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<!DOCTYPE html> <html lang="en"> <head> <title>minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or 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="index.html">Documentation </a> <br><a href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Macaulay2 documentation" href="index.html">Macaulay2Doc</a> » <a href="_modules.html">modules</a> » <a title="including degree and betti numbers" href="___Hilbert_spfunctions_spand_spfree_spresolutions.html">Hilbert functions and free resolutions</a> » <a title="display or modify a Betti diagram" href="_betti.html">betti</a> » <a title="minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module" href="_minimal__Betti.html">minimalBetti</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="_betti_lp__Betti__Tally_rp.html">previous</a> | <a href="_basis.html">forward</a> | <a href="_betti_lp__Betti__Tally_rp.html">backward</a> | <a href="_betti.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">B = minimalBetti I</code></dd> <dd><code class="language-macaulay2">B = minimalBetti(I, DegreeLimit=>d, LengthLimit=>len, Weights=>h)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="___Ideal.html">ideal</a></span> or <span>a <a title="the class of all modules" href="___Module.html">module</a></span>, a homogeneous ideal or module in a singly graded polynomial ring or skew commuting polynomial ring over a finite prime field</span></li> </ul> </li> <li><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">DegreeLimit</span><span class="tt"> => </span><span>an <a title="the class of all integers" href="___Z__Z.html">integer</a></span>, <span>default value null</span>, if given, only compute enough to determine the Betti diagram up to and including the row labelled <span class="tt">d</span></span></li> <li><span><span class="tt">LengthLimit</span><span class="tt"> => </span><span>an <a title="the class of all integers" href="___Z__Z.html">integer</a></span>, <span>default value infinity</span>, if given, only compute enough to determine the Betti diagram up to and including the column labelled <span class="tt">len</span></span></li> <li><span><span class="tt">Weights</span><span class="tt"> => </span><span>a <a title="the class of all lists -- {...}" href="___List.html">list</a></span>, <span>default value null</span>, see <a title="view and set the weight vector of a Betti diagram" href="_betti_lp__Betti__Tally_rp.html">betti(...,Weights=>...)</a></span></li> <li><span><span class="tt">ParallelizeByDegree</span><span class="tt"> => </span><span>a <a title="the class of boolean values" href="___Boolean.html">Boolean value</a></span>, <span>default value false</span>, Use additional parallelism to compute different (homological degree, internal degree) pairs in parallel when possible. To control parallelism, see <a title="parallelism in engine computations" href="_parallelism_spin_spengine_spcomputations.html">parallelism in engine computations</a></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all Betti tallies" href="___Betti__Tally.html">Betti tally</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Given a singly-graded module, this function computes the minimal betti numbers of the module. If the input is an ideal $I \subset S$, it computes the minimal betti numbers of $S^1/I$.</p> <p>The algorithm used is based on the <a title="value of an optional argument" href="___Fast__Nonminimal.html">FastNonminimal</a> algorithm, except that the complex is not constructed, resulting in a smaller memory footprint and often reduced computation time.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : I = Grassmannian(1,6, CoefficientRing => ZZ/101); ZZ o1 : Ideal of ---[p ..p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p ] 101 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 0,5 1,5 2,5 3,5 4,5 0,6 1,6 2,6 3,6 4,6 5,6</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S = ring I o2 = S o2 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : elapsedTime C = minimalBetti I -- 1.59113s elapsed 0 1 2 3 4 5 6 7 8 9 10 o3 = total: 1 35 140 385 819 1080 819 385 140 35 1 0: 1 . . . . . . . . . . 1: . 35 140 189 84 . . . . . . 2: . . . 196 735 1080 735 196 . . . 3: . . . . . . 84 189 140 35 . 4: . . . . . . . . . . 1 o3 : BettiTally</code></pre> </td> </tr> </table> <div> <p>One can compute smaller parts of the Betti table, by using <a title="an optional argument" href="___Degree__Limit.html">DegreeLimit</a> and/or <a title="an optional argument" href="___Length__Limit.html">LengthLimit</a>.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : I = ideal I_*; o4 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2) -- .664584s elapsed 0 1 2 3 4 5 6 7 o5 = total: 1 35 140 385 819 1080 735 196 0: 1 . . . . . . . 1: . 35 140 189 84 . . . 2: . . . 196 735 1080 735 196 o5 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : I = ideal I_*; o6 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5) -- .027232s elapsed 0 1 2 3 4 o7 = total: 1 35 140 189 84 0: 1 . . . . 1: . 35 140 189 84 o7 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : I = ideal I_*; o8 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5) -- 1.09817s elapsed 0 1 2 3 4 5 o9 = total: 1 35 140 385 819 1080 0: 1 . . . . . 1: . 35 140 189 84 . 2: . . . 196 735 1080 o9 : BettiTally</code></pre> </td> </tr> </table> <div> <p>This function computes only as much of the non-minimal resolution as needed to compute the desired Betti numbers. Further calls will generally not recompute previously computed parts of the resolution, except that if you ask for a longer resolution than previously, it currently will recompute the resolution. This behavior might change in later releases.</p> </div> <div> <p>If one has already computed the non-minimal free resolution using <a title="value of an optional argument" href="___Fast__Nonminimal.html">FastNonminimal</a>, then one can use <a title="minimal betti numbers of a non-minimal free resolution" href="../../OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html">betti(...,Minimize=>...)</a>, except that it doesn't currently have support for <span class="tt">DegreeLimit</span> and <span class="tt">LengthLimit</span>, and probably still computes more than is needed (it is still experimental).</p> </div> </div> <div> <h2>Caveat</h2> <div> <p>Only works over finite prime field. If the ideal or module is a non-homogeneous or multi-homogeneous object, then this function will result in an error.</p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="display or modify a Betti diagram" href="_betti.html">betti</a> -- display or modify a Betti diagram</span></li> <li><span><a title="minimal betti numbers of a non-minimal free resolution" href="../../OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html">betti(...,Minimize=>...)</a> -- minimal betti numbers of a non-minimal free resolution</span></li> <li><span><a title="projective resolution" href="../../OldChainComplexes/html/_resolution.html">resolution</a> -- projective resolution</span></li> <li><span><a title="compute a non-minimal graded free resolution" href="../../OldChainComplexes/html/___Fast__Nonminimal.html">resolution(...,FastNonminimal=>...)</a> -- compute a non-minimal graded free resolution</span></li> <li><span><a title="algorithm for computing nonminimal free resolutions" href="../../Complexes/html/___Strategy_spfor_spnonminimal_spfree_spresolutions.html">Nonminimal</a> -- algorithm for computing nonminimal free resolutions</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">minimalBetti</span>:</h2> <ul> <li><kbd>minimalBetti(Ideal)</kbd></li> <li><kbd>minimalBetti(Module)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module" href="_minimal__Betti.html">minimalBetti</a> is <span>a <a title="a type of method function" href="___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">Macaulay2Doc/functions/minimalBetti-doc.m2:71:0</span>.</p> </div> </div> </div> </body> </html>