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
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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
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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
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i4 : I==J
o4 = true
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i5 : betti I==betti J
o5 = false
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i6 : minimalBettiNumbersIdeal I==minimalBettiNumbersIdeal J
o6 = true
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