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D.4.7.20 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:
 

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