#### D.4.8.6 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

  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).

  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;

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