<?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>betti(Module) -- show the degrees of the generators and relations of a module or a coherent sheaf</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Betti__Tally.html">next</a> | <a href="_betti_lp__Matrix_rp.html">previous</a> | <a href="___Betti__Tally.html">forward</a> | <a href="_betti_lp__Matrix_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>betti(Module) -- show the degrees of the generators and relations of a module or a coherent sheaf</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>betti M</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_betti.html" title="display degrees">betti</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, graded.</span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="___Betti__Tally.html">Betti tally</a></span>, a diagram showing the degrees of the generators and the relations in the module <tt>M</tt></span></li> </ul> </div> </li> <li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_betti_lp..._cm_sp__Weights_sp_eq_gt_sp..._rp.html">Weights => ...</a>, </span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>The diagram lists the zeroth and first graded and total Betti numbers of the module.<table class="examples"><tr><td><pre>i1 : S = ZZ/10007[x,y];</pre> </td></tr> <tr><td><pre>i2 : betti coker matrix{{x^3,x*y^2},{y*x^2,y^3}} 0 1 o2 = total: 2 2 0: 2 . 1: . . 2: . 2 o2 : BettiTally</pre> </td></tr> </table> Another example:<table class="examples"><tr><td><pre>i3 : betti coker map(S^{0,-1},,matrix{{x^2,y},{y^3,x^2}}) 0 1 o3 = total: 2 2 0: 1 . 1: 1 . 2: . 1 3: . 1 o3 : BettiTally</pre> </td></tr> </table> For an explanation of the diagram, see <a href="_betti_lp__Graded__Module_rp.html" title="display of degrees in a graded module">betti(GradedModule)</a>.</div> </div> </div> </body> </html>