<?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>dim(Face,PolynomialRing) -- Compute the dimension of a face.</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_dim_lp__First__Order__Deformation_rp.html">next</a> | <a href="_dim_lp__Face_cm__Complex_rp.html">previous</a> | <a href="_dim_lp__First__Order__Deformation_rp.html">forward</a> | <a href="_dim_lp__Face_cm__Complex_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>dim(Face,PolynomialRing) -- Compute the dimension of a face.</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>dim(F,R)</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="../../Macaulay2Doc/html/_dim.html" title="compute the Krull dimension">dim</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="___Face.html">face</a></span></span></li> <li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, with coker grading.</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>, bigger or equal to -1</span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><p>Computes the dimension of a face. If F.indices is present (usually the case by construction) this requires no computations.</p> <p>If F.indices is not present but a polynomial ring R can be associated to F (which is the case if F.ofComplex is present (or given as a second argument) or F is non-empty) then R.grading (which can be installed by <a href="_add__Coker__Grading.html" title="Stores a cokernel grading in a polynomial ring.">addCokerGrading</a>) is used to compute the dimension of the plane spanned by F.</p> <div/> <table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4] o1 = R o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : addCokerGrading R o2 = | -1 -1 -1 -1 | | 1 0 0 0 | | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | 5 4 o2 : Matrix ZZ <--- ZZ</pre> </td></tr> <tr><td><pre>i3 : C=simplex R o3 = 4: x x x x x 0 1 2 3 4 o3 : complex of dim 4 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0</pre> </td></tr> <tr><td><pre>i4 : bC=boundaryOfPolytope C o4 = 3: x x x x x x x x x x x x x x x x x x x x 0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 1 2 3 4 o4 : complex of dim 3 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1</pre> </td></tr> <tr><td><pre>i5 : F=bC.fc_2_0 o5 = x x x 0 1 2 o5 : face with 3 vertices</pre> </td></tr> <tr><td><pre>i6 : dim F o6 = 2</pre> </td></tr> </table> </div> </div> <div class="single"><h2>Caveat</h2> <div><div>If F.indices is not present this returns a dimension as explained above but note that this does not check whether F is a face of the convex hull of the rows of R.grading.</div> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="___Complex.html" title="The class of all embedded complexes.">Complex</a> -- The class of all embedded complexes.</span></li> <li><span><a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a> -- The class of all embedded co-complexes.</span></li> </ul> </div> </div> </body> </html>