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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 with
- 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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