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<!DOCTYPE html> <html lang="en"> <head> <title>beilinson -- Vector bundle map associated to the Beilinson monad</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="Bernstein-Gel'fand-Gel'fand correspondence" href="index.html">BGG</a> :: <a title="Vector bundle map associated to the Beilinson monad" href="_beilinson.html">beilinson</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="_bgg.html">next</a> | <a href="index.html">previous</a> | <a href="_bgg.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>beilinson -- Vector bundle map associated to the Beilinson monad</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">beilinson(m,S)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">m</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a presentation matrix for a module over an exterior algebra <span class="tt">E</span></span></li> <li><span><span class="tt">S</span>, <span>a <a title="the class of all ordered monoid rings" href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, polynomial ring with the same number of variables as <span class="tt">E</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, vector bundle map</span></li> </ul> </li> </ul> <div> <h2>Description</h2> The BGG correspondence is an equivalence between complexes of modules over exterior algebras and complexes of coherent sheaves over projective spaces. This function takes as input a map between two free <span class="tt">E</span>-modules, and returns the associate map between direct sums of exterior powers of cotangent bundles. In particular, it is useful to construct the Belinson monad for a coherent sheaf. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = ZZ/32003[x_0..x_2]; </code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : alphad = map(E^1,E^{2:-1},{{e_1,e_2}}); 1 2 o3 : Matrix E <-- E</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : alpha = map(E^{2:-1},E^{1:-2},{{e_1},{e_2}}); 2 1 o4 : Matrix E <-- E</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : alphad' = beilinson(alphad,S) o5 = | x_0 0 -x_2 0 x_0 x_1 | o5 : Matrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : alpha' = beilinson(alpha,S) o6 = {1} | 0 | {1} | 1 | {1} | 0 | {1} | -1 | {1} | 0 | {1} | 0 | o6 : Matrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : F = prune homology(alphad',alpha') o7 = cokernel {1} | x_1^2-x_2^2 | {1} | x_1x_2 | {2} | -x_0 | 3 o7 : S-module, quotient of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : betti F 0 1 o8 = total: 3 1 1: 2 . 2: 1 1 o8 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : cohomologyTable(presentation F,E,-2,3) -2 -1 0 1 2 3 4 o9 = 2: 7 2 . . . . . 1: . 1 2 1 . . . 0: . . . 2 7 14 23 o9 : CohomologyTally</code></pre> </td> </tr> </table> As the next example, we construct the monad of the Horrock-Mumford bundle: <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i10 : S = ZZ/32003[x_0..x_4]; </code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : E = ZZ/32003[e_0..e_4, SkewCommutative=>true];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : alphad = map(E^5,E^{2:-2},{{e_4*e_1,e_2*e_3},{e_0*e_2,e_3*e_4},{e_1*e_3,e_4*e_0},{e_2*e_4,e_0*e_1},{e_3*e_0,e_1*e_2}}) o12 = | -e_1e_4 e_2e_3 | | e_0e_2 e_3e_4 | | e_1e_3 -e_0e_4 | | e_2e_4 e_0e_1 | | -e_0e_3 e_1e_2 | 5 2 o12 : Matrix E <-- E</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : alpha = syz alphad o13 = {2} | e_0e_1 e_2e_3 e_0e_4 e_1e_2 -e_3e_4 | {2} | -e_2e_4 e_1e_4 e_1e_3 e_0e_3 e_0e_2 | 2 5 o13 : Matrix E <-- E</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i14 : alphad' = beilinson(alphad,S) o14 = | 0 0 0 0 x_0 0 -x_2 0 -x_3 0 0 0 -x_0 -x_1 0 | x_1 0 -x_3 0 0 -x_4 0 0 0 0 0 0 0 0 0 | 0 -x_0 0 x_2 0 0 0 0 -x_4 0 0 0 0 0 -x_1 | 0 0 0 0 0 -x_0 -x_1 0 0 x_3 -x_2 -x_3 0 0 -x_4 | 0 -x_1 -x_2 0 0 0 0 x_4 0 0 -x_0 0 0 -x_3 0 ----------------------------------------------------------------------- 0 0 0 0 -x_4 | 0 0 -x_0 -x_1 -x_2 | -x_2 0 -x_3 0 0 | 0 0 0 0 0 | 0 -x_4 0 0 0 | o14 : Matrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i15 : alpha' = beilinson(alpha,S) o15 = {1} | 0 0 0 0 -1 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 -1 0 0 | {1} | 0 1 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 1 0 | {1} | 0 0 0 0 0 | {1} | 1 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 1 0 0 0 0 | {1} | 0 1 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 0 | {1} | 0 0 -1 0 0 | {1} | 0 0 0 1 0 | {1} | 0 0 0 0 0 | {1} | 0 0 0 0 -1 | {1} | 0 0 0 0 0 | o15 : Matrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i16 : F = prune homology(alphad',alpha');</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i17 : betti freeResolution F 0 1 2 3 o17 = total: 19 35 20 2 3: 4 . . . 4: 15 35 20 . 5: . . . 2 o17 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i18 : regularity F o18 = 5</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i19 : cohomologyTable(presentation F,E,-6,6) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 o19 = 4: 210 100 35 4 . . . . . . . . . . 3: . . 2 10 10 5 . . . . . . . . 2: . . . . . . 2 . . . . . . . 1: . . . . . . . 5 10 10 2 . . . 0: . . . . . . . . . 4 35 100 210 380 o19 : CohomologyTally</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="the first differential of the complex R(M)" href="_sym__Ext.html">symExt</a> -- the first differential of the complex R(M)</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">beilinson</span>:</h2> <ul> <li><kbd>beilinson(Matrix,PolynomialRing)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Vector bundle map associated to the Beilinson monad" href="_beilinson.html">beilinson</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">BGG.m2:856:0</span>.</p> </div> </div> </div> </body> </html>