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D.4.3.5 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- G1be 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).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)).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)- 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; |