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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_symmetric_spalgebras.html" title="">symmetric algebras</a></div>
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<div><h1>symmetric algebras</h1>
<div>Polynomial rings are symmetric algebras with explicit generators, and we have already seen how to construct them. But if you have a module, then its symmetric algebra can be constructed with <a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : symmetricAlgebra R^3
o2 = R[p , p , p ]
0 1 2
o2 : PolynomialRing</pre>
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</table>
Maps between symmetric algebras can be constructed functorially.<table class="examples"><tr><td><pre>i3 : vars R
o3 = | a b c d |
1 4
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : symmetricAlgebra vars R
o4 = map(R[p ],R[p , p , p , p ],{a*p , b*p , c*p , d*p , a, b, c, d})
0 0 1 2 3 0 0 0 0
o4 : RingMap R[p ] <--- R[p , p , p , p ]
0 0 1 2 3</pre>
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<tr><td><pre>i5 : symmetricAlgebra transpose vars R
o5 = map(R[p , p , p , p ],R[p ],{a*p + b*p + c*p + d*p , a, b, c, d})
0 1 2 3 0 0 1 2 3
o5 : RingMap R[p , p , p , p ] <--- R[p ]
0 1 2 3 0</pre>
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Until the ring is used with <a href="_use.html" title="install or activate object">use</a> or assigned to a global variable, its generators are not assigned to global variables.<table class="examples"><tr><td><pre>i6 : a
o6 = a
o6 : R</pre>
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<tr><td><pre>i7 : p_0
o7 = p
0
o7 : IndexedVariable</pre>
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<tr><td><pre>i8 : S = o2;</pre>
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<tr><td><pre>i9 : a
o9 = a
o9 : R</pre>
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<tr><td><pre>i10 : p_0
o10 = p
0
o10 : S</pre>
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To specify the names of the variables when creating the ring, use the <a href="___Variables.html" title="name for an optional argument">Variables</a> option or the <a href="_monoid.html" title="make or retrieve a monoid">VariableBaseName</a> option.<table class="examples"><tr><td><pre>i11 : symmetricAlgebra(R^3, Variables => {t,u,v})
o11 = R[t, u, v]
o11 : PolynomialRing</pre>
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<tr><td><pre>i12 : symmetricAlgebra(R^3, VariableBaseName => t)
o12 = R[t , t , t ]
0 1 2
o12 : PolynomialRing</pre>
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We can construct the symmetric algebra of a module that isn't necessarily free.<table class="examples"><tr><td><pre>i13 : use R
o13 = R
o13 : PolynomialRing</pre>
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<tr><td><pre>i14 : symmetricAlgebra(R^1/(a,b^3))
R[p ]
0
o14 = ------------
3
(a*p , b p )
0 0
o14 : QuotientRing</pre>
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