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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
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:
 


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