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5.1.3 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 $R^n/M$, if $R$ denotes the basering and $M$ a homogeneous submodule of $R^n$ and the argument represents a resolution of $R^n/M$.
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 $R^n/M$ (the 0th (resp. 1st) syzygy module of $R^n/M$ is $R^n$ (resp. $M$)). The argument is considered to be the result of a res/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.

The optional second argument is a switch for the minimization of the Betti numbers. If it is 0 then the Betti numbers correspond exactly to the input - otherwise the command is identical to the one-argument form.

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

  • 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,

where the generators are the columns of the displayed matrix and degrees are assigned such that the corresponding maps have degree 0:

\begin{displaymath} 0 \longleftarrow r/j \longleftarrow r(1) \buildrel{T[1]}\ove... ...ldrel{T[3]}\over{\longleftarrow} r(5) \longleftarrow 0 \quad . \end{displaymath}

See Syzygies and resolutions; hres; lres; mres; print; res; resolution; sres.

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