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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp1.4.9.html" title="global versus local rings">Singular Book 1.4.9</a></div>
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<div><h1>Singular Book 1.4.9 -- global versus local rings</h1>
<div>Consider the union of a line and a plane in affine 3-space.<table class="examples"><tr><td><pre>i1 : S = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : I = ideal(y*(x-1), z*(x-1));
o2 : Ideal of S</pre>
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The dimension is 2, the maximum of the dimensions of the two components. In order to find the dimension, Macaulay2 requires the Groebner basis of I. It computes this behind the scenes, and caches the value with I.<table class="examples"><tr><td><pre>i3 : dim I
o3 = 2</pre>
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<tr><td><pre>i4 : gens gb I
o4 = | xz-z xy-y |
1 2
o4 : Matrix S <--- S</pre>
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Notice that <i>y</i> is not in <i>I</i>.<table class="examples"><tr><td><pre>i5 : y % I
o5 = y
o5 : S</pre>
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Now let's use a local order.<table class="examples"><tr><td><pre>i6 : R = QQ[x,y,z,MonomialOrder=>{Weights=>{-1,-1,-1},RevLex},Global=>false];</pre>
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<tr><td><pre>i7 : J = substitute(I,R)
o7 = ideal (- y + x*y, - z + x*z)
o7 : Ideal of R</pre>
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<tr><td><pre>i8 : gens gb J
o8 = | y-xy z-xz |
1 2
o8 : Matrix R <--- R</pre>
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The dimension in this case is 1.<table class="examples"><tr><td><pre>i9 : dim J
o9 = 1</pre>
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The following is WRONG. In this local ring, <i>y</i> is in the ideal <i>J</i>.<table class="examples"><tr><td><pre>i10 : y % J
o10 = 0
o10 : R</pre>
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Translate the origin to <i>(1,0,0)</i>. The plane <i>x-1 = 0</i> goes through this new origin.<table class="examples"><tr><td><pre>i11 : J = substitute(J, {x=>x+1})
o11 = ideal (x*y, x*z)
o11 : Ideal of R</pre>
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<tr><td><pre>i12 : dim J
o12 = 2</pre>
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<p/>
Compute the global dimension after translation.<table class="examples"><tr><td><pre>i13 : use ring I
o13 = S
o13 : PolynomialRing</pre>
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<tr><td><pre>i14 : I1 = substitute(I, {x=>x+1})
o14 = ideal (x*y, x*z)
o14 : Ideal of S</pre>
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<tr><td><pre>i15 : dim I1
o15 = 2</pre>
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See also <a href="_dim.html" title="compute the Krull dimension">dim</a>.</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_dim.html" title="compute the Krull dimension">dim</a> -- compute the Krull dimension</span></li>
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