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<head><title>HH^ZZ Module -- local cohomology of a module</title>
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<div><h1>HH^ZZ Module -- local cohomology of a 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>HH^i(M)</tt></div>
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<li><span>Function: <a href="_cohomology.html" title="general cohomology functor">cohomology</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span>, which is non negative</span></li>
<li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, which is graded over its base polynomial ring</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_cohomology_lp..._cm_sp__Degree_sp_eq_gt_sp..._rp.html">Degree => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>The command computes the local cohomology of the graded module <tt>M</tt> with respect to the maximal irrelevant ideal (the ideal of variables in the base ring of <tt>M</tt>).<p/>
The package <a href="../../Dmodules/html/index.html" title="algorithms for D-modules">Dmodules</a> has alternative code to compute local cohomology (even in the non homogeneous case)<p/>
A very simple example:<table class="examples"><tr><td><pre>i1 : R = QQ[a,b];</pre>
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<tr><td><pre>i2 : HH^2 (R^{-3})
o2 = cokernel | b a 0 |
| 0 -b a |
2
o2 : R-module, quotient of R</pre>
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<tr><td><pre>i3 : HH^2 (R^{-4})
o3 = cokernel | b a 0 0 |
| 0 -b a 0 |
| 0 0 -b a |
3
o3 : R-module, quotient of R</pre>
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Another example, a singular surface in projective fourspace (with one apparent double point):<table class="examples"><tr><td><pre>i4 : R = ZZ/101[x_0..x_4];</pre>
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<tr><td><pre>i5 : I = ideal(x_1*x_4-x_2*x_3, x_1^2*x_3+x_1*x_2*x_0-x_2^2*x_0, x_3^3+x_3*x_4*x_0-x_4^2*x_0)
2 2 3 2
o5 = ideal (- x x + x x , x x x - x x + x x , x + x x x - x x )
2 3 1 4 0 1 2 0 2 1 3 3 0 3 4 0 4
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : M = R^1/module(I)
o6 = cokernel | -x_2x_3+x_1x_4 x_0x_1x_2-x_0x_2^2+x_1^2x_3 x_3^3+x_0x_3x_4-x_0x_4^2 |
1
o6 : R-module, quotient of R</pre>
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<tr><td><pre>i7 : HH^1(M)
o7 = cokernel | x_4 x_3 x_2 x_1 x_0^3 |
1
o7 : R-module, quotient of R</pre>
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<tr><td><pre>i8 : HH^2(M)
o8 = cokernel | x_4 x_3 x_2 x_1 x_0 |
1
o8 : R-module, quotient of R</pre>
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<div class="single"><h2>Caveat</h2>
<div>There is no check made if the given module is graded over the base polynomial ring</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="../../Dmodules/html/index.html" title="algorithms for D-modules">Dmodules</a> -- algorithms for D-modules</span></li>
<li><span><a href="___H__H^__Z__Z_sp__Sum__Of__Twists.html" title="coherent sheaf cohomology module">HH^ZZ SumOfTwists</a> -- coherent sheaf cohomology module</span></li>
<li><span><a href="___H__H^__Z__Z_sp__Coherent__Sheaf.html" title="cohomology of a coherent sheaf on a projective variety">HH^ZZ CoherentSheaf</a> -- cohomology of a coherent sheaf on a projective variety</span></li>
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