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7.3.1 betti (plural)
Syntax:
betti ( list_expression )
betti ( resolution_expression )
betti ( list_expression , int_expression )
betti ( resolution_expression , int_expression )
Type:
- intmat
Note:
- in the non-commutative case, computing Betti numbers makes sense only if the basering
 has homogeneous relations.
The output of the command can be pretty-printed using print( , "betti"), i.e., with "betti" as second argument; see below example.
Purpose:
- with 1 argument: computes the graded Betti numbers of a minimal resolution of
 , if denotes the basering and
 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 (the 0th (resp. 1st) syzygy module of is
(resp. )).The argument is considered to be the result of a mres or nres
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:
| | int i;int N=2;
ring r=0,(x(1..N),d(1..N),q(1..N)),Dp;
matrix D[3*N][3*N];
for (i=1;i<=N;i++)
{ D[i,N+i]=q(i)^2; }
def W=nc_algebra(1,D); setring W;
// this algebra is a kind of homogenized Weyl algebra
W;
==> // characteristic : 0
==> // number of vars : 6
==> // block 1 : ordering Dp
==> // : names x(1) x(2) d(1) d(2) q(1) q(2)
==> // block 2 : ordering C
==> // noncommutative relations:
==> // d(1)x(1)=x(1)*d(1)+q(1)^2
==> // d(2)x(2)=x(2)*d(2)+q(2)^2
ideal I = x(1),x(2),d(1),d(2),q(1),q(2);
option(redSB);
option(redTail);
resolution R = mres(I,0);
// thus R will be the full length minimal resolution
print(betti(R),"betti");
==> 0 1 2 3 4 5 6
==> ------------------------------------------------
==> 0: 1 6 15 20 15 6 1
==> ------------------------------------------------
==> total: 1 6 15 20 15 6 1
==>
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