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TSpreadIdeals > minimalBettiNumbersIdeal

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

Synopsis

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 :

For the programmer

The object minimalBettiNumbersIdeal is a method function.