<?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>eagonNorthcott(Matrix) -- Eagon-Northcott complex of a matrix of linear forms</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_echo__Off.html">next</a> | <a href="_dual_lp__Monomial__Ideal_cm__Ring__Element_rp.html">previous</a> | <a href="_echo__Off.html">forward</a> | <a href="_dual_lp__Monomial__Ideal_cm__Ring__Element_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>eagonNorthcott(Matrix) -- Eagon-Northcott complex of a matrix of linear forms</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>eagonNorthcott f</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_eagon__Northcott_lp__Matrix_rp.html" title="Eagon-Northcott complex of a matrix of linear forms">eagonNorthcott</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a matrix of linear forms</span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><tt>C</tt>, <span>a <a href="___Chain__Complex.html">chain complex</a></span>, the Eagon-Northcott complex of <tt>f</tt></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>The Eagon-Northcott complex is an explicit chain complex that gives a minimal projective resolution of the cokernel of the matrix maximal minors of a generic matrix of linear forms.<table class="examples"><tr><td><pre>i1 : R = QQ[a..z] o1 = R o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : f = genericMatrix(R,3,5) o2 = | a d g j m | | b e h k n | | c f i l o | 3 5 o2 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i3 : M = coker gens minors_3 f o3 = cokernel | -ceg+bfg+cdh-afh-bdi+aei -cej+bfj+cdk-afk-bdl+ael -chj+bij+cgk-aik-bgl+ahl -fhj+eij+fgk-dik-egl+dhl -cem+bfm+cdn-afn-bdo+aeo -chm+bim+cgn-ain-bgo+aho -fhm+eim+fgn-din-ego+dho -ckm+blm+cjn-aln-bjo+ako -fkm+elm+fjn-dln-ejo+dko -ikm+hlm+ijn-gln-hjo+gko | 1 o3 : R-module, quotient of R</pre> </td></tr> <tr><td><pre>i4 : C = res M 1 10 15 6 o4 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o4 : ChainComplex</pre> </td></tr> <tr><td><pre>i5 : D = eagonNorthcott f 1 10 15 6 o5 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o5 : ChainComplex</pre> </td></tr> <tr><td><pre>i6 : H = prune HH D o6 = 0 : cokernel | ikm-hlm-ijn+gln+hjo-gko fkm-elm-fjn+dln+ejo-dko ckm-blm-cjn+aln+bjo-ako fhm-eim-fgn+din+ego-dho chm-bim-cgn+ain+bgo-aho cem-bfm-cdn+afn+bdo-aeo fhj-eij-fgk+dik+egl-dhl chj-bij-cgk+aik+bgl-ahl cej-bfj-cdk+afk+bdl-ael ceg-bfg-cdh+afh+bdi-aei | 1 : 0 2 : 0 3 : 0 4 : 0 o6 : GradedModule</pre> </td></tr> <tr><td><pre>i7 : assert( H_0 == M and H_1 == 0 and H_2 == 0 and H_3 == 0 )</pre> </td></tr> </table> This function was written by Greg Smith.</div> </div> </div> </body> </html>