<?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>codim(CoherentSheaf) -- codimension of the support of a coherent sheaf on a projective variety</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_codim_lp__Ideal_rp.html">next</a> | <a href="_codim.html">previous</a> | <a href="_codim_lp__Ideal_rp.html">forward</a> | <a href="_codim.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>codim(CoherentSheaf) -- codimension of the support of a coherent sheaf on a projective variety</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>codim F</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_codim.html" title="compute the codimension">codim</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>, a coherent sheaf over a <a href="___Projective__Variety.html" title="the class of all projective varieties">ProjectiveVariety</a><tt> X</tt></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span></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="_codim_lp__Ideal_cm_sp__Generic_sp_eq_gt_sp..._rp.html">Generic => ...</a>, </span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>Computes the codimension of the support of <tt>F</tt> as given by <tt>dim(R) - dim(M)</tt> where <tt>M</tt> is the module representing <tt>F</tt> over the homogeneous coordinate ring <tt>R</tt> of <tt>X</tt>.<table class="examples"><tr><td><pre>i1 : R = ZZ/31991[a,b,c,d];</pre> </td></tr> <tr><td><pre>i2 : I = monomialCurveIdeal(R,{1,3,5}) 2 2 2 3 2 o2 = ideal (c - b*d, b c - a d, b - a c) o2 : Ideal of R</pre> </td></tr> <tr><td><pre>i3 : projplane = Proj(R) o3 = projplane o3 : ProjectiveVariety</pre> </td></tr> <tr><td><pre>i4 : II = sheaf module I o4 = image | c2-bd b2c-a2d b3-a2c | 1 o4 : coherent sheaf on projplane, subsheaf of OO projplane</pre> </td></tr> <tr><td><pre>i5 : can = sheafExt^1(II,OO_projplane^1(-4)) o5 = cokernel | c b a2 | | d c b2 | 2 o5 : coherent sheaf on projplane, quotient of OO projplane</pre> </td></tr> <tr><td><pre>i6 : codim can o6 = 2</pre> </td></tr> </table> </div> </div> <div class="single"><h2>Caveat</h2> <div>The returned value is the usual codimension if <tt>R</tt> is an integral domain or, more generally, equidimensional.</div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_dim_lp__Module_rp.html" title="compute the Krull dimension">dim(Module)</a> -- compute the Krull dimension</span></li> </ul> </div> </div> </body> </html>