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<!DOCTYPE html> <html lang="en"> <head> <title>symExt -- the first differential of the complex R(M)</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="Bernstein-Gel'fand-Gel'fand correspondence" href="index.html">BGG</a> :: <a title="the first differential of the complex R(M)" href="_sym__Ext.html">symExt</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="_tate__Resolution.html">next</a> | <a href="___Regularity.html">previous</a> | <a href="_tate__Resolution.html">forward</a> | <a href="___Regularity.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>symExt -- the first differential of the complex R(M)</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">symExt(m,E)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">m</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a presentation matrix for a positively graded module M over a polynomial ring</span></li> <li><span><span class="tt">E</span>, <span>a <a title="the class of all ordered monoid rings" href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, exterior algebra</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a matrix representing the map <span class="tt">M_1 ** omega_E <-- M_0 ** omega_E</span></span></li> </ul> </li> </ul> <div> <h2>Description</h2> This function takes as input a matrix <span class="tt">m</span> with linear entries, which we think of as a presentation matrix for a positively graded <span class="tt">S</span>-module <span class="tt">M</span> matrix representing the map <span class="tt">M_1 ** omega_E <-- M_0 ** omega_E</span> which is the first differential of the complex <span class="tt">R(M)</span>. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = ZZ/32003[x_0..x_2]; </code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : M = coker matrix {{x_0^2, x_1^2}};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : m = presentation truncate(regularity M,M); 4 8 o4 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : symExt(m,E) o5 = {-1} | e_2 0 0 0 | {-1} | e_1 e_2 0 0 | {-1} | e_0 0 e_2 0 | {-1} | 0 e_0 e_1 e_2 | 4 4 o5 : Matrix E <-- E</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> This function is a quick-and-dirty tool which requires little computation. However if it is called on two successive truncations of a module, then the maps it produces may NOT compose to zero because the choice of bases is not consistent. </div> <div> <h2>See also</h2> <ul> <li><span><a title="the ith differential of the complex R(M)" href="_bgg.html">bgg</a> -- the ith differential of the complex R(M)</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">symExt</span>:</h2> <ul> <li><kbd>symExt(Matrix,PolynomialRing)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="the first differential of the complex R(M)" href="_sym__Ext.html">symExt</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">BGG.m2:697:0</span>.</p> </div> </div> </div> </body> </html>