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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_exterior_spalgebras.html" title="">exterior algebras</a></div>
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<div><h1>exterior algebras</h1>
<div>An exterior algebra is a polynomial ring where multiplication is mildly non-commutative, in that, for every x and y in the ring, y*x = (-1)^(deg(x) deg(y)) x*y, and that for every x of odd degree, x*x = 0.In Macaulay2, deg(x) is the degree of x, or the first degree of x, in case a multi-graded ring is being used. The default degree for each variable is 1, so in this case, y*x = -x*y, if x and y are variables in the ring.<p/>
Create an exterior algebra with explicit generators by creating a polynomial ring with the option <a href="_monoid.html" title="make or retrieve a monoid">SkewCommutative</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z, SkewCommutative => true]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : y*x
o2 = -x*y
o2 : R</pre>
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<tr><td><pre>i3 : (x+y+z)^2
o3 = 0
o3 : R</pre>
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<tr><td><pre>i4 : basis R
o4 = | 1 x xy xyz xz y yz z |
1 8
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : basis(2,R)
o5 = | xy xz yz |
1 3
o5 : Matrix R <--- R</pre>
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<table class="examples"><tr><td><pre>i6 : S = QQ[a,b,r,s,t, SkewCommutative=>true, Degrees=>{2,2,1,1,1}];</pre>
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<tr><td><pre>i7 : r*a == a*r
o7 = false</pre>
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<tr><td><pre>i8 : a*a
o8 = 0
o8 : S</pre>
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<tr><td><pre>i9 : f = a*r+b*s; f^2
o10 = -2a*b*r*s
o10 : S</pre>
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<tr><td><pre>i11 : basis(2,S)
o11 = | a b rs rt st |
1 5
o11 : Matrix S <--- S</pre>
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All modules over exterior algebras are right modules. This means that matrices multiply from the opposite side:<table class="examples"><tr><td><pre>i12 : x*y
o12 = x*y
o12 : R</pre>
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<tr><td><pre>i13 : matrix{{x}} * matrix{{y}}
o13 = | -xy |
1 1
o13 : Matrix R <--- R</pre>
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You may compute Gröbner bases, syzygies, and form quotient rings of these skew commutative rings.</div>
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