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<!DOCTYPE html> <html lang="en"> <head> <title>localCohom(Ideal,Module) -- local cohomology 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="local cohomology of a D-module" href="_local__Cohom_lp__Ideal_cm__Module_rp.html">localCohom(Ideal,Module)</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="_local__Cohom_lp__List_cm__Ideal_rp.html">next</a> | <a href="_local__Cohom_lp__Ideal_rp.html">previous</a> | <a href="_local__Cohom_lp__List_cm__Ideal_rp.html">forward</a> | <a href="_local__Cohom_lp__Ideal_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>localCohom(Ideal,Module) -- local cohomology of a D-module</h1> <ul> <li><span>Function: <a title="local cohomology" href="_local__Cohom.html">localCohom</a></span></li> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">H = localCohom(I,M)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, an ideal of <em>R = k[x<sub>1</sub>,...,x<sub>n</sub>]</em></span></li> <li><span><span class="tt">M</span>, <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a holonomic module over Weyl algebra <em>A<sub>n</sub>(k)</em></span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="specify localization strategy for local cohomology" href="_local__Cohom_lp..._cm__Loc__Strategy_eq_gt..._rp.html">LocStrategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value null</span>, <span>specify localization strategy for local cohomology</span></span></li> <li><span><a title="specify strategy for local cohomology" href="_local__Cohom_lp..._cm__Strategy_eq_gt..._rp.html">Strategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value Walther</span>, <span>specify strategy for local cohomology</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">H</span>, <span>a <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, each entry of <span class="tt">H</span> has an integer key and contains the cohomology module in the corresponding degree.</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : W = QQ[X, dX, Y, dY, Z, dZ, WeylAlgebra=>{X=>dX, Y=>dY, Z=>dZ}] o1 = W o1 : PolynomialRing, 3 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal (X*(Y-Z), X*Y*Z) o2 = ideal (X*Y - X*Z, X*Y*Z) o2 : Ideal of W</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : h = localCohom(I, W^1 / ideal{dX,dY,dZ}) o3 = HashTable{0 => subquotient (| dZ dY dX |, | dX dY dZ |) } 1 => subquotient (| 0 0 -dY-dZ 0 0 0 0 0 dXdY+dXdZ 0 0 -dYdZ-dZ^2 0 0 XdX+2 XdX+4YdZ-4ZdZ-6 0 0 -dX^2dY-dX^2dZ XdXdZ+2dZ 0 3dXdYdZ+3dXdZ^2 0 0 -2XdXY+3XdXZ-4Y+6Z -XdX^2-3dX -XdX^2-6dXYdZ+6dXZdZ+9dX -3dX^2YdZ+3dX^2ZdZ+6dX^2 2dX^2YdZ-2dX^2ZdZ-4dX^2 -2XdXYdZ+3XdXZdZ+2XdX+4YdY-6dYZ+4 2XdXYdZ-3XdXZdZ-6XdX-2YdZ -XdX^2dZ-3dXdZ -dX^2dYdZ-dX^2dZ^2 2XdXYZ-3XdXZ2+8Y2-12YZ+2Z2 2dX^2YdY-3dX^2dYZ+2dX^2YdZ-3dX^2ZdZ 4dX^2Y2-8dX^2YZ+4dX^2Z2 2XdX^2Y-3XdX^2Z+6dXY-9dXZ 4dX^2YdYdZ-6dX^2dYZdZ+4dX^2YdZ^2-6dX^2ZdZ^2-6dX^2dY-6dX^2dZ 4dX^2Y2dZ-6dX^2YZdZ+2dX^2Z2dZ-8dX^2Y+4dX^2Z -2XdX^2YdZ+3XdX^2ZdZ+6XdX^2+3dXYdZ 2XdX^2YdZ-3XdX^2ZdZ-2XdX^2-6dXYdY+9dXdYZ-6dX -2XdX^2YZ+3XdX^2Z2-12dXY2+18dXYZ-3dXZ2 2dX^2Y2dY-3dX^2YdYZ+2dX^2YZdZ-3dX^2Z2dZ+4dX^2Y-6dX^2Z |, | X2Y2-2X2YZ+X2Z2 dY+dZ -YdZ+ZdZ+2 -XdX-2 0 0 0 |) | -XdX-2 -XdX+YdY -Z2dZ-2Z -dYZdZ-2dY XdXdZ-YdYdZ -XdXdY-2dY XdX^2+3dX 2XdX^2-3dXYdY dXZ2dZ+2dXZ dXdYZdZ+2dXdY dX^2ZdZ+2dX^2 -XdXZdZ+YdYZdZ-dYZ2dZ-Z2dZ^2-2dYZ-4ZdZ-2 XdXdYdZ+2dYdZ -2XdX^2dZ+3dXYdYdZ XdXZ2+2Z2 XdXZ2+4YZ2dZ+4Z3dZ+8YZ+10Z2 dX^2YdY+2dX^2 XdX^2dY+3dXdY -dX^2Z2dZ-2dX^2Z XdXZ2dZ+2XdXZ+2Z2dZ+4Z -dX^2dYZdZ-2dX^2dY 2XdX^2ZdZ-3dXYdYZdZ+3dXdYZ2dZ+3dXZ2dZ^2+6dXdYZ+12dXZdZ+6dX -dX^2YdYdZ-2dX^2dZ -XdX^2dYdZ-3dXdYdZ -2XdXYZ2-XdXZ3-4YZ2-2Z3 -XdX^2Z2-3dXZ2 -XdX^2Z2-6dXYZ2dZ-6dXZ3dZ-12dXYZ-15dXZ2 -3dX^2YZ2dZ-dX^2Z3dZ-6dX^2YZ-2dX^2Z2 2dX^2YZ2dZ+2dX^2Z3dZ+4dX^2YZ+4dX^2Z2 -2XdXYZ2dZ-XdXZ3dZ-4XdXYZ-4XdXZ2+4YdYZ2+4Z2 2XdXYZ2dZ+XdXZ3dZ+4XdXYZ+2XdXZ2-2YZ2dZ-4YZ -XdX^2Z2dZ-2XdX^2Z-3dXZ2dZ-6dXZ dX^2YdYZdZ-dX^2dYZ2dZ-dX^2Z2dZ^2-2dX^2dYZ-2dX^2ZdZ-2dX^2 2XdXYZ3+XdXZ4+8Y2Z2+4YZ3+2Z4 2dX^2YdYZ2+2dX^2YZ2dZ+dX^2Z3dZ+4dX^2YZ+6dX^2Z2 4dX^2Y2Z2 2XdX^2YZ2+XdX^2Z3+6dXYZ2+3dXZ3 4dX^2YdYZ2dZ+3dX^2dYZ3dZ+4dX^2YZ2dZ^2+2dX^2Z3dZ^2+8dX^2YdYZ+6dX^2dYZ2+16dX^2YZdZ+18dX^2Z2dZ+8dX^2Y+24dX^2Z 4dX^2Y2Z2dZ+2dX^2YZ3dZ+2dX^2Z4dZ+8dX^2Y2Z+4dX^2YZ2+4dX^2Z3 -2XdX^2YZ2dZ-XdX^2Z3dZ-4XdX^2YZ-2XdX^2Z2+3dXYZ2dZ+6dXYZ 2XdX^2YZ2dZ+XdX^2Z3dZ+4XdX^2YZ+4XdX^2Z2-6dXYdYZ2-6dXZ2 -2XdX^2YZ3-XdX^2Z4-12dXY2Z2-6dXYZ3-3dXZ4 2dX^2Y2dYZ2+dX^2YdYZ3+2dX^2YZ3dZ+dX^2Z4dZ+8dX^2YZ2+4dX^2Z3 | | X2Y2Z2 0 0 0 -ZdZ-2 -YdY-2 -XdX-2 | 2 => cokernel | -X2Y2Z2 X2Y2-2X2YZ+X2Z2 -YdY-ZdZ-6 -XdX-4 YZdZ-Z2dZ+2Y-4Z | o3 : HashTable</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : pruneLocalCohom h o4 = HashTable{0 => 0 } 1 => | dZ dY XdX+3 X3 | 2 => | dYZ+YdZ+2 YdY+ZdZ+6 Y2-2YZ+Z2 XdX+4 YZdZ-Z2dZ+2Y-4Z 2YZ3-Z4 Z4dZ+2YZ2+4Z3 Z5 | o4 : HashTable</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> The modules returned are not simplified, use <a title="prunes local cohomology modules" href="_prune__Local__Cohom_lp__Hash__Table_rp.html">pruneLocalCohom</a>. </div> <div> <h2>See also</h2> <ul> <li><span><a title="prunes local cohomology modules" href="_prune__Local__Cohom_lp__Hash__Table_rp.html">pruneLocalCohom</a> -- prunes local cohomology modules</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use this method:</h2> <ul> <li><span><a title="local cohomology of a D-module" href="_local__Cohom_lp__Ideal_cm__Module_rp.html">localCohom(Ideal,Module)</a> -- local cohomology of a D-module</span></li> </ul> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/localCohom.m2:173:0</span>.</p> </div> </div> </div> </body> </html>
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