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<head><title>exterior power of a module</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_modules.html" title="">modules</a> > <a href="_exterior_sppower_spof_spa_spmodule.html" title="">exterior power of a module</a></div>
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<div><h1>exterior power of a module</h1>
<div>The <tt>k</tt>-th exterior power of a module <tt>M</tt> is the <tt>k</tt>-fold tensor product of <tt>M</tt> together with the equivalence relation:<pre>
m_1 ** m_2 ** .. ** m_k = 0 if m_i = m_j for i != j
</pre>
If <tt>M</tt> is a free <tt>R</tt>-module of rank <tt>n</tt>, then the <tt>k</tt>-th exterior power of <tt>M</tt> is a free <tt>R</tt>-module of rank <tt>binomial(n,k)</tt>. Macaulay2 computes the <tt>k</tt>-th exterior power of a module <tt>M</tt> with the command exteriorPower.<table class="examples"><tr><td><pre>i1 : R = ZZ/2[x,y]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : exteriorPower(3,R^6)
20
o2 = R
o2 : R-module, free</pre>
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<tr><td><pre>i3 : binomial(6,3)
o3 = 20</pre>
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Macaulay2 can compute exterior powers of modules that are not free as well.<table class="examples"><tr><td><pre>i4 : exteriorPower(2,R^1)
o4 = 0
o4 : R-module</pre>
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<tr><td><pre>i5 : I = module ideal (x,y)
o5 = image | x y |
1
o5 : R-module, submodule of R</pre>
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<tr><td><pre>i6 : exteriorPower(2,I)
o6 = cokernel {2} | x y |
1
o6 : R-module, quotient of R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_exterior_sppower_spof_spa_spmatrix.html" title="">exterior power of a matrix</a></span></li>
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