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<head><title>^ -- a binary operator, usually used for powers</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="___The_sp__Macaulay2_splanguage.html" title="">The Macaulay2 language</a> > <a href="_operators.html" title="">operators</a> > <a href="_^.html" title="a binary operator, usually used for powers">^</a></div>
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<div><h1>^ -- a binary operator, usually used for powers</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>x ^ y</tt></div>
</dd></dl>
</div>
</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><p/>
This operator is used for exponentiation, making free modules and sheaves, for shifting complexes left or right, for projection maps involving direct sums, and for making nets.</div>
</div>
<div class="waystouse"><h2>Ways to use <tt>^</tt> :</h2>
<ul><li>CC ^ ZZ</li>
<li>InexactFieldFamily ^ ZZ</li>
<li><span><a href="___Chain__Complex_sp^_sp__Z__Z.html" title="access member, cohomological degree">ChainComplex ^ ZZ</a> -- access member, cohomological degree</span></li>
<li><span><a href="___Chain__Complex__Map_sp^_sp__Z__Z.html" title="iterated composition">ChainComplexMap ^ ZZ</a> -- iterated composition</span></li>
<li><span>GradedModuleMap ^ ZZ, see <span><a href="___Chain__Complex__Map_sp^_sp__Z__Z.html" title="iterated composition">ChainComplexMap ^ ZZ</a> -- iterated composition</span></span></li>
<li><span><a href="___Coherent__Sheaf_sp^_sp__Z__Z.html" title="direct sum">CoherentSheaf ^ ZZ</a> -- direct sum</span></li>
<li><span>SheafOfRings ^ ZZ, see <span><a href="___Coherent__Sheaf_sp^_sp__Z__Z.html" title="direct sum">CoherentSheaf ^ ZZ</a> -- direct sum</span></span></li>
<li><span>Constant ^ Constant, see <span><a href="___Constant.html" title="">Constant</a></span></span></li>
<li><span>Constant ^ InexactNumber, see <span><a href="___Constant.html" title="">Constant</a></span></span></li>
<li><span>Constant ^ Number, see <span><a href="___Constant.html" title="">Constant</a></span></span></li>
<li><span>InexactNumber ^ Constant, see <span><a href="___Constant.html" title="">Constant</a></span></span></li>
<li><span>Number ^ Constant, see <span><a href="___Constant.html" title="">Constant</a></span></span></li>
<li><span>Expression ^ Expression, see <span><a href="___Expression.html" title="the class of all expressions">Expression</a> -- the class of all expressions</span></span></li>
<li><span><a href="___Ideal_sp^_sp__Z__Z.html" title="power">Ideal ^ ZZ</a> -- power</span></li>
<li><span>MonomialIdeal ^ ZZ, see <span><a href="___Ideal_sp^_sp__Z__Z.html" title="power">Ideal ^ ZZ</a> -- power</span></span></li>
<li><span>Constant ^ Ring, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>Constant ^ RingFamily, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>Number ^ Ring, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>Number ^ RingFamily, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>RingElement ^ Ring, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>RingElement ^ RingFamily, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>ChainComplexMap ^ Array, see <span><a href="___Matrix_sp^_sp__Array.html" title="component of map corresponding to summand of target">Matrix ^ Array</a> -- component of map corresponding to summand of target</span></span></li>
<li><span>GradedModuleMap ^ Array, see <span><a href="___Matrix_sp^_sp__Array.html" title="component of map corresponding to summand of target">Matrix ^ Array</a> -- component of map corresponding to summand of target</span></span></li>
<li><span><a href="___Matrix_sp^_sp__Array.html" title="component of map corresponding to summand of target">Matrix ^ Array</a> -- component of map corresponding to summand of target</span></li>
<li><span><a href="___Matrix_sp^_sp__List.html" title="select rows">Matrix ^ List</a> -- select rows</span></li>
<li><span><a href="___Matrix_sp^_sp__Z__Z.html" title="power">Matrix ^ ZZ</a> -- power</span></li>
<li><span>ChainComplex ^ Array, see <span><a href="___Module_sp^_sp__Array.html" title="projection onto summand">Module ^ Array</a> -- projection onto summand</span></span></li>
<li><span><a href="___Module_sp^_sp__Array.html" title="projection onto summand">Module ^ Array</a> -- projection onto summand</span></li>
<li><span><a href="___Module_sp^_sp__List.html" title="projection onto summand">Module ^ List</a> -- projection onto summand</span></li>
<li><span><a href="___Module_sp^_sp__Z__Z.html" title="direct sum">Module ^ ZZ</a> -- direct sum</span></li>
<li><span><a href="___Net_sp^_sp__Z__Z.html" title="raise a net or string">Net ^ ZZ</a> -- raise a net or string</span></li>
<li><span>String ^ ZZ, see <span><a href="___Net_sp^_sp__Z__Z.html" title="raise a net or string">Net ^ ZZ</a> -- raise a net or string</span></span></li>
<li><span>RingMap ^ ZZ, see <span><a href="_powers.html" title="">powers</a></span></span></li>
<li><span><a href="___Ring_sp^_sp__List.html" title="make a free module">Ring ^ List</a> -- make a free module</span></li>
<li><span><a href="___Ring_sp^_sp__Z__Z.html" title="make a free module">Ring ^ ZZ</a> -- make a free module</span></li>
<li><span><a href="___Ring__Element_sp^_sp__Z__Z.html" title="power">RingElement ^ ZZ</a> -- power</span></li>
<li><span>ScriptedFunctor ^ Thing, see <span><a href="___Scripted__Functor.html" title="the class of all scripted functors">ScriptedFunctor</a> -- the class of all scripted functors</span></span></li>
<li><span><a href="___Sheaf__Of__Rings_sp^_sp__List.html" title="make a graded free coherent sheaf">SheafOfRings ^ List</a> -- make a graded free coherent sheaf</span></li>
<li><span><a href="___String_sp^_sp__Sequence.html" title="vertically stacked copies of a string">String ^ Sequence</a> -- vertically stacked copies of a string</span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="_^.html" title="a binary operator, usually used for powers">^</a> is <span>a <a href="___Keyword.html">keyword</a></span>.</p>
<div><div><p>This operator may be used as a binary operator in an expression like <tt>x^y</tt>. The user may install <a href="_binary_spmethods.html" title="">binary methods</a> for handling such expressions with code such as</p>
<pre> X ^ Y := (x,y) -> ...</pre>
<p>where <tt>X</tt> is the class of <tt>x</tt> and <tt>Y</tt> is the class of <tt>y</tt>.</p>
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