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_rational__Function__Ext.html
<!DOCTYPE html> <html lang="en"> <head> <title>rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</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="algorithms for b-functions, local cohomology, and intersection cohomology" href="index.html">BernsteinSato</a> :: <a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</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="_rational__Function__Ext_lp..._cm__Strategy_eq_gt..._rp.html">next</a> | <a href="_rational__Function__Annihilator.html">previous</a> | <a href="_rational__Function__Ext_lp..._cm__Strategy_eq_gt..._rp.html">forward</a> | <a href="_rational__Function__Annihilator.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">rationalFunctionExt M, rationalFunctionExt I; rationalFunctionExt(M,f), rationalFunctionExt(I,f);</code></dd> <dd><code class="language-macaulay2">rationalFunctionExt(i,M), rationalFunctionExt(i,I); rationalFunctionExt(i,M,f), rationalFunctionExt(i,I,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>, over the Weyl algebra <em>D</em></span></li> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which represents the module <em>M = D/I</em></span></li> <li><span><span class="tt">f</span>, <span>a <a title="the class of all ring elements handled by the engine" href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a polynomial</span></li> <li><span><span class="tt">i</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, nonnegative</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a href="_rational__Function__Ext_lp..._cm__Strategy_eq_gt..._rp.html">Strategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value Schreyer</span>, <span></span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span> or <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, the Ext^i group(s) between holonomic <em>M</em> and the polynomial ring localized at the singular locus of <em>M</em> (or at <em>f</em> if specified)</span></li> </ul> </li> </ul> <div> <h2>Description</h2> The Ext groups between M and N are the derived functors of Hom, and are finite-dimensional vector spaces over the ground field when M and N are holonomic. <p>The algorithm used appears in the paper 'Polynomial and rational solutions of holonomic systems' by Oaku-Takayama-Tsai (2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}] o1 = W o1 : PolynomialRing, 1 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : M = W^1/ideal(x*D+5) o2 = cokernel | xD+5 | 1 o2 : W-module, quotient of W</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : rationalFunctionExt M 1 o3 = HashTable{0 => QQ } 1 1 => QQ o3 : HashTable</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> Input modules M or D/I should be holonomic. </div> <div> <h2>See also</h2> <ul> <li><span><a title="resolution of a D-module" href="___Dresolution.html">Dresolution</a> -- resolution of a D-module</span></li> <li><span><a title="integration modules of a D-module" href="___Dintegration.html">Dintegration</a> -- integration modules of a D-module</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">rationalFunctionExt</span>:</h2> <ul> <li><kbd>rationalFunctionExt(Ideal)</kbd></li> <li><kbd>rationalFunctionExt(Ideal,RingElement)</kbd></li> <li><kbd>rationalFunctionExt(Module)</kbd></li> <li><kbd>rationalFunctionExt(Module,RingElement)</kbd></li> <li><kbd>rationalFunctionExt(ZZ,Ideal)</kbd></li> <li><kbd>rationalFunctionExt(ZZ,Ideal,RingElement)</kbd></li> <li><kbd>rationalFunctionExt(ZZ,Module)</kbd></li> <li><kbd>rationalFunctionExt(ZZ,Module,RingElement)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/DHom.m2:224:0</span>.</p> </div> </div> </div> </body> </html>
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