Description
let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and let
I a graded ideal of $S$. A graded Betti number of the ideal $\beta_{k,k+\ell}(I)\ne 0$ is called extremal if $\beta_{i,i+j}(I)=0$ for all $i\ge k,j\ge\ell,(i,j)\ne(k,\ell).$
Let $\beta_{k,k+\ell}(I)$ be an extremal Betti number of
I, the pair $(k,\ell)$ is called a corner of
I. If $(k_1,\ell_1),\dots,(k_r,\ell_r)$, with $n-1\ge k_1>k_2>\dots>k_r\ge1 \text{ and } 1\le \ell_1<\ell_2<\dots<\ell_r$, are all the corners of a graded ideal
I of $S$, the set $\mathrm{Corn}(I)=\Big\{(k_1,\ell_1),(k_2,\ell_2),\dots,(k_r,\ell_r)\Big\}$ is called the corner sequence of
I.
We recall that
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_10]
o1 = S
o1 : PolynomialRing
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i2 : I=ideal(x_1*x_3,x_1*x_4,x_1*x_5,x_2*x_4*x_6,x_2*x_4*x_7,x_2*x_5*x_7*x_9,x_3*x_5*x_7*x_9)
o2 = ideal (x x , x x , x x , x x x , x x x , x x x x , x x x x )
1 3 1 4 1 5 2 4 6 2 4 7 2 5 7 9 3 5 7 9
o2 : Ideal of S
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i3 : tExtremalBettiCorners(I,2)
o3 = {(2, 4)}
o3 : List
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i4 : minimalBettiNumbersIdeal I
0 1 2
o4 = total: 7 10 4
2: 3 3 1
3: 2 3 1
4: 2 4 2
o4 : BettiTally
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