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<head><title>localCohom(Ideal,Module) -- local cohomology of a D-module</title>
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<div><h1>localCohom(Ideal,Module) -- local cohomology of a D-module</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>H = localCohom(I,M)</tt></div>
</dd></dl>
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</li>
<li><span>Function: <a href="_local__Cohom.html" title="local cohomology">localCohom</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a 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><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a holonomic module over Weyl algebra <em>A<sub>n</sub>(k)</em></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>H</tt>, <span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, each entry of <tt>H</tt> has an integer key and contains the cohomology module in the corresponding degree.</span></li>
</ul>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_local__Cohom_lp..._cm_sp__Loc__Strategy_sp_eq_gt_sp..._rp.html">LocStrategy => ...</a>, -- specify localization strategy for local cohomology</span></li>
<li><span><a href="_local__Cohom_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- specify strategy for local cohomology</span></li>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : W = QQ[X, dX, Y, dY, Z, dZ, WeylAlgebra=>{X=>dX, Y=>dY, Z=>dZ}]
o1 = W
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I = ideal (X*(Y-Z), X*Y*Z)
o2 = ideal (X*Y - X*Z, X*Y*Z)
o2 : Ideal of W</pre>
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<tr><td><pre>i3 : h = localCohom(I, W^1 / ideal{dX,dY,dZ})
WARNING! Dlocalization is an obsolete name for Dlocalize
WARNING! Dlocalization is an obsolete name for Dlocalize
WARNING! Dlocalization is an obsolete name for Dlocalize
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 4Y2-8YZ+4Z2 -3XdX-6YdZ+6ZdZ+6 -XdX+4YdY-6dYZ+4YdZ-6ZdZ-2 XdX+2 XdX+4YdZ-4ZdZ-6 0 0 XdXdZ+2dZ 0 3dXdYdZ+3dXdZ^2 0 0 -4dXY2+8dXYZ-4dXZ2 -2XdXY+3XdXZ-4Y+6Z -XdX^2-3dX -XdX^2-6dXYdZ+6dXZdZ+9dX 3XdX^2+9dXYdZ-9dXZdZ-9dX XdX^2-6dXYdY+9dXdYZ-6dXYdZ+9dXZdZ+3dX -dX^2dY-dX^2dZ -4dX^2YdZ+4dX^2ZdZ+8dX^2 2dX^2YdZ-2dX^2ZdZ-4dX^2 -XdX^2dZ-3dXdZ -dX^2dYdZ-dX^2dZ^2 2dX^2YdY-3dX^2dYZ+2dX^2YdZ-3dX^2ZdZ 4dX^2Y2-8dX^2YZ+4dX^2Z2 2XdX^2Y-3XdX^2Z+6dXY-9dXZ |, | 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 4Y2Z2 -XdXZ2-6YZ2dZ-2Z3dZ-12YZ-6Z2 -XdXZ2+4YdYZ2+4YZ2dZ+2Z3dZ+8YZ+10Z2 XdXZ2+2Z2 XdXZ2+4YZ2dZ+4Z3dZ+8YZ+10Z2 dX^2YdY+2dX^2 XdX^2dY+3dXdY XdXZ2dZ+2XdXZ+2Z2dZ+4Z -dX^2dYZdZ-2dX^2dY 2XdX^2ZdZ-3dXYdYZdZ+3dXdYZ2dZ+3dXZ2dZ^2+6dXdYZ+12dXZdZ+6dX -dX^2YdYdZ-2dX^2dZ -XdX^2dYdZ-3dXdYdZ -4dXY2Z2 -2XdXYZ2-XdXZ3-4YZ2-2Z3 -XdX^2Z2-3dXZ2 -XdX^2Z2-6dXYZ2dZ-6dXZ3dZ-12dXYZ-15dXZ2 XdX^2Z2+9dXYZ2dZ+3dXZ3dZ+18dXYZ+9dXZ2 XdX^2Z2-6dXYdYZ2-6dXYZ2dZ-3dXZ3dZ-12dXYZ-15dXZ2 -2dX^2YdYZ-dX^2Z2dZ-6dX^2Z -4dX^2YZ2dZ-8dX^2YZ 2dX^2YZ2dZ+2dX^2Z3dZ+4dX^2YZ+4dX^2Z2 -XdX^2Z2dZ-2XdX^2Z-3dXZ2dZ-6dXZ dX^2YdYZdZ-dX^2dYZ2dZ-dX^2Z2dZ^2-2dX^2dYZ-2dX^2ZdZ-2dX^2 2dX^2YdYZ2+2dX^2YZ2dZ+dX^2Z3dZ+4dX^2YZ+6dX^2Z2 4dX^2Y2Z2 2XdX^2YZ2+XdX^2Z3+6dXYZ2+3dXZ3 | | 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 dYZ2dZ+Z2dZ^2+4dYZ+8ZdZ+10 |
o3 : HashTable</pre>
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<tr><td><pre>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</pre>
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<div class="single"><h2>Caveat</h2>
<div>The modules returned are not simplified, use <a href="_prune__Local__Cohom_lp__Hash__Table_rp.html" title="prunes local cohomology modules">pruneLocalCohom</a>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_prune__Local__Cohom_lp__Hash__Table_rp.html" title="prunes local cohomology modules">pruneLocalCohom</a> -- prunes local cohomology modules</span></li>
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