<?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>randomSocleModule(List,ZZ) -- random finite length module with prescribed number of socle elements in single degree</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_support__Functional.html">next</a> | <a href="_random__Module_lp__List_cm__Z__Z_rp.html">previous</a> | <a href="_support__Functional.html">forward</a> | <a href="_random__Module_lp__List_cm__Z__Z_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>randomSocleModule(List,ZZ) -- random finite length module with prescribed number of socle elements in single degree</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>randomSocleModule(L,m)</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_random__Socle__Module_lp__List_cm__Z__Z_rp.html" title="random finite length module with prescribed number of socle elements in single degree">randomSocleModule</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a strictly increasing degree sequence of integers</span></li> <li><span><tt>m</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, randomly generated having m b_c socle generators in degree L_c and m b_0 generators, where b = pureBetti L, and c = length L</span></li> </ul> </div> </li> <li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>CoefficientRing => </tt><span><span>default value ZZ/101</span>, The base field for the resulting module</span></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>There are many cases where these produce pure resolutions of the minimal size.<table class="examples"><tr><td><pre>i1 : L={0,2,3,7} o1 = {0, 2, 3, 7} o1 : List</pre> </td></tr> <tr><td><pre>i2 : B = pureBetti L o2 = {10, 42, 35, 3} o2 : List</pre> </td></tr> <tr><td><pre>i3 : betti res randomSocleModule(L,1) 0 1 2 3 o3 = total: 10 42 35 3 0: 10 . . . 1: . 42 35 . 2: . . . . 3: . . . . 4: . . . 3 o3 : BettiTally</pre> </td></tr> <tr><td><pre>i4 : betti res randomModule(L,1) 0 1 2 3 o4 = total: 10 42 50 18 0: 10 . . . 1: . 42 26 . 2: . . 24 18 o4 : BettiTally</pre> </td></tr> </table> <p/> The method used is roughly the following: Given a strictly increasing degree sequence L and a number of generators m, this routine produces a generic module of finite length with the m generators and number of socle elements and regularity corresponding to the pure resolution with degree sequence L. The module is constructed by taking a certain number of generic elements inside an appropriate direct sum of copies of a zero-dimensional complete intersection. We use the fact that in a polynomial ring in c variables, modulo the r+1 st power of each variable, the part of generated in degree (c-1)r looks like the part of the injective hull of the residue class field generated in degree -r.</div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_random__Module_lp__List_cm__Z__Z_rp.html" title="module with random relations in prescribed degrees">randomModule</a> -- module with random relations in prescribed degrees</span></li> <li><span><a href="_pure__Betti_lp__List_rp.html" title="list of smallest integral Betti numbers corresponding to a degree sequence">pureBetti</a> -- list of smallest integral Betti numbers corresponding to a degree sequence</span></li> </ul> </div> </div> </body> </html>