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<!DOCTYPE html> <html lang="en"> <head> <title>Tom -- The Kustin-Miller complex for Tom</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="Unprojection and the Kustin-Miller complex construction" href="index.html">KustinMiller</a> :: <a title="The Kustin-Miller complex for Tom" href="___Tom.html">Tom</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="_unprojection__Homomorphism.html">next</a> | <a href="_substitute_lp__Face_cm__Polynomial__Ring_rp.html">previous</a> | <a href="_unprojection__Homomorphism.html">forward</a> | <a href="_substitute_lp__Face_cm__Polynomial__Ring_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Tom -- The Kustin-Miller complex for Tom</h1> <div> <h2>Description</h2> <div> <p>The Kustin-Miller complex construction for the Tom example which can be found in Section 5.5 of</p> <p>Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geometry 13 (2004) 249-268, <a href="http://arxiv.org/abs/math/0111195">http://arxiv.org/abs/math/0111195</a></p> <p>Here we pass from a Pfaffian to a codimension 4 variety.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x_1..x_4,z_1..z_4] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : b2 = matrix {{0,x_1,x_2,x_3,x_4},{-x_1,0,0,z_1,z_2},{-x_2,0,0,z_3,z_4},{-x_3,-z_1,-z_3,0,0},{-x_4,-z_2,-z_4,0,0}} o2 = | 0 x_1 x_2 x_3 x_4 | | -x_1 0 0 z_1 z_2 | | -x_2 0 0 z_3 z_4 | | -x_3 -z_1 -z_3 0 0 | | -x_4 -z_2 -z_4 0 0 | 5 5 o2 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : betti(cI=resBE b2) 0 1 2 3 o3 = total: 1 5 5 1 0: 1 . . . 1: . 5 5 . 2: . . . 1 o3 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : b1 = cI.dd_1 o4 = | z_2z_3-z_1z_4 -x_4z_3+x_3z_4 x_4z_1-x_3z_2 x_2z_2-x_1z_4 ------------------------------------------------------------------------ -x_2z_1+x_1z_3 | 1 5 o4 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : J = ideal (z_1..z_4); o5 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : betti(cJ=freeResolution J) 0 1 2 3 4 o6 = total: 1 4 6 4 1 0: 1 4 6 4 1 o6 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : betti(cU=kustinMillerComplex(cI,cJ,QQ[T])) 0 1 2 3 4 o7 = total: 1 9 16 9 1 -8: . . . . 1 -7: . . . . . -6: . . . . . -5: . . 5 9 . -4: . . . . . -3: . 5 11 . . -2: . . . . . -1: . 4 . . . 0: 1 . . . . o7 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : S=ring cU o8 = S o8 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : isExactRes cU o9 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : print cU.dd_1 | z_2z_3-z_1z_4 -x_4z_3+x_3z_4 x_4z_1-x_3z_2 x_2z_2-x_1z_4 -x_2z_1+x_1z_3 -x_1x_3+Tz_1 -x_1x_4+Tz_2 -x_2x_3+Tz_3 -x_2x_4+Tz_4 |</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : print cU.dd_2 {-2} | 0 x_1 x_2 x_3 x_4 0 0 0 0 0 0 T 0 0 0 0 | {-2} | -x_1 0 0 z_1 z_2 0 0 -x_1 0 0 x_2 0 T 0 0 0 | {-2} | -x_2 0 0 z_3 z_4 -x_1 0 0 -x_2 0 0 0 0 T 0 0 | {-2} | -x_3 -z_1 -z_3 0 0 0 0 -x_3 -x_3 -x_4 0 -x_3 0 0 T 0 | {-2} | -x_4 -z_2 -z_4 0 0 0 x_3 0 0 0 0 0 0 0 0 T | {0} | 0 0 0 0 0 z_2 z_3 0 z_4 0 0 z_4 0 -x_4 0 x_2 | {0} | 0 0 0 0 0 -z_1 0 z_3 0 z_4 0 0 0 x_3 -x_2 0 | {0} | 0 0 0 0 0 0 -z_1 -z_2 0 0 z_4 -z_2 x_4 0 0 -x_1 | {0} | 0 0 0 0 0 0 0 0 -z_1 -z_2 -z_3 0 -x_3 0 x_1 0 |</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : print cU.dd_3 {-3} | 0 -z_2 0 z_4 -T 0 0 x_3 0 | {-3} | x_3 x_4 0 0 0 -T 0 0 0 | {-3} | 0 0 -x_3 -x_4 0 0 -T 0 0 | {-3} | -x_1 0 x_2 0 0 0 0 -T 0 | {-3} | 0 -x_1 0 x_2 0 0 0 0 -T | {-1} | -z_3 -z_4 0 0 0 x_2 0 0 0 | {-1} | z_2 0 -z_4 0 0 0 0 x_4 0 | {-1} | -z_1 0 0 -z_4 0 0 -x_2 -x_3 0 | {-1} | 0 z_2 z_3 0 0 -x_1 0 -x_3 0 | {-1} | 0 -z_1 0 z_3 0 0 0 0 -x_3 | {-1} | 0 0 -z_1 -z_2 0 0 -x_1 0 0 | {-1} | 0 0 0 0 0 x_1 x_2 x_3 x_4 | {-1} | 0 0 0 0 -x_1 0 0 z_1 z_2 | {-1} | 0 0 0 0 -x_2 0 0 z_3 z_4 | {-1} | 0 0 0 0 -x_3 -z_1 -z_3 0 0 | {-1} | 0 0 0 0 -x_4 -z_2 -z_4 0 0 |</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : print cU.dd_4 {-2} | -x_2x_4+Tz_4 | {-2} | x_2x_3-Tz_3 | {-2} | -x_1x_4+Tz_2 | {-2} | x_1x_3-Tz_1 | {-2} | z_2z_3-z_1z_4 | {-2} | -x_4z_3+x_3z_4 | {-2} | x_4z_1-x_3z_2 | {-2} | x_2z_2-x_1z_4 | {-2} | -x_2z_1+x_1z_3 |</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="Compute Kustin-Miller resolution of the unprojection of I in J" href="_kustin__Miller__Complex.html">kustinMillerComplex</a> -- Compute Kustin-Miller resolution of the unprojection of I in J</span></li> <li><span><a title="compute a free resolution of a module or ideal" href="../../Complexes/html/_free__Resolution.html">freeResolution</a> -- compute a free resolution of a module or ideal</span></li> <li><span><a title="display or modify a Betti diagram" href="../../Macaulay2Doc/html/_betti.html">betti</a> -- display or modify a Betti diagram</span></li> <li><span><a title="The Kustin-Miller complex for Jerry" href="___Jerry.html">Jerry</a> -- The Kustin-Miller complex for Jerry</span></li> </ul> </div> <div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="The Kustin-Miller complex for Tom" href="___Tom.html">Tom</a> is <span>a <a title="the class of all symbols" href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">KustinMiller.m2:1790:0</span>.</p> </div> </div> </div> </body> </html>