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 $i\ge k,j\ge\ell,(i,j)\ne(k,\ell).$
The basic monomials related to an extremal Betti number configuration (values as well as positions) are defined by the following characterization. Let $I$ be a $t$-spread strongly stable ideal of
S and let $\beta_{k,k+\ell}(I)$ be an extremal Betti number of $I$. Then $\beta_{k,k+\ell}(I)=\Big|\Big\{ u\in G(I)_\ell:\max(u)=k+t(\ell-1)+1 \Big\}\Big|.$
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_12]
o1 = S
o1 : PolynomialRing
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i2 : corners={{4,2},{3,4},{2,5}}
o2 = {{4, 2}, {3, 4}, {2, 5}}
o2 : List
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i3 : a={1,2,1}
o3 = {1, 2, 1}
o3 : List
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i4 : l=tExtremalBettiMonomials(S,corners,a,2)
o4 = {x x , x x x x , x x x x , x x x x x }
1 7 2 4 6 10 2 4 7 10 2 5 7 9 11
o4 : List
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i5 : I=tStronglyStableIdeal(ideal l,2)
o5 = ideal (x x , 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 1 6 1 7 2 4 6 8 2 4 6 9 2 4 6 10
------------------------------------------------------------------------
x x x x , x x x x , x x x x x )
2 4 7 9 2 4 7 10 2 5 7 9 11
o5 : Ideal of S
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i6 : minimalBettiNumbersIdeal I
0 1 2 3 4
o6 = total: 11 23 19 7 1
2: 5 10 10 5 1
3: . . . . .
4: 5 11 8 2 .
5: 1 2 1 . .
o6 : BettiTally
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