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<!DOCTYPE html> <html lang="en"> <head> <title>Dresolution -- resolution of a D-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="algorithms for b-functions, local cohomology, and intersection cohomology" href="index.html">BernsteinSato</a> :: <a title="resolution of a D-module" href="___Dresolution.html">Dresolution</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="___Dresolution_lp..._cm__Length__Limit_eq_gt..._rp.html">next</a> | <a href="___Dlocalize__Map.html">previous</a> | <a href="___Dresolution_lp..._cm__Length__Limit_eq_gt..._rp.html">forward</a> | <a href="___Dlocalize__Map.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Dresolution -- resolution of a D-module</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">Dresolution M, Dresolution I, Dresolution(M,w), Dresolution(I,w)</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">w</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, a weight vector</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="the limit for the length of a resolution of a D-module" href="___Dresolution_lp..._cm__Length__Limit_eq_gt..._rp.html">LengthLimit</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value infinity</span>, <span>the limit for the length of a resolution of a D-module</span></span></li> <li><span><a title="strategy for computing a resolution of a D-module" href="___Dresolution_lp..._cm__Strategy_eq_gt..._rp.html">Strategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value Schreyer</span>, <span>strategy for computing a resolution of a D-module</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all chain complexes" href="../../OldChainComplexes/html/___Chain__Complex.html">chain complex</a></span>, a Schreyer resolution of the D-module <em>M</em> or a resolution adapted to a weight vector <em>w</em> of the form <em>(-u,u)</em></span></li> </ul> </li> </ul> <div> <h2>Description</h2> This routine computes various resolutions of a D-module. If no weight vector is specified, then the command produces a resolution by using the Schreyer order implemented in the engine. If a weight vector <em>w</em> of the form <em>(-u,u)</em> is specified, then the command produces a resolution with shifts which is adapted to the weight vector <em>w</em>. These <em>w</em>-adapted resolutions are compatible with b-functions and used in the restriction algorithm. For ordinary resolutions, the user may use the command <span class="tt">resolution</span>. Note that the notion of a minimal resolution is well-defined only in case of homogenized Weyl algebra. <p></p> There are two strategies for constructing w-adapted resolutions. The first strategy is to construct a Schreyer resolution in the homogenized Weyl algebra and then dehomogenize. The second strategy is to homogenize with respect to the weight vector. These strategies are described in the paper 'Algorithms for D-modules' by Oaku-Takayama(1999). <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x_1,x_2,D_1,D_2,WeylAlgebra=>{x_1=>D_1,x_2=>D_2}] o1 = R o1 : PolynomialRing, 2 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal(x_1*D_1+3*x_2*D_2-1, D_1^3-D_2) 3 o2 = ideal (x D + 3x D - 1, D - D ) 1 1 2 2 1 2 o2 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : Dresolution(I,{-1,-1,1,1}) 1 5 6 2 o3 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o3 : ChainComplex</code></pre> </td> </tr> </table> Abbreviations : <ul> <li>Dres</li> </ul> </div> <div> <h2>See also</h2> <ul> <li><span><a title="compute a Gröbner basis with respect to a weight vector" href="../../WeylAlgebras/html/_gbw.html">gbw</a> -- compute a Gröbner basis with respect to a weight vector</span></li> <li><span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">Dresolution</span>:</h2> <ul> <li><kbd>Dresolution(Ideal)</kbd></li> <li><kbd>Dresolution(Ideal,List)</kbd></li> <li><kbd>Dresolution(Module)</kbd></li> <li><kbd>Dresolution(Module,List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="resolution of a D-module" href="___Dresolution.html">Dresolution</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/Drestriction.m2:71:0</span>.</p> </div> </div> </div> </body> </html>