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<!DOCTYPE html> <html lang="en"> <head> <title>soc -- compute the $\text{Soc}_\mathfrak{p}^*(I)$, where $I$ is a monomial ideal</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="a package for computing the v-number and v-function" href="index.html">VNumber</a> :: <a title="compute the $\\text{Soc}_\\mathfrak{p}^*(I)$, where $I$ is a monomial ideal" href="_soc.html">soc</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_stable__Max.html">next</a> | <a href="_rees__Map.html">previous</a> | <a href="_stable__Max.html">forward</a> | <a href="_rees__Map.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>soc -- compute the $\text{Soc}_\mathfrak{p}^*(I)$, where $I$ is a monomial ideal</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">soc(I,p)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a monomial ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field</span></li> <li><span><span class="tt">p</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a stable prime of $I$</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">M</span>, <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>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 $I^{k+1}$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = QQ[x_1..x_3];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : p = ideal(x_1,x_2) o3 = ideal (x , x ) 1 2 o3 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : soc(I,p) o4 = subquotient (| x_3 x_1x_2 |, | x_2x_3 x_1x_3 x_1x_2 x_3y_1 |) QQ[x ..x , y ..y ] / QQ[x ..x , y ..y ] \ 1 3 1 3 | 1 3 1 3 |1 o4 : -------------------------------module, subquotient of |------------------------------| (- x y + x y , - x y + x y ) |(- x y + x y , - x y + x y )| 1 3 2 2 1 3 3 1 \ 1 3 2 2 1 3 3 1 /</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="compute the $\\text{v}_\\mathfrak{p}$-function of monomial ideal $I$" href="_v__Function__P.html">vFunctionP</a> -- compute the $\text{v}_\mathfrak{p}$-function of monomial ideal $I$</span></li> <li><span><a title="compute the $\\text{v}$-function of monomial ideal $I$" href="_v__Function.html">vFunction</a> -- compute the $\text{v}$-function of monomial ideal $I$</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">soc</span>:</h2> <ul> <li><kbd>soc(Ideal,Ideal)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="compute the $\\text{Soc}_\\mathfrak{p}^*(I)$, where $I$ is a monomial ideal" href="_soc.html">soc</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">VNumber.m2:547:0</span>.</p> </div> </div> </div> </body> </html>