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D.4.3.4 Ext_R
Procedure from library homolog.lib (see homolog_lib).
- Usage:
Ext_R(v,M[,p]); v int resp. intvec , M module, p int
- Compute:
A presentation of Ext^k(M’,R); for k=v[1],v[2],..., M’=coker(M). Let
0 <-- M' <-- F0 <-M-- F1 <-- F2 <-- ...
be a free resolution of M’. If
0 --> F0* -A1-> F1* -A2-> F2* -A3-> ...
is the dual sequence, Fi*=Hom(Fi,R), then Ext^k = ker(Ak+1)/im(Ak) is presented as in the following exact sequences:
R^p --syz(Ak+1)-> Fk* ---Ak+1----> Fk+1* , R^q ----Ext^k---> R^p --syz(Ak+1)-> Fk*/im(Ak).Hence, Ext^k=modulo(syz(Ak+1),Ak) presents Ext^k(M’,R).
- Return:
- module Ext, a presentation of Ext^k(M’,R) 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 type int return a list l:l[1] = module Ext^k resp. list of Ext^k l[2] = SB of Ext^k resp. list of SB of Ext^k l[3] = matrix resp. list of matrices, each representing a kbase of Ext^k (if finite dimensional)- Display:
printlevel >=0: (affine) dimension of Ext^k for each k (default) printlevel >=1: Ak, Ak+1 and kbase of Ext^k in Fk*
- Note:
In order to compute Ext^k(M,R) use the command Ext_R(k,syz(M)); or the 2 commands: list L=mres(M,2); Ext_R(k,L[2]);
Example:
LIB "homolog.lib"; int p = printlevel; printlevel = 1; ring r = 0,(x,y,z),dp; ideal i = x2y,y2z,z3x; module E = Ext_R(1,i); //computes Ext^1(r/i,r) → // Computing Ext^1: → // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of M, → // then F1*-->F2* is given by: → x2, -yz,0, → 0, z3, -xy, → xz2,0, -y2 → // and F0*-->F1* is given by: → y2z, → x2y, → xz3 → → // dimension of Ext^1: -1 → is_zero(E); → 1 qring R = std(x2+yz); intvec v = 0,2; printlevel = 2; //shows what is going on ideal i = x,y,z; //computes Ext^i(r/(x,y,z),r/(x2+yz)), i=0,2 list L = Ext_R(v,i,1); //over the qring R=r/(x2+yz), std and kbase → // Computing Ext^0: → // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of M, → // then F0*-->F1* is given by: → z, → y, → x → // and F-1*-->F0* is given by: → 0 → → // dimension of Ext^0: -1 → → // columns of matrix are kbase of Ext^0 in F0*: → 0 → → // Computing Ext^2: → // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of M, → // then F2*-->F3* is given by: → x,-y,z, 0, → z,x, 0, z, → 0,0, x, y, → 0,0, -z,x → // and F1*-->F2* is given by: → y,-z,0, → x,0, -z, → 0,x, -y, → 0,z, x → → // dimension of Ext^2: 0 → // vdim of Ext^2: 1 → → // columns of matrix are kbase of Ext^2 in F2*: → x, → -z, → 0, → 0 → printlevel = p; |