minimalBettiNumbersIdeal -- return the minimal Betti numbers of a given graded ideal

Synopsis

• Usage:
minimalBettiNumbersIdeal I
• Inputs:
• I, a graded ideal of a polynomial ring
• Outputs:
• , the Betti table of the ideal I computed using its minimal generators

Description

let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and let I a graded ideal of $S$. Then I has a minimal graded free $S$ resolution:$F_{\scriptscriptstyle\bullet}:0\rightarrow \bigoplus_{j\in\mathbb{Z}}S(-j)^{\beta_{r,j}}\rightarrow \cdots\rightarrow \bigoplus_{j\in\mathbb{Z}}S(-j)^{\beta_{1,j}}\rightarrow \bigoplus_{j\in\mathbb{Z}}S(-j)^{\beta_{0,j}}\rightarrow I\rightarrow 0.$
The integer $\beta_{i,j}$ is a graded Betti number of I and it represents the dimension as a $K$-vector space of the $j$-th graded component of the $i$-th free module of the resolution. Each of the numbers $\beta_i=\sum_{j\in\mathbb{Z}}\beta_{i,j}$ is called the $i$-th Betti number of I.

Example:

 i1 : S=QQ[x_1..x_4] o1 = S o1 : PolynomialRing i2 : I=ideal(x_1*x_2,x_1*x_3,x_2*x_3) o2 = ideal (x x , x x , x x ) 1 2 1 3 2 3 o2 : Ideal of S i3 : J=ideal(join(flatten entries gens I,{x_1*x_2*x_3})) o3 = ideal (x x , x x , x x , x x x ) 1 2 1 3 2 3 1 2 3 o3 : Ideal of S i4 : I==J o4 = true i5 : betti I==betti J o5 = false i6 : minimalBettiNumbersIdeal I==minimalBettiNumbersIdeal J o6 = true

Ways to use minimalBettiNumbersIdeal :

• "minimalBettiNumbersIdeal(Ideal)"

For the programmer

The object minimalBettiNumbersIdeal is .