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# soc -- compute the $\text{Soc}_\mathfrak{p}^*(I)$, where $I$ is a monomial ideal

## Synopsis

• Usage:
soc(I,p)
• Inputs:
• I, an ideal, a monomial ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
• p, an ideal, a stable prime of $I$
• Outputs:
• M, ,

## Description

This method computes the module $\text{Soc}_\mathfrak{p}^*(I)$. Let $\mathcal{R}(I)=\bigoplus_{k\ge0}I^k$ be the Rees algebra of $I$. For the prime $\mathfrak{p}\in\text{Ass}^\infty(I)$, we set $$X_\mathfrak{p}=\begin{cases}S&\text{if}\ \mathfrak{p}\in\text{Max}^\infty(I),\\ \prod\{\mathfrak{q}\in\text{Ass}^\infty(I):\mathfrak{p}\subsetneq\mathfrak{q}\}&\text{otherwise}.\end{cases}$$ Let $\mathcal{R}'=\bigoplus_{k\ge0}I^{k+1}$. We set $$\text{Soc}_\mathfrak{p}^*(I)=\frac{(\mathcal{R}':_{\mathcal{R}(I)}\mathfrak{p}\mathcal{R}(I))}{(\mathcal{R}':_{\mathcal{R}(I)}(\mathfrak{p}+X_\mathfrak{p}^\infty)\mathcal{R}(I))}.$$ Here $X_\mathfrak{p}^\infty=\bigcup_{k\ge0}(\mathfrak{p}:\mathfrak{m}^k)$ is the saturation of $\mathfrak{p}$ with respect to the maximal ideal $\mathfrak{m}=(x_1,\dots,x_n)$. As proved by Conca, for all $k\gg0$ we have $$\text{Soc}_\mathfrak{p}^*(I)_{(*,k)}=\frac{(I^{k+1}:\mathfrak{p})}{I^{k+1}:(\mathfrak{p}+X_\mathfrak{p}^\infty)},$$ and the initial degree of this module is the $\text{v}_\mathfrak{p}$-number of $I^{k+1}$.

 i1 : S = QQ[x_1..x_3]; 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 : p = ideal(x_1,x_2) o3 = ideal (x , x ) 1 2 o3 : Ideal of S i4 : soc(I,p) o4 = image | x_3 | QQ[x ..x , y ..y ] / QQ[x ..x , y ..y ] \ 1 3 1 3 | 1 3 1 3 |1 o4 : -------------------------------------------------module, submodule of |------------------------------------------------| (x x , x x , x x , - x y + x y , - x y + x y ) |(x x , x x , x x , - x y + x y , - x y + x y )| 1 2 1 3 2 3 1 3 2 2 1 3 3 1 \ 1 2 1 3 2 3 1 3 2 2 1 3 3 1 /

• vFunctionP -- compute the $\text{v}_\mathfrak{p}$-function of monomial ideal $I$
• vFunction -- compute the $\text{v}$-function of monomial ideal $I$