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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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