<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>pureTwoInvariant -- first betti number of specific exact complex</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_pure__Weyman.html">next</a> | <a href="_pure__Cohomology__Table_lp__List_cm__Z__Z_cm__Z__Z_rp.html">previous</a> | <a href="_pure__Weyman.html">forward</a> | <a href="_pure__Cohomology__Table_lp__List_cm__Z__Z_cm__Z__Z_rp.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>pureTwoInvariant -- first betti number of specific exact complex</h1> <div class="single"><h2>Synopsis</h2> <ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>pureTwoInvariant</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a strictly increasing sequence of degrees</span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, The zero-th betti number of the corresponding pure resolution construction</span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><a href="_pure__Two__Invariant.html" title="first betti number of specific exact complex">pureTwoInvariant</a> corresponds to the construction in math.AC/0709.1529v3 "The Existence of Pure Free Resolutions", Section 3.<table class="examples"><tr><td><pre>i1 : L = {0,2,3,9} o1 = {0, 2, 3, 9} o1 : List</pre> </td></tr> <tr><td><pre>i2 : B = pureBettiDiagram L 0 1 2 3 o2 = total: 7 27 21 1 0: 7 . . . 1: . 27 21 . 2: . . . . 3: . . . . 4: . . . . 5: . . . . 6: . . . 1 o2 : BettiTally</pre> </td></tr> <tr><td><pre>i3 : pureTwoInvariant L o3 = 196</pre> </td></tr> <tr><td><pre>i4 : L1 = {0,4,5,7} o4 = {0, 4, 5, 7} o4 : List</pre> </td></tr> <tr><td><pre>i5 : B1 = pureBettiDiagram L1 0 1 2 3 o5 = total: 3 35 42 10 0: 3 . . . 1: . . . . 2: . . . . 3: . 35 42 . 4: . . . 10 o5 : BettiTally</pre> </td></tr> <tr><td><pre>i6 : pureTwoInvariant L1 o6 = 15</pre> </td></tr> </table> Thus, for large enough multiples m, m*B occurs as the Betti diagram of a module from the pureTwoInvariant construction<p/> However, B itself occurs as the betti table of a module:<table class="examples"><tr><td><pre>i7 : betti res randomSocleModule(L,1) 0 1 2 3 o7 = total: 7 27 21 1 0: 7 . . . 1: . 27 21 . 2: . . . . 3: . . . . 4: . . . . 5: . . . . 6: . . . 1 o7 : BettiTally</pre> </td></tr> <tr><td><pre>i8 : betti res randomModule(L,1) 0 1 2 3 o8 = total: 7 27 35 15 0: 7 . . . 1: . 27 11 . 2: . . 24 15 o8 : BettiTally</pre> </td></tr> <tr><td><pre>i9 : betti res randomModule({0,6,7,9},1) 0 1 2 3 o9 = total: 1 21 27 7 0: 1 . . . 1: . . . . 2: . . . . 3: . . . . 4: . . . . 5: . 21 27 . 6: . . . 7 o9 : BettiTally</pre> </td></tr> <tr><td><pre>i10 : betti res randomSocleModule(L1,1) 0 1 2 3 o10 = total: 3 35 42 10 0: 3 . . . 1: . . . . 2: . . . . 3: . 35 42 . 4: . . . 10 o10 : BettiTally</pre> </td></tr> <tr><td><pre>i11 : betti res randomModule(L1,1) 0 1 2 3 o11 = total: 3 35 42 10 0: 3 . . . 1: . . . . 2: . . . . 3: . 35 42 . 4: . . . 10 o11 : BettiTally</pre> </td></tr> <tr><td><pre>i12 : betti res randomModule({0,2,3,7},1) 0 1 2 3 o12 = total: 10 42 50 18 0: 10 . . . 1: . 42 26 . 2: . . 24 18 o12 : BettiTally</pre> </td></tr> <tr><td><pre>i13 : betti res randomSocleModule({0,2,3,7},1) 0 1 2 3 o13 = total: 10 42 35 3 0: 10 . . . 1: . 42 35 . 2: . . . . 3: . . . . 4: . . . 3 o13 : BettiTally</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_pure__All.html" title="Vector of first betti number of our three specific exact complexes">pureAll</a> -- Vector of first betti number of our three specific exact complexes</span></li> <li><span><a href="_pure__Weyman.html" title="first betti number of specific exact complex">pureWeyman</a> -- first betti number of specific exact complex</span></li> <li><span><a href="_pure__Char__Free.html" title="first betti number of specific exact complex">pureCharFree</a> -- first betti number of specific exact complex</span></li> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>pureTwoInvariant</tt> :</h2> <ul><li>pureTwoInvariant(List)</li> </ul> </div> </div> </body> </html>