Top
Back: depth
Forward: Ext
FastBack: grwalk_lib
FastForward: integralbasis_lib
Up: homolog_lib
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

D.4.7.5 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));
By default, the procedure uses the mres command. If called with the additional parameter "sres", the sres command is used instead.
If the attribute "isHomog" has been set for the input module, it is also set for the returned module (accordingly).

Example:
 


Top Back: depth Forward: Ext FastBack: grwalk_lib FastForward: integralbasis_lib Up: homolog_lib Top: Singular Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 3-1-6, Dec 2012, generated by texi2html.