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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 $R$ 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 $R^n/M$, 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 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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