 0, 0, 0, x2y, x2, 0, 0, 0, 0, x2y, 0, 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: → 0,1,0,0, → 0,0,0,0, → 0,0,0,0, → 0,0,0,0 → //----component 2: → 0,0,-1,0, → 0,1,0, 0, → 0,0,0, 0, → 0,0,0, 0 → //----component 3: → 0,0,0,-1, → 0,0,0,0, → 0,1,0,0, → 0,0,0,0 → //----component 4: → 0,0,0, 0, → 1,0,0, 1, → 0,0,-1,0, → 0,1,0, 0 → ////// end level 3 ////// show(l[1]);show(l[2]); → // matrix, 4x17 → 0,1,0, 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, → 0,0,0, -1,0,0,0,0,0,1,0, 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; |