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<!DOCTYPE html> <html lang="en"> <head> <title>BGG -- Bernstein-Gel'fand-Gel'fand correspondence</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="Bernstein-Gel'fand-Gel'fand correspondence" href="index.html">BGG</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="_beilinson.html">next</a> | previous | <a href="_beilinson.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>BGG -- Bernstein-Gel'fand-Gel'fand correspondence</h1> <div> <h2>Description</h2> The Bernstein-Gel'fand-Gel'fand correspondence is an isomorphism between the derived category of bounded complexes of finitely generated modules over a polynomial ring and the derived category of bounded complexes of finitely generated module over an exterior algebra (or of certain Tate resolutions). This package implements routines for investigating the BGG correspondence. <p></p> More details can be found in <a href="https://macaulay2.com/Book/">Sheaf Algorithms Using Exterior Algebra</a>. </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="http://www.webpages.uidaho.edu/~abo/">Hirotachi Abo</a><span> <<a href="mailto:abo%40uidaho.edu">abo@uidaho.edu</a>></span></li> <li><a href="http://www.math.uni-sb.de/ag/decker/">Wolfram Decker</a><span> <<a href="mailto:decker%40math.uni-sb.de">decker@math.uni-sb.de</a>></span></li> <li><a href="http://www.msri.org/~de/">David Eisenbud</a><span> <<a href="mailto:de%40msri.org">de@msri.org</a>></span></li> <li><a href="http://www.math.uni-sb.de/ag/schreyer/">Frank-Olaf Schreyer</a><span> <<a href="mailto:schreyer%40math.uni-sb.de">schreyer@math.uni-sb.de</a>></span></li> <li><a href="http://www.mast.queensu.ca/~ggsmith/">Gregory G. Smith</a><span> <<a href="mailto:ggsmith%40mast.queensu.ca">ggsmith@mast.queensu.ca</a>></span></li> <li><a href="http://www.math.cornell.edu/~mike/">Mike Stillman</a><span> <<a href="mailto:mike%40math.cornell.edu">mike@math.cornell.edu</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.4.2</b> of BGG, released <b>Jan 11, 2016</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{BGGSource, title = {{BGG: Bernstein-Gelfand-Gelfand correspondence. Version~1.4.2}}, author = {Hirotachi Abo and Wolfram Decker and David Eisenbud and Frank-Olaf Schreyer and Gregory G. Smith and Mike Stillman}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><a title="Vector bundle map associated to the Beilinson monad" href="_beilinson.html">beilinson</a> -- Vector bundle map associated to the Beilinson monad</span></li> <li><span><a title="the ith differential of the complex R(M)" href="_bgg.html">bgg</a> -- the ith differential of the complex R(M)</span></li> <li><span><a title="dimensions of cohomology groups" href="_cohomology__Table.html">cohomologyTable</a> -- dimensions of cohomology groups</span></li> <li><span><a title="direct image complex" href="_direct__Image__Complex.html">directImageComplex</a> -- direct image complex</span></li> <li><span><a title="Makes a product of projective spaces and a system of parameters" href="_projective__Product.html">projectiveProduct</a> -- Makes a product of projective spaces and a system of parameters</span></li> <li><span><a title="creates a pure resolution as an iterated direct image" href="_pure__Resolution.html">pureResolution</a> -- creates a pure resolution as an iterated direct image</span></li> <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> <li><span><a title="finite piece of the Tate resolution" href="_tate__Resolution.html">tateResolution</a> -- finite piece of the Tate resolution</span></li> <li><span><a title="Universal extension of vector bundles on P^1" href="_universal__Extension.html">universalExtension</a> -- Universal extension of vector bundles on P^1</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>beilinson(Matrix,PolynomialRing)</kbd> -- see <span><a title="Vector bundle map associated to the Beilinson monad" href="_beilinson.html">beilinson</a> -- Vector bundle map associated to the Beilinson monad</span></span></li> <li><span><kbd>bgg(ZZ,Module,PolynomialRing)</kbd> -- see <span><a title="the ith differential of the complex R(M)" href="_bgg.html">bgg</a> -- the ith differential of the complex R(M)</span></span></li> <li><span><kbd>cohomologyTable(Matrix,PolynomialRing,ZZ,ZZ)</kbd> -- see <span><a title="dimensions of cohomology groups" href="_cohomology__Table.html">cohomologyTable</a> -- dimensions of cohomology groups</span></span></li> <li><span><a title="direct image of a chain complex" href="_direct__Image__Complex_lp__Complex_rp.html">directImageComplex(Complex)</a> -- direct image of a chain complex</span></li> <li><span><a title="map of direct image complexes" href="_direct__Image__Complex_lp__Matrix_rp.html">directImageComplex(Matrix)</a> -- map of direct image complexes</span></li> <li><span><a title="Complex representing the direct image" href="_direct__Image__Complex_lp__Module_rp.html">directImageComplex(Module)</a> -- Complex representing the direct image</span></li> <li><span><kbd>projectiveProduct(Matrix,List)</kbd> -- see <span><a title="Makes a product of projective spaces and a system of parameters" href="_projective__Product.html">projectiveProduct</a> -- Makes a product of projective spaces and a system of parameters</span></span></li> <li><span><kbd>projectiveProduct(Ring,List)</kbd> -- see <span><a title="Makes a product of projective spaces and a system of parameters" href="_projective__Product.html">projectiveProduct</a> -- Makes a product of projective spaces and a system of parameters</span></span></li> <li><span><kbd>pureResolution(Matrix,List)</kbd> -- see <span><a title="creates a pure resolution as an iterated direct image" href="_pure__Resolution.html">pureResolution</a> -- creates a pure resolution as an iterated direct image</span></span></li> <li><span><kbd>pureResolution(Ring,List)</kbd> -- see <span><a title="creates a pure resolution as an iterated direct image" href="_pure__Resolution.html">pureResolution</a> -- creates a pure resolution as an iterated direct image</span></span></li> <li><span><kbd>pureResolution(ZZ,List)</kbd> -- see <span><a title="creates a pure resolution as an iterated direct image" href="_pure__Resolution.html">pureResolution</a> -- creates a pure resolution as an iterated direct image</span></span></li> <li><span><kbd>pureResolution(ZZ,ZZ,List)</kbd> -- see <span><a title="creates a pure resolution as an iterated direct image" href="_pure__Resolution.html">pureResolution</a> -- creates a pure resolution as an iterated direct image</span></span></li> <li><span><kbd>symExt(Matrix,PolynomialRing)</kbd> -- see <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></span></li> <li><span><kbd>tateResolution(Matrix,PolynomialRing,ZZ,ZZ)</kbd> -- see <span><a title="finite piece of the Tate resolution" href="_tate__Resolution.html">tateResolution</a> -- finite piece of the Tate resolution</span></span></li> <li><span><kbd>universalExtension(List,List)</kbd> -- see <span><a title="Universal extension of vector bundles on P^1" href="_universal__Extension.html">universalExtension</a> -- Universal extension of vector bundles on P^1</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><a title="dual exterior algebra cached in a polynomial ring" href="___Exterior.html">Exterior</a> -- dual exterior algebra cached in a polynomial ring</span></li> <li><span><a title="Option for directImageComplex" href="___Regularity.html">Regularity</a> -- Option for directImageComplex</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Bernstein-Gel'fand-Gel'fand correspondence" href="index.html">BGG</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">BGG.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BGG.m2:669:0</span>.</p> </div> </div> </div> </body> </html>