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D.4.7.6 Ext
Procedure from library homolog.lib (see homolog_lib).
- Usage:
- Ext(v,M,N[,any]); v int resp. intvec, M,N modules
- Compute:
- A presentation of Ext^k(M',N'); for k=v[1],v[2],... where
M'=coker(M) and N'=coker(N). Let
| | 0 <-- M' <-- F0 <-M-- F1 <-- F2 <--... ,
0 <-- N' <-- G0 <--N- G1
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be a free resolution of M', resp. a presentation of N'. Consider
the commutative diagram
| | 0 0 0
|^ |^ |^
--> Hom(Fk-1,N') -Ak-> Hom(Fk,N') -Ak+1-> Hom(Fk+1,N')
|^ |^ |^
--> Hom(Fk-1,G0) -Ak-> Hom(Fk,G0) -Ak+1-> Hom(Fk+1,G0)
|^ |^
|C |B
Hom(Fk,G1) ------> Hom(Fk+1,G1)
(Ak,Ak+1 induced by M and B,C induced by N).
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Let K=modulo(Ak+1,B), J=module(Ak)+module(C) and Ext=modulo(K,J),
then we have exact sequences
| | R^p --K-> Hom(Fk,G0) --Ak+1-> Hom(Fk+1,G0)/im(B),
R^q -Ext-> R^p --K-> Hom(Fk,G0)/(im(Ak)+im(C)).
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Hence, Ext presents Ext^k(M',N').
- Return:
- - module Ext, a presentation of Ext^k(M',N') if v is of type int
- a list of Ext^k (k=v[1],v[2],...) if v is of type intvec.
- In case of a third argument of any type return a list l:
| | l[1] = module Ext/list of Ext^k
l[2] = SB of Ext/list of SB of Ext^k
l[3] = matrix/list of matrices, each representing a kbase of Ext^k
(if finite dimensional)
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- Display:
- printlevel >=0: dimension, vdim of Ext^k for each k (default).
printlevel >=1: matrices Ak, Ak+1 and kbase of Ext^k in Hom(Fk,G0)
(if finite dimensional)
- Note:
- In order to compute Ext^k(M,N) use the command Ext(k,syz(M),syz(N));
or: list P=mres(M,2); list Q=mres(N,2); Ext(k,P[2],Q[2]);
Example:
| | LIB "homolog.lib";
int p = printlevel;
printlevel = 1;
ring r = 0,(x,y),dp;
ideal i = x2-y3;
ideal j = x2-y5;
list E = Ext(0..2,i,j); // Ext^k(r/i,r/j) for k=0,1,2 over r
==> // Computing Ext^0 (help Ext; gives an explanation):
==> // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of coker(M),
==> // and 0<--coker(N)<--G0<--G1 a presentation of coker(N),
==> // then Hom(F0,G0)-->Hom(F1,G0) is given by:
==> y3-x2
==> // and Hom(F-1,G0) + Hom(F0,G1)-->Hom(F0,G0) is given by:
==> 0,-y5+x2
==>
==> // dimension of Ext^0: -1
==>
==> // Computing Ext^1 (help Ext; gives an explanation):
==> // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of coker(M),
==> // and 0<--coker(N)<--G0<--G1 a presentation of coker(N),
==> // then Hom(F1,G0)-->Hom(F2,G0) is given by:
==> 0
==> // and Hom(F0,G0) + Hom(F1,G1)-->Hom(F1,G0) is given by:
==> y3-x2,-y5+x2
==>
==> // dimension of Ext^1: 0
==> // vdim of Ext^1: 10
==>
==> // Computing Ext^2 (help Ext; gives an explanation):
==> // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of coker(M),
==> // and 0<--coker(N)<--G0<--G1 a presentation of coker(N),
==> // then Hom(F2,G0)-->Hom(F3,G0) is given by:
==> 1
==> // and Hom(F1,G0) + Hom(F2,G1)-->Hom(F2,G0) is given by:
==> 0,-y5+x2
==>
==> // dimension of Ext^2: -1
==>
qring R = std(i);
ideal j = fetch(r,j);
module M = [-x,y],[-y2,x];
printlevel = 2;
module E1 = Ext(1,M,j); // Ext^1(R^2/M,R/j) over R=r/i
==> // Computing Ext^1 (help Ext; gives an explanation):
==> // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of coker(M),
==> // and 0<--coker(N)<--G0<--G1 a presentation of coker(N),
==> // then Hom(F1,G0)-->Hom(F2,G0) is given by:
==> x, -y,
==> y2,-x
==> // and Hom(F0,G0) + Hom(F1,G1)-->Hom(F1,G0) is given by:
==> x, -y,-y5+x2,0,
==> y2,-x,0, -y5+x2
==>
==> // dimension of Ext^1: -1
==>
list l = Ext(4,M,M,1); // Ext^4(R^2/M,R^2/M) over R=r/i
==> // Computing Ext^4 (help Ext; gives an explanation):
==> // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of coker(M),
==> // and 0<--coker(N)<--G0<--G1 a presentation of coker(N),
==> // then Hom(F4,G0)-->Hom(F5,G0) is given by:
==> x, -y,0, 0,
==> y2,-x,0, 0,
==> 0, 0, x, -y,
==> 0, 0, y2,-x
==> // and Hom(F3,G0) + Hom(F4,G1)-->Hom(F4,G0) is given by:
==> x, -y,0, 0, -x,0, -y2,0,
==> y2,-x,0, 0, 0, -x,0, -y2,
==> 0, 0, x, -y,y, 0, x, 0,
==> 0, 0, y2,-x,0, y, 0, x
==>
==> // dimension of Ext^4: 0
==> // vdim of Ext^4: 2
==>
==> // columns of matrix are kbase of Ext^4 in Hom(F4,G0)
==> 1,0,
==> 0,y,
==> 0,1,
==> 1,0
==>
==> // element 1 of kbase of Ext^4 in Hom(F4,G0)
==> // as matrix: F4-->G0
==> 1,0,
==> 0,1
==> // element 2 of kbase of Ext^4 in Hom(F4,G0)
==> // as matrix: F4-->G0
==> 0,y,
==> 1,0
==>
printlevel = p;
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