#### D.4.8.20 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 with

           l[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;

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