<?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>numgens(CoherentSheaf) -- the number of generators of the underlying module</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_numgens_lp__General__Ordered__Monoid_rp.html">next</a> | <a href="_numgens.html">previous</a> | <a href="_numgens_lp__General__Ordered__Monoid_rp.html">forward</a> | <a href="_numgens.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>numgens(CoherentSheaf) -- the number of generators of the underlying module</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>numgens F</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_numgens.html" title="the number of generators">numgens</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span>, number of generators of the underlying module <tt>M</tt> of <tt>F</tt></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>In Macaulay2, each coherent sheaf comes equipped with a module over the coordinate ring. In the homogeneous case, this is not necessarily the number of generators of the sum of twists <tt>H^0(F(d))</tt>, summed over all d, which in fact could be infinitely generated.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d]/(a^3+b^3+c^3+d^3) o1 = R o1 : QuotientRing</pre> </td></tr> <tr><td><pre>i2 : X = Proj R;</pre> </td></tr> <tr><td><pre>i3 : T' = cotangentSheaf X o3 = cokernel {2} | c 0 0 d 0 a2 b2 0 | {2} | a d 0 0 b2 -c2 0 0 | {2} | -b 0 d 0 a2 0 c2 0 | {2} | 0 b a 0 -d2 0 0 c2 | {2} | 0 -c 0 a 0 -d2 0 b2 | {2} | 0 0 -c -b 0 0 d2 a2 | 6 o3 : coherent sheaf on X, quotient of OO (-2) X</pre> </td></tr> <tr><td><pre>i4 : numgens T' o4 = 6</pre> </td></tr> <tr><td><pre>i5 : module T' o5 = cokernel {2} | c 0 0 d 0 a2 b2 0 | {2} | a d 0 0 b2 -c2 0 0 | {2} | -b 0 d 0 a2 0 c2 0 | {2} | 0 b a 0 -d2 0 0 c2 | {2} | 0 -c 0 a 0 -d2 0 b2 | {2} | 0 0 -c -b 0 0 d2 a2 | 6 o5 : R-module, quotient of R</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_module_lp__Coherent__Sheaf_rp.html" title="get the module defining a coherent sheaf">module(CoherentSheaf)</a> -- get the module defining a coherent sheaf</span></li> <li><span><a href="_tangent__Sheaf.html" title="tangent sheaf of a projective variety">tangentSheaf</a> -- tangent sheaf of a projective variety</span></li> </ul> </div> </div> </body> </html>