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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
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.

If the optional second argument is non-zero, the Betti numbers will be minimized.

Example:
 

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:

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


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