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<!DOCTYPE html> <html lang="en"> <head> <title>golodBetti -- list the ranks of the free modules in the resolution of a Golod module</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="list the ranks of the free modules in the resolution of a Golod module" href="_golod__Betti.html">golodBetti</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="_has__Minimal__Mult.html">next</a> | <a href="_burke__Resolution.html">previous</a> | <a href="_has__Minimal__Mult.html">forward</a> | <a href="_extract__Blocks.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>golodBetti -- list the ranks of the free modules in the resolution of a Golod module</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">B = golodBetti(M,b)</code></dd> </dl> </li> <li>Inputs: <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>, R-module</span></li> <li><span><span class="tt">b</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, homological degree to which to carry the computation</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">B</span>, <span>a <a title="the class of all Betti tallies" href="../../Macaulay2Doc/html/___Betti__Tally.html">Betti tally</a></span>, This would be betti table of the free res of M over R, if M were a Golod module over R</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Let S be a standard graded polynomial ring. A module M over R = S/I is Golod if the resolution H of M has maximal betti numbers given the betti numbers of the S-free resolutions F of R and K of M. This resolution, H, has underlying graded module H = R**K**T(B), where B is the truncated resolution F_1 <- F_2... and T(B) is the tensor algebra.</p> <p>Since the component modules of H are given, the computation only requires the computation of the minimal S-free resolution of M, and then is purely numeric; the differentials in the R-free resolution of M are not computed.</p> <p>In case M = coker vars R, the result is the Betti table of the Golod-Shamash-Eagon resolution of the residue field.</p> <p>We say that M is a Golod module (over R) if the ranks of the free modules in a minimal R-free resolution of M are equal to the numbers produced by golodBetti. Theorems of Levin and Lescot assert that if R has a Golod module, then R is a Golod ring; and that if R is Golod, then the d-th syzygy of any R-module M is Golod for all d greater than or equal to the projective dimension of M as an S-module (more generally, the co-depth of M) (Avramov, 6 lectures, 5.3.2).</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = (ideal(a,b,c^2))^2 2 2 2 2 4 o2 = ideal (a , a*b, a*c , b , b*c , c ) o2 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : F = res(S^1/I) 1 6 8 3 o3 = S <-- S <-- S <-- S 0 1 2 3 o3 : Complex</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 : F = burkeResolution (coker vars R, 6) 1 3 9 27 81 243 729 o5 = R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 o5 : Complex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : golodBetti(coker vars R,6) 0 1 2 3 4 5 6 o6 = total: 1 3 9 27 81 243 729 0: 1 3 6 12 24 48 96 1: . . 2 10 32 88 224 2: . . 1 5 20 72 232 3: . . . . 4 28 128 4: . . . . 1 7 42 5: . . . . . . 6 6: . . . . . . 1 o6 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : betti res (coker vars R, LengthLimit => 6) 0 1 2 3 4 5 6 o7 = total: 1 3 9 27 81 243 729 0: 1 3 6 12 24 48 96 1: . . 2 10 32 88 224 2: . . 1 5 20 72 232 3: . . . . 4 28 128 4: . . . . 1 7 42 5: . . . . . . 6 6: . . . . . . 1 o7 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : betti F 0 1 2 3 4 5 6 o8 = total: 1 3 9 27 81 243 729 0: 1 3 6 12 24 48 96 1: . . 2 10 32 88 224 2: . . 1 5 20 72 232 3: . . . . 4 28 128 4: . . . . 1 7 42 5: . . . . . . 6 6: . . . . . . 1 o8 : BettiTally</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="compute a resolution from A-infinity structures" href="_burke__Resolution.html">burkeResolution</a> -- compute a resolution from A-infinity structures</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">golodBetti</span>:</h2> <ul> <li><kbd>golodBetti(Module,ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="list the ranks of the free modules in the resolution of a Golod module" href="_golod__Betti.html">golodBetti</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.m2:1388:0</span>.</p> </div> </div> </div> </body> </html>