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<!DOCTYPE html> <html lang="en"> <head> <title>localCohom(ZZ,Ideal) -- local cohomology of a polynomial ring</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 polynomial ring" href="_local__Cohom_lp__Z__Z_cm__Ideal_rp.html">localCohom(ZZ,Ideal)</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__Z__Z_cm__Ideal_cm__Module_rp.html">next</a> | <a href="_local__Cohom_lp__List_cm__Ideal_cm__Module_rp.html">previous</a> | <a href="_local__Cohom_lp__Z__Z_cm__Ideal_cm__Module_rp.html">forward</a> | <a href="_local__Cohom_lp__List_cm__Ideal_cm__Module_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>localCohom(ZZ,Ideal) -- local cohomology of a polynomial ring</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">localCohom(d,I)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">d</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></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></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>a <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, the local cohomology of <span class="tt">I</span> in degree <em>d</em></span></li> </ul> </li> </ul> <div> <h2>Description</h2> See <a title="local cohomology of a polynomial ring" href="_local__Cohom_lp__Ideal_rp.html">localCohom(Ideal)</a> for the full description. <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 (2,I) o3 = cokernel | -XYZ XY-XZ 3XdX-2YdY-2ZdZ YdY+ZdZ+3 Y2dY-2YdYZ-2YZdZ+Z2dZ | 1 o3 : W-module, quotient of W</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : Dprune h o4 = cokernel | Y-Z Z2 dYZ+ZdZ+2 XdX+2 | 1 o4 : W-module, quotient of W</code></pre> </td> </tr> </table> </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 polynomial ring" href="_local__Cohom_lp__Z__Z_cm__Ideal_rp.html">localCohom(ZZ,Ideal)</a> -- local cohomology of a polynomial ring</span></li> </ul> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/localCohom.m2:142:0</span>.</p> </div> </div> </div> </body> </html>