# Singular

### 5.1.4 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 , if denotes the basering, is 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 , i.e. the 0th (resp. 1st) syzygy module of is (resp. ).The argument is considered to be the result of a res/fres/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.
betti sets the attribute rowShift.

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); ==> 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); ==> 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); ==> -b, ==> a, ==> -c, ==> d print(T); ==> 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 (which is j) has 4 generators (one in degree 2 and three in degree 3),
• the 2nd syzygy module T has 4 generators (all in degree 4),
• the 3rd syzygy module T 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: ### Misc 