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<!DOCTYPE html> <html lang="en"> <head> <title>resolutionPoset -- generates a poset from a 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="a package for working with partially ordered sets" href="index.html">Posets</a> :: <a title="generates a poset from a resolution" href="_resolution__Poset.html">resolutionPoset</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="_set__Partition.html">next</a> | <a href="_remove__Isomorphic__Posets.html">previous</a> | <a href="_set__Partition.html">forward</a> | <a href="_remove__Isomorphic__Posets.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>resolutionPoset -- generates a poset from a resolution</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">P = resolutionPoset C</code></dd> <dd><code class="language-macaulay2">P = resolutionPoset I</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">C</span>, <span>a <a title="the class of all chain complexes" href="../../Complexes/html/___Complex.html">complex</a></span>, </span></li> <li><span><span class="tt">I</span>, <span>a <a title="the class of all monomial ideals handled by the engine" href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span>, </span></li> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">P</span>, <span>an instance of the type <a title="a class for partially ordered sets (posets)" href="___Poset.html">Poset</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Given a resolution $C$, a poset can be defined by the non-zero entries of the matrices of each component of the resolution.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x,y,z];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : C = freeResolution ideal(y*z,x*z,x^2*y) 1 3 2 o2 = R <-- R <-- R 0 1 2 o2 : Complex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : resolutionPoset C o3 = Relation Matrix: | 1 1 1 1 1 1 | | 0 1 0 0 1 0 | | 0 0 1 0 1 1 | | 0 0 0 1 0 1 | | 0 0 0 0 1 0 | | 0 0 0 0 0 1 | o3 : Poset</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : (resolutionPoset C).GroundSet o4 = {{0, 0}, {1, 0}, {1, 1}, {1, 2}, {2, 0}, {2, 1}} o4 : List</code></pre> </td> </tr> </table> <div> <p>Moreover, the resolution-poset of a <a title="the class of all monomial ideals handled by the engine" href="../../Macaulay2Doc/html/___Monomial__Ideal.html">MonomialIdeal</a> can be labeled as the <a title="least common multiple" href="../../Macaulay2Doc/html/_lcm.html">lcm</a> of the generators involved at each level. As the lcm needn't be unique at each step, we simply append it to the base labeling, as above.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : P = resolutionPoset monomialIdeal(y*z,x*z,x^2*y) o5 = P o5 : Poset</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : P.GroundSet 2 2 o6 = {{0, 0, {0, 0}}, {1, 0, x y}, {1, 1, x*z}, {1, 2, y*z}, {2, 0, x y*z}, ------------------------------------------------------------------------ {2, 1, x*y*z}} o6 : List</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">resolutionPoset</span>:</h2> <ul> <li><kbd>resolutionPoset(Complex)</kbd></li> <li><kbd>resolutionPoset(Ideal)</kbd></li> <li><kbd>resolutionPoset(MonomialIdeal)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="generates a poset from a resolution" href="_resolution__Poset.html">resolutionPoset</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">Posets.m2:3862:0</span>.</p> </div> </div> </div> </body> </html>