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D.4.3.19 Tor
Procedure from library homolog.lib (see homolog_lib).
- Compute:
a presentation of Tor_k(M’,N’), for k=v[1],v[2],... , where M’=coker(M) and N’=coker(N): let
0 <-- M' <-- G0 <-M-- G1 0 <-- N' <-- F0 <--N- F1 <-- F2 <--...be a presentation of M’, resp. a free resolution of N’, and consider the commutative diagram
0 0 0 |^ |^ |^ Tensor(M',Fk+1) -Ak+1-> Tensor(M',Fk) -Ak-> Tensor(M',Fk-1) |^ |^ |^ Tensor(G0,Fk+1) -Ak+1-> Tensor(G0,Fk) -Ak-> Tensor(G0,Fk-1) |^ |^ |C |B Tensor(G1,Fk) ----> Tensor(G1,Fk-1) (Ak,Ak+1 induced by N and B,C induced by M).Let K=modulo(Ak,B), J=module(C)+module(Ak+1) and Tor=modulo(K,J), then we have exact sequences
R^p --K-> Tensor(G0,Fk) --Ak-> Tensor(G0,Fk-1)/im(B), R^q -Tor-> R^p --K-> Tensor(G0,Fk)/(im(C)+im(Ak+1)).Hence, Tor presents Tor_k(M’,N’).
- Return:
- if v is of type int: module Tor, a presentation of Tor_k(M’,N’);
- if v is of type intvec: a list of Tor_k(M’,N’) (k=v[1],v[2],...);
- in case of a third argument of any type: list l withl[1] = module Tor/list of Tor_k(M’,N’), l[2] = SB of Tor/list of SB of Tor_k(M’,N’), l[3] = matrix/list of matrices, each representing a kbase of Tor_k(M’,N’) (if finite dimensional), or 0.- Display:
printlevel >=0: (affine) dimension of Tor_k for each k (default).
printlevel >=1: matrices Ak, Ak+1 and kbase of Tor_k in Tensor(G0,Fk) (if finite dimensional).- Note:
In order to compute Tor_k(M,N) use the command Tor(k,syz(M),syz(N)); or: list P=mres(M,2); list Q=mres(N,2); Tor(k,P[2],Q[2]);
Example:
LIB "homolog.lib"; int p = printlevel; printlevel = 1; ring r = 0,(x,y),dp; ideal i = x2,y; ideal j = x; list E = Tor(0..2,i,j); // Tor_k(r/i,r/j) for k=0,1,2 over r → // dimension of Tor_0: 0 → // vdim of Tor_0: 1 → → // Computing Tor_1 (help Tor; gives an explanation): → // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M), → // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of → // coker(N), then Tensor(G0,F1)-->Tensor(G0,F0) is given by: → x → // and Tensor(G0,F2) + Tensor(G1,F1)-->Tensor(G0,F1) is given by: → 0,x2,y → → // dimension of Tor_1: 0 → // vdim of Tor_1: 1 → → // Computing Tor_2 (help Tor; gives an explanation): → // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M), → // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of → // coker(N), then Tensor(G0,F2)-->Tensor(G0,F1) is given by: → 0 → // and Tensor(G0,F3) + Tensor(G1,F2)-->Tensor(G0,F2) is given by: → 1,x2,y → → // dimension of Tor_2: -1 → qring R = std(i); ideal j = fetch(r,j); module M = [x,0],[0,x]; printlevel = 2; module E1 = Tor(1,M,j); // Tor_1(R^2/M,R/j) over R=r/i → // Computing Tor_1 (help Tor; gives an explanation): → // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M), → // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of → // coker(N), then Tensor(G0,F1)-->Tensor(G0,F0) is given by: → x,0, → 0,x → // and Tensor(G0,F2) + Tensor(G1,F1)-->Tensor(G0,F1) is given by: → x,0,x,0, → 0,x,0,x → → // dimension of Tor_1: 0 → // vdim of Tor_1: 2 → list l = Tor(3,M,M,1); // Tor_3(R^2/M,R^2/M) over R=r/i → // Computing Tor_3 (help Tor; gives an explanation): → // Let 0 <- coker(M) <- G0 <-M- G1 be the present. of coker(M), → // and 0 <- coker(N) <- F0 <-N- F1 <- F2 <- ... a resolution of → // coker(N), then Tensor(G0,F3)-->Tensor(G0,F2) is given by: → x,0,0,0, → 0,x,0,0, → 0,0,x,0, → 0,0,0,x → // and Tensor(G0,F4) + Tensor(G1,F3)-->Tensor(G0,F3) is given by: → x,0,0,0,x,0,0,0, → 0,x,0,0,0,x,0,0, → 0,0,x,0,0,0,x,0, → 0,0,0,x,0,0,0,x → → // dimension of Tor_3: 0 → // vdim of Tor_3: 4 → → // columns of matrix are kbase of Tor_3 in Tensor(G0,F3) → 1,0,0,0, → 0,1,0,0, → 0,0,1,0, → 0,0,0,1 → printlevel = p; |