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_is__Golod__A__Inf.html
<!DOCTYPE html> <html lang="en"> <head> <title>isGolodAInf -- Determines if the ring is Golod or not</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-infinity algebra and module structures on free resolutions" href="index.html">AInfinity</a> :: <a title="Determines if the ring is Golod or not" href="_is__Golod__A__Inf.html">isGolodAInf</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> next | <a href="_has__Minimal__Mult.html">previous</a> | forward | <a href="_has__Minimal__Mult.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>isGolodAInf -- Determines if the ring is Golod or not</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">h = isGolod R</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">R</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">h</span>, <span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, Whether or not the ring is Golod</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This function computes the A-infinity multiplications to all required orders, and reduces them modulo the maximal ideal. If all reductions are zero, then the ring R is Golod.</p> <p>Below is an example of an artinian ring R (based on an example of Roos and Katthan) which has minimal multiplications of order two, but is not Golod.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S = kk[x,y,z,u] o2 = S o2 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : I = ideal(u^3, x*y^2, (x+y)*z^2, x^2*u+z*u^2, y^2*u+x*z*u, y^2*z+y*z^2) 3 2 2 2 2 2 2 2 2 o3 = ideal (u , x*y , x*z + y*z , x u + z*u , y u + x*z*u, y z + y*z ) o3 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : J = trim (I + (ideal vars S)^6) 3 2 2 2 2 2 2 2 2 5 o4 = ideal (u , y u + x*z*u, x u + z*u , x*z + y*z , y z + y*z , x*y , z u, ------------------------------------------------------------------------ 6 4 5 6 5 6 z , x y*z, x z, y , x y, x ) o4 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : hasMinimalMult(quotient J, 2) o5 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : isGolodAInf quotient J o6 = false</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">isGolodAInf</span>:</h2> <ul> <li><kbd>isGolodAInf(Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Determines if the ring is Golod or not" href="_is__Golod__A__Inf.html">isGolodAInf</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">AInfinity.m2:1518:0</span>.</p> </div> </div> </div> </body> </html>
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