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<head><title>Ext^ZZ(Module,Module) -- Ext module</title>
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<div><h1>Ext^ZZ(Module,Module) -- Ext 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>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="___Module.html">module</a></span></span></li>
<li><span><tt>N</tt>, <span>a <a href="___Module.html">module</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 <tt>i</tt>-th <tt>Ext</tt> module of <tt>M</tt> and <tt>N</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>If <tt>M</tt> or <tt>N</tt> is an ideal or ring, it is regarded as a module in the evident way.<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[a..d];</pre>
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<tr><td><pre>i2 : I = monomialCurveIdeal(R,{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 R</pre>
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<tr><td><pre>i3 : M = R^1/I
o3 = cokernel | bc-ad c3-bd2 ac2-b2d b3-a2c |
1
o3 : R-module, quotient of R</pre>
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<tr><td><pre>i4 : Ext^1(M,R)
o4 = 0
o4 : R-module</pre>
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<tr><td><pre>i5 : Ext^2(M,R)
o5 = cokernel {-3} | c a 0 b 0 |
{-3} | -d -b c 0 a |
{-3} | 0 0 d c b |
3
o5 : R-module, quotient of R</pre>
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<tr><td><pre>i6 : Ext^3(M,R)
o6 = cokernel {-5} | d c b a |
1
o6 : R-module, quotient of R</pre>
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<tr><td><pre>i7 : Ext^1(I,R)
o7 = cokernel {-3} | c 0 -d 0 -b |
{-3} | b c 0 a 0 |
{-3} | 0 d c b a |
3
o7 : R-module, quotient of R</pre>
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As an efficiency consideration, it is generally much more efficient to compute Ext^i(R^1/I,N) rather than Ext^(i-1)(I,N). The latter first computes a presentation of the ideal I, and then a free resolution of that. For many examples, the difference in time and space required can be very large.</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="_monomial__Curve__Ideal.html" title="make the ideal of a monomial curve">monomialCurveIdeal</a> -- make the ideal of a monomial curve</span></li>
<li><span><a href="___Ext^__Z__Z_lp__Matrix_cm__Module_rp.html" title="map between Ext modules">Ext^ZZ(Matrix,Module)</a> -- map between Ext modules</span></li>
<li><span><a href="___Ext^__Z__Z_lp__Module_cm__Matrix_rp.html" title="map between Ext modules">Ext^ZZ(Module,Matrix)</a> -- map between Ext modules</span></li>
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