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<head><title>dim(Ideal) -- compute the Krull dimension</title>
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<div><h1>dim(Ideal) -- compute the Krull dimension</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>dim I</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_dim.html" title="compute the Krull dimension">dim</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span></span></li>
</ul>
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<div class="single"><h2>Description</h2>
<div>Computes the Krull dimension of the base ring of <tt>I</tt> mod <tt>I</tt>.<p/>
The ideal of 3x3 commuting matrices:<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x_(0,0)..x_(2,2),y_(0,0)..y_(2,2)]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : M = genericMatrix(R,x_(0,0),3,3)
o2 = | x_(0,0) x_(1,0) x_(2,0) |
| x_(0,1) x_(1,1) x_(2,1) |
| x_(0,2) x_(1,2) x_(2,2) |
3 3
o2 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i3 : N = genericMatrix(R,y_(0,0),3,3)
o3 = | y_(0,0) y_(1,0) y_(2,0) |
| y_(0,1) y_(1,1) y_(2,1) |
| y_(0,2) y_(1,2) y_(2,2) |
3 3
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : I = ideal flatten(M*N-N*M);
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : dim I
o5 = 12</pre>
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The dimension of a Stanley-Reisner monomial ideal associated to a simplicial complex.<p/>
A hollow tetrahedron:<table class="examples"><tr><td><pre>i6 : needsPackage "SimplicialComplexes"
o6 = SimplicialComplexes
o6 : Package</pre>
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<tr><td><pre>i7 : R = QQ[a..d]
o7 = R
o7 : PolynomialRing</pre>
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<tr><td><pre>i8 : D = simplicialComplex {a*b*c,a*b*d,a*c*d,b*c*d}
o8 = | bcd acd abd abc |
o8 : SimplicialComplex</pre>
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<tr><td><pre>i9 : I = monomialIdeal D
o9 = monomialIdeal(a*b*c*d)
o9 : MonomialIdeal of R</pre>
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<tr><td><pre>i10 : facets D
o10 = | bcd acd abd abc |
1 4
o10 : Matrix R <--- R</pre>
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<tr><td><pre>i11 : dim D
o11 = 2</pre>
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<tr><td><pre>i12 : dim I
o12 = 3</pre>
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</table>
Note that the dimension of the zero ideal is <tt>-1</tt>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_ideal.html" title="make an ideal">ideal</a> -- make an ideal</span></li>
<li><span><a href="_monomial__Ideal.html" title="make a monomial ideal">monomialIdeal</a> -- make a monomial ideal</span></li>
<li><span><a href="../../SimplicialComplexes/html/index.html" title="simplicial complexes">SimplicialComplexes</a> -- simplicial complexes</span></li>
</ul>
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