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<head><title>CoherentSheaf ^** ZZ -- tensor power</title>
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<div><h1>CoherentSheaf ^** ZZ -- tensor power</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>M^**i</tt></div>
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<li><span>Operator: <a href="_^_st_st.html" title="a binary operator, usually used for tensor or Cartesian power">^**</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span></span></li>
<li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span>, the <tt>i</tt>-th tensor power of <tt>M</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The second symmetric power of the canonical sheaf of the rational quartic:<table class="examples"><tr><td><pre>i1 : R = QQ[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 : X = variety I
o3 = X
o3 : ProjectiveVariety</pre>
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<tr><td><pre>i4 : KX = sheaf(Ext^1(I,R^{-4}) ** ring X)
o4 = cokernel {1} | c 0 -d 0 -b |
{1} | b c 0 a 0 |
{1} | 0 d c b a |
3
o4 : coherent sheaf on X, quotient of OO (-1)
X</pre>
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<tr><td><pre>i5 : K2 = KX^**2
o5 = cokernel {2} | c 0 -d 0 -b 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b 0 0 0 0 0 0 0 0 0 0 |
{2} | b c 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b 0 0 0 0 0 |
{2} | 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b |
{2} | 0 0 0 0 0 c 0 -d 0 -b 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b c 0 a 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b 0 d c b a 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 d c b a |
9
o5 : coherent sheaf on X, quotient of OO (-2)
X</pre>
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<tr><td><pre>i6 : prune K2
o6 = cokernel {1} | c2 bd ac b2 |
{2} | -d -c -b -a |
1 1
o6 : coherent sheaf on X, quotient of OO (-1) ++ OO (-2)
X X</pre>
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Notice that the resulting sheaf is not always presented in the most economical manner. Use <a href="_prune.html" title="prune, e.g., compute a minimal presentation">prune</a> to improve the presentation.</div>
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<div class="single"><h2>See also</h2>
<ul><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.html" title="compute an Ext module">Ext</a> -- compute an Ext module</span></li>
<li><span><a href="_variety.html" title="get the variety">variety</a> -- get the variety</span></li>
<li><span><a href="_sheaf.html" title="make a coherent sheaf">sheaf</a> -- make a coherent sheaf</span></li>
<li><span><a href="_prune.html" title="prune, e.g., compute a minimal presentation">prune</a> -- prune, e.g., compute a minimal presentation</span></li>
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