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<!DOCTYPE html> <html lang="en"> <head> <title>bruns -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module</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="Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module" href="_bruns.html">bruns</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="_bruns__Ideal.html">next</a> | <a href="index.html">previous</a> | <a href="_bruns__Ideal.html">forward</a> | <a href="index.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>bruns -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">j= bruns M or j= bruns f</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">M</span>, <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a second syzygy (graded) module</span></li> <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 cokernel is a second syzygy (graded) module</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">j</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a homogeneous ideal generated by three elements whose second syzygy module is isomorphic to M, or image f</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This function takes a graded module M over a polynomial ring S that is a second syzygy, and returns a three-generator ideal j whose second syzygy is M, so that the resolution of S/j, from the third step, is isomorphic to the resolution of M. Alternately <span class="tt">bruns</span> takes a matrix whose cokernel is a second syzygy, and finds a 3-generator ideal whose second syzygy is the image of that matrix.</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 : i=ideal(a^2,b^2,c^2, d^2) 2 2 2 2 o3 = ideal (a , b , c , d ) o3 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : betti (F=res i) 0 1 2 3 4 o4 = total: 1 4 6 4 1 0: 1 . . . . 1: . 4 . . . 2: . . 6 . . 3: . . . 4 . 4: . . . . 1 o4 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : M = image F.dd_3 o5 = image {4} | c2 d2 0 0 | {4} | -b2 0 d2 0 | {4} | a2 0 0 d2 | {4} | 0 -b2 -c2 0 | {4} | 0 a2 0 -c2 | {4} | 0 0 a2 b2 | 6 o5 : S-module, submodule of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : f=F.dd_3 o6 = {4} | c2 d2 0 0 | {4} | -b2 0 d2 0 | {4} | a2 0 0 d2 | {4} | 0 -b2 -c2 0 | {4} | 0 a2 0 -c2 | {4} | 0 0 a2 b2 | 6 4 o6 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : j=bruns M; o7 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : betti res j -- the ideal has 3 generators 0 1 2 3 4 o8 = total: 1 3 5 4 1 0: 1 . . . . 1: . . . . . 2: . . . . . 3: . 3 . . . 4: . . . . . 5: . . . . . 6: . . 5 . . 7: . . . 4 . 8: . . . . 1 o8 : BettiTally</code></pre> </td> </tr> </table> <div> <p>Here is a more complicated example, also involving a complete intersection. You can see that columns three and four in the two Betti diagrams are the same.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i9 : kk=ZZ/32003 o9 = kk o9 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : S=kk[a..d] o10 = S o10 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : i=ideal(a^2,b^3,c^4, d^5) 2 3 4 5 o11 = ideal (a , b , c , d ) o11 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : betti (F=res i) 0 1 2 3 4 o12 = total: 1 4 6 4 1 0: 1 . . . . 1: . 1 . . . 2: . 1 . . . 3: . 1 1 . . 4: . 1 1 . . 5: . . 2 . . 6: . . 1 1 . 7: . . 1 1 . 8: . . . 1 . 9: . . . 1 . 10: . . . . 1 o12 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : M = image F.dd_3 o13 = 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 o13 : S-module, submodule of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i14 : f=F.dd_3 o14 = {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 o14 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i15 : j1=bruns f; o15 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i16 : betti res j1 0 1 2 3 4 o16 = total: 1 3 5 4 1 0: 1 . . . . 1: . . . . . 2: . . . . . 3: . . . . . 4: . . . . . 5: . . . . . 6: . . . . . 7: . . . . . 8: . 1 . . . 9: . 2 . . . 10: . . . . . 11: . . . . . 12: . . . . . 13: . . . . . 14: . . . . . 15: . . 1 . . 16: . . 1 . . 17: . . 2 . . 18: . . 1 1 . 19: . . . 1 . 20: . . . 1 . 21: . . . 1 . 22: . . . . 1 o16 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i17 : j=bruns M; o17 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i18 : betti res j 0 1 2 3 4 o18 = total: 1 3 5 4 1 0: 1 . . . . 1: . . . . . 2: . . . . . 3: . . . . . 4: . . . . . 5: . . . . . 6: . . . . . 7: . . . . . 8: . 1 . . . 9: . 2 . . . 10: . . . . . 11: . . . . . 12: . . . . . 13: . . . . . 14: . . . . . 15: . . 1 . . 16: . . 1 . . 17: . . 2 . . 18: . . 1 1 . 19: . . . 1 . 20: . . . 1 . 21: . . . 1 . 22: . . . . 1 o18 : BettiTally</code></pre> </td> </tr> </table> <div> <p>In the next example, we perform the "Brunsification" of a rational curve.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i19 : kk=ZZ/32003 o19 = kk o19 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i20 : S=kk[a..e] o20 = S o20 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i21 : i=monomialCurveIdeal(S, {1,3,4,5}) 2 2 2 3 2 o21 = ideal (d - c*e, b*d - a*e, c - b*e, b*c - a*d, a*c*d - b e, b - a c) o21 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i22 : betti (F=res i) 0 1 2 3 4 o22 = total: 1 5 8 5 1 0: 1 . . . . 1: . 4 2 . . 2: . 1 6 5 1 o22 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i23 : time j=bruns F.dd_3; -- used 0.187433s (cpu); 0.187593s (thread); 0s (gc) o23 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i24 : betti res j 0 1 2 3 4 o24 = total: 1 3 6 5 1 0: 1 . . . . 1: . . . . . 2: . . . . . 3: . . . . . 4: . 3 . . . 5: . . . . . 6: . . . . . 7: . . 2 . . 8: . . 4 5 1 o24 : BettiTally</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal" href="_bruns__Ideal.html">brunsIdeal</a> -- Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">bruns</span>:</h2> <ul> <li><kbd>bruns(Matrix)</kbd></li> <li><kbd>bruns(Module)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module" href="_bruns.html">bruns</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.m2:288:0</span>.</p> </div> </div> </div> </body> </html>