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<head><title>Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext</title>
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<div><h1>Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext</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>Ext^i(M,N)</tt></div>
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<li><span>Scripted functor: <a href="___Ext.html" title="compute an Ext module">Ext</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>M</tt>, <span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span></span></li>
<li><span><tt>N</tt>, <span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, The global Ext module <i>Ext<sup>i</sup><sub>X</sub>(M,N)</i></span></li>
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<div class="single"><h2>Description</h2>
<div>If <tt>M</tt> or <tt>N</tt> is a sheaf of rings, it is regarded as a sheaf of modules in the evident way.<p/>
<tt>M</tt> and <tt>N</tt> must be coherent sheaves on the same projective variety or scheme <tt>X</tt>.<p/>
As an example, we compute Hom_X(I_X,OO_X), and Ext^1_X(I_X,OO_X), for the rational quartic curve in <i>P<sup>3</sup></i>.<table class="examples"><tr><td><pre>i1 : S = QQ[a..d];</pre>
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<tr><td><pre>i2 : I = monomialCurveIdeal(S,{1,3,4})
3 2 2 2 3 2
o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o2 : Ideal of S</pre>
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<tr><td><pre>i3 : R = S/I
o3 = R
o3 : QuotientRing</pre>
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<tr><td><pre>i4 : X = Proj R
o4 = X
o4 : ProjectiveVariety</pre>
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<tr><td><pre>i5 : IX = sheaf (module I ** R)
o5 = cokernel {2} | c2 bd ac b2 |
{3} | -b -a 0 0 |
{3} | d c -b -a |
{3} | 0 0 -d -c |
1 3
o5 : coherent sheaf on X, quotient of OO (-2) ++ OO (-3)
X X</pre>
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<tr><td><pre>i6 : Ext^1(IX,OO_X)
o6 = 0
o6 : QQ-module</pre>
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<tr><td><pre>i7 : Hom(IX,OO_X)
16
o7 = QQ
o7 : QQ-module, free</pre>
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The Ext^1 being zero says that the point corresponding to I on the Hilbert scheme is smooth (unobstructed), and vector space dimension of Hom tells us that the dimension of the component at the point I is 16.<p/>
The method used may be found in: Smith, G., <em>Computing global extension modules</em>, J. Symbolic Comp (2000) 29, 729-746<p/>
If the module <i>⊕<sub>d≥0</sub> Ext<sup>i</sup>(M,N(d))</i> is desired, see <a href="___Ext^__Z__Z_lp__Coherent__Sheaf_cm__Sum__Of__Twists_rp.html" title="global Ext">Ext^ZZ(CoherentSheaf,SumOfTwists)</a>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_resolution.html" title="projective resolution">resolution</a> -- projective resolution</span></li>
<li><span><a href="___Tor.html" title="Tor module">Tor</a> -- Tor module</span></li>
<li><span><a href="___Hom.html" title="module of homomorphisms">Hom</a> -- module of homomorphisms</span></li>
<li><span><a href="___H__H.html" title="general homology and cohomology functor">HH</a> -- general homology and cohomology functor</span></li>
<li><span><a href="_sheaf__Ext^__Z__Z_lp__Coherent__Sheaf_cm__Coherent__Sheaf_rp.html" title="sheaf Ext of coherent sheaves">sheafExt</a> -- sheaf Ext of coherent sheaves</span></li>
<li><span><a href="___Ext^__Z__Z_lp__Coherent__Sheaf_cm__Sum__Of__Twists_rp.html" title="global Ext">Ext^ZZ(CoherentSheaf,SumOfTwists)</a> -- global Ext</span></li>
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