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<!DOCTYPE html> <html lang="en"> <head> <title>eagonBeta -- print the eagonBeta maps in the Eagon resolution</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="Construct the Eagon double complex, which contains a resolution of the residue field" href="index.html">EagonResolution</a> :: <a title="print the eagonBeta maps in the Eagon resolution" href="_eagon__Beta.html">eagonBeta</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="___Eagon__Data.html">next</a> | <a href="_eagon.html">previous</a> | <a href="___Eagon__Data.html">forward</a> | <a href="_eagon.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>eagonBeta -- print the eagonBeta maps in the Eagon resolution</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">N = eagonBeta E</code></dd> <dd><code class="language-macaulay2">N = eagonBeta(E,n)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">E</span>, <span>an instance of the type <a title="HashTable storing output of eagon" href="___Eagon__Data.html">EagonData</a></span>, created by eagon(R,b)</span></li> <li><span><span class="tt">n</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, which eagonBeta to show</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">Display</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value "picture"</span></span></li> <li><span><span class="tt">Verbose</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">N</span>, <span>a <a title="the class of all nets and strings" href="../../Macaulay2Doc/html/___Net.html">net</a></span>, either a "Display" display (with Display => "picture", the default) or a "displayBlocks" display with Display => "DisplayBlocks" or a plain matrix if Display => <anything else>. With Verbose => true, the display includes (rank target eagonBeta,rank source eagonBeta)</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The eagonBeta maps are the components of the Eagon resolution, starting from the 2nd differential that may or may not be minimal, and are therefore most interesting. With the default option</p> <p>Display => "picture"</p> <p>the pictures (which blocks are 0,nonzero, nonminimal) are shown; or the displayBlocks output with Display => "DisplayBlocks" or a plain matrix if Display => <any other string>.</p> <p>In the notes of Gulliksen-Levin it is proven that R is Golod if and only if the maps eagonBeta can be taken with values in the Koszul complex; thus in particular, if R is Golod, then there are no "new" eagonBetas after eagonBeta(E,numgens R+1). Since R is Golod iff all the eagonBeta matrices have all entries in the maximal ideal, this proves in particular that R is Golod if and only if the Betti numbers of the resolution of coker vars R agree up to the step numgens R with the Betti numbers of the Eagon resolution.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = ZZ/101[a,b,c,d] o1 = S o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal(a,b,c)*ideal(a,b,c,d) 2 2 2 o2 = ideal (a , a*b, a*c, a*d, a*b, b , b*c, b*d, a*c, b*c, c , c*d) o2 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : I = ideal"a3,b3,c3" 3 3 3 o3 = ideal (a , b , c ) o3 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : R = S/I o4 = R o4 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : E = eagon(R,4);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : eagonBeta(E,4) +--------+--------+-----------+ o6 = | |(2, {1})|(0, {1, 1})| +--------+--------+-----------+ | (3, {})| * | . | +--------+--------+-----------+ |(0, {2})| . | 3,3 | +--------+--------+-----------+</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : eagonBeta(E,4,Display => "DisplayBlocks") +--------+--------------------------------------------------------+----------------------------+ o7 = | | (2, {1}) | (0, {1, 1}) | +--------+--------------------------------------------------------+----------------------------+ | (3, {})|{3} | 0 0 c2 0 -b2 0 a2 0 0 0 0 0 0 0 0 0 0 0 || . | | |{3} | 0 0 0 0 0 0 0 0 0 0 -b2 0 a2 0 0 0 0 0 || | | |{3} | 0 0 0 0 0 0 0 0 0 0 0 -c2 0 0 0 a2 0 0 || | | |{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c2 0 b2 0 || | +--------+--------------------------------------------------------+----------------------------+ |(0, {2})| . |{6} | 0 1 0 -1 0 0 0 0 0 || | | |{6} | 0 0 1 0 0 0 -1 0 0 || | | |{6} | 0 0 0 0 0 1 0 -1 0 || +--------+--------------------------------------------------------+----------------------------+</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : eagonBeta(E,4,Display => "") o8 = {3} | 0 0 c2 0 -b2 0 a2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 {3} | 0 0 0 0 0 0 0 0 0 0 -b2 0 a2 0 0 0 0 0 0 0 0 0 0 0 0 {3} | 0 0 0 0 0 0 0 0 0 0 0 -c2 0 0 0 a2 0 0 0 0 0 0 0 0 0 {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c2 0 b2 0 0 0 0 0 0 0 0 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ------------------------------------------------------------------------ 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | -1 0 | 7 27 o8 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : eagonBeta E +-------------------------------+ |+-------+--------+ | o9 = || |(0, {1})| | |+-------+--------+ | ||(1, {})| * | | |+-------+--------+ | +-------------------------------+ |+-------+--------+ | || |(1, {1})| | |+-------+--------+ | ||(2, {})| * | | |+-------+--------+ | +-------------------------------+ |+--------+--------+-----------+| || |(2, {1})|(0, {1, 1})|| |+--------+--------+-----------+| || (3, {})| * | . || |+--------+--------+-----------+| ||(0, {2})| . | 3,3 || |+--------+--------+-----------+| +-------------------------------+</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="compute the Eagon double complex" href="_eagon.html">eagon</a> -- compute the Eagon double complex</span></li> <li><span><a title="information about components of a labeled Matrix or Complex" href="_picture.html">picture</a> -- information about components of a labeled Matrix or Complex</span></li> <li><span><a title="Display => "DisplayBlocks" option for picture" href="___Display__Blocks.html">DisplayBlocks</a> -- Display => "DisplayBlocks" option for picture</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">eagonBeta</span>:</h2> <ul> <li><kbd>eagonBeta(EagonData)</kbd></li> <li><kbd>eagonBeta(EagonData,ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="print the eagonBeta maps in the Eagon resolution" href="_eagon__Beta.html">eagonBeta</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">EagonResolution.m2:1039:0</span>.</p> </div> </div> </div> </body> </html>