|  |  5.1.4 betti 
Syntax:betti (list_expression)
 betti (resolution_expression)
 betti (list_expression,int_expression)
 betti (resolution_expression,int_expression)Type:intmat
Purpose:with 1 argument: computes the graded Betti numbers of a minimal resolution of
 , if  denotes the basering,  is a homogeneous submodule of  and the argument represents a
resolution of  . The entry d of the intmat at place (i,j) is the minimal number of
generators in degree i+j of the j-th syzygy module (= module of
relations) of
  , i.e. the 0th (resp. 1st) syzygy module of  is  (resp.  ).The argument is considered to be the result of a res/fres/sres/mres/nres/lres
command. This implies that a zero is only allowed (and counted) as a
generator in the first module. For the computation betti uses only the initial monomials. This could lead
to confusing results for a non-homogeneous input.
 
If the optional second argument is non-zero, the Betti numbers will be minimized.
 bettisets the attributerowShift.Example:|  |   ring r=32003,(a,b,c,d),dp;
  ideal j=bc-ad,b3-a2c,c3-bd2,ac2-b2d;
  list T=mres(j,0); // 0 forces a full resolution
  // a minimal set of generators for j:
  print(T[1]);
==> bc-ad,
==> c3-bd2,
==> ac2-b2d,
==> b3-a2c
  // second syzygy module of r/j which is the first
  // syzygy module of j (minimal generating set):
  print(T[2]);
==> bd,c2,ac,b2,
==> -a,-b,0, 0, 
==> c, d, -b,-a,
==> 0, 0, -d,-c 
  // the second syzygy module (minimal generating set):
  print(T[3]);
==> -b,
==> a, 
==> -c,
==> d  
  print(T[4]);
==> 0
  betti(T);
==> 1,0,0,0,
==> 0,1,0,0,
==> 0,3,4,1 
  // most useful for reading off the graded Betti numbers:
  print(betti(T),"betti");
==>            0     1     2     3
==> ------------------------------
==>     0:     1     -     -     -
==>     1:     -     1     -     -
==>     2:     -     3     4     1
==> ------------------------------
==> total:     1     4     4     1
==> 
 | 
 
Hence,
 
where the generators are the columns of the
displayed matrix and degrees are assigned such that the corresponding maps
have degree 0:the 0th syzygy module of r/j (which is r) has 1 generator in
degree 0 (which is 1),
the 1st syzygy module T[1](which is j) has 4
generators (one in degree 2 and three in degree 3),the 2nd syzygy
module T[2]has 4 generators (all in degree 4),the 3rd syzygy module T[3]has
1 generator in degree 5, 
 
 
See
 Syzygies and resolutions;
 fres;
 hres;
 lres;
 mres;
 print;
 res;
 resolution;
 sres.
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