Calculating syzygies with Macaulay2

Question

I'm trying to calculate the syzygies of a set of elements on the polinomial ring of 6 variables. But I'm trying to specify the number of generator in each degree the syzygies have. I know that Macaulay2 can give me the syzygies of the sistem very fast, but it is returning some columns with mixed degree, and because of this I don`t know how to determinate how many generator of each degree I have.

I thought in define the set of polynomials as a map, and find a resolution for the kernel of this map, but it seems that even with Macaulay2, if I go that way I will have aplenty of calculations to do... My question is if that is any other faster way to solve this.

Thanks in any advance.

Answer 1

I think the problem here is understanding Macaulay2's notation. Let's take an example:

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i7 : I = ideal(random(2,R), random(3,R), random(2,R));

o7 : Ideal of R

i8 : betti res I

            0 1 2 3
o8 = total: 1 3 3 1
         0: 1 . . .
         1: . 2 . .
         2: . 1 1 .
         3: . . 2 .
         4: . . . 1

Here the last table gives us all the numerical information about the resolution of some ideal in $\mathbb Q[x,y,z]$. The way to read this is as follows: In the first syzygy $1$ there's two generators of degree $2$ and one of degree $3=1+2$. In the second syzygy there's one generator of degree $4=2+2$ and two of degree $5=3+2$.

In general, the number of generators of degree $i$ in the $j$th syzygy is found by reading the $(i,j-i)$ position in the Betti table.