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<!DOCTYPE html> <html lang="en"> <head> <title>elementary -- Elementary moves are used to reduce the target of a syzygy matrix</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="produces an ideal with three generators whose 2nd syzygy module is isomorphic to a given module" href="index.html">Bruns</a> :: <a title="Elementary moves are used to reduce the target of a syzygy matrix" href="_elementary.html">elementary</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_evans__Griffith.html">next</a> | <a href="_bruns__Ideal.html">previous</a> | <a href="_evans__Griffith.html">forward</a> | <a href="_bruns__Ideal.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>elementary -- Elementary moves are used to reduce the target of a syzygy matrix</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">g= elementary(f,k,m)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">f</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose target degrees are in ascending order</span></li> <li><span><span class="tt">k</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, whose value is strictly less than the number of rows of f</span></li> <li><span><span class="tt">m</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, positive</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">g</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, obtained from f by adding random multiples of the last row by polynomials in the first m variables to the k preceding rows, and then deleting the last row.</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Factors out a general element, reducing the rank of f. More precisely, the routine adds random multiples of the last row, whose coefficients are polynomials in the first m variables, to the k preceding rows and drops the last row. For this to be effective, the target degrees of f must be in ascending order.</p> <p>This is a fundamental operation in the theory of basic elements, see D. Eisenbud and E. G. Evans, <em>Basic elements: theorems from algebraic k-theory</em>, Bulletin of the AMS, <b>78</b>, No.4, 1972, 546-549.</p> <p>Here is a basic example:</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : kk=ZZ/32003 o1 = kk o1 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S=kk[a..d] o2 = S o2 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : M=matrix{{a,0,0,0},{0,b,0,0},{0,0,c,0},{0,0,0,d}} o3 = | a 0 0 0 | | 0 b 0 0 | | 0 0 c 0 | | 0 0 0 d | 4 4 o3 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : elementary(M,0,1)-- since k=0, this command simply eliminates the last row of M. o4 = | a 0 0 0 | | 0 b 0 0 | | 0 0 c 0 | 3 4 o4 : Matrix S <-- S</code></pre> </td> </tr> </table> <div> <p>Here is a more involved example. This is also how this function is used within the package.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : kk=ZZ/32003 o5 = kk o5 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : S=kk[a..d] o6 = S o6 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : I=ideal(a^2,b^3,c^4, d^5) 2 3 4 5 o7 = ideal (a , b , c , d ) o7 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : F=res I 1 4 6 4 1 o8 = S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 o8 : ChainComplex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : M=image F.dd_3 o9 = image {5} | c4 d5 0 0 | {6} | -b3 0 d5 0 | {7} | a2 0 0 d5 | {7} | 0 -b3 -c4 0 | {8} | 0 a2 0 -c4 | {9} | 0 0 a2 b3 | 6 o9 : S-module, submodule of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : f=matrix gens M o10 = {5} | c4 d5 0 0 | {6} | -b3 0 d5 0 | {7} | a2 0 0 d5 | {7} | 0 -b3 -c4 0 | {8} | 0 a2 0 -c4 | {9} | 0 0 a2 b3 | 6 4 o10 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : fascending=transpose sort(transpose f, DegreeOrder=>Descending) -- this is f with rows sorted so that the degrees are ascending. o11 = {5} | c4 d5 0 0 | {6} | -b3 0 d5 0 | {7} | 0 -b3 -c4 0 | {7} | a2 0 0 d5 | {8} | 0 a2 0 -c4 | {9} | 0 0 a2 b3 | 6 4 o11 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : g=elementary(fascending,1,1) --k=1, so add random multiples of the last row to the preceding row o12 = {5} | c4 d5 0 0 | {6} | -b3 0 d5 0 | {7} | 0 -b3 -c4 0 | {7} | a2 0 0 d5 | {8} | 0 a2 107a3 107ab3-c4 | 5 4 o12 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : g1=elementary(fascending,1,3) o13 = {5} | c4 d5 0 0 | {6} | -b3 0 d5 0 | {7} | 0 -b3 -c4 0 | {7} | a2 0 0 d5 | {8} | 0 a2 4376a3-5570a2b+3187a2c 4376ab3-5570b4+3187b3c-c4 | 5 4 o13 : Matrix S <-- S</code></pre> </td> </tr> </table> <div> <p>This method is called by <a title="Reduces the rank of a syzygy" href="_evans__Griffith.html">evansGriffith</a>.</p> </div> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">elementary</span>:</h2> <ul> <li><kbd>elementary(Matrix,ZZ,ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Elementary moves are used to reduce the target of a syzygy matrix" href="_elementary.html">elementary</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Bruns.m2:383:0</span>.</p> </div> </div> </div> </body> </html>