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Macaulay2Doc :: dim(Ideal)

dim(Ideal) -- compute the Krull dimension

Synopsis

Description

Computes the Krull dimension of the base ring of I mod I.

The ideal of 3x3 commuting matrices:

i1 : R = ZZ/101[x_(0,0)..x_(2,2),y_(0,0)..y_(2,2)]

o1 = R

o1 : PolynomialRing
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
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
i4 : I = ideal flatten(M*N-N*M);

o4 : Ideal of R
i5 : dim I

o5 = 12
The dimension of a Stanley-Reisner monomial ideal associated to a simplicial complex.

A hollow tetrahedron:

i6 : needsPackage "SimplicialComplexes"

o6 = SimplicialComplexes

o6 : Package
i7 : R = QQ[a..d]

o7 = R

o7 : PolynomialRing
i8 : D = simplicialComplex {a*b*c,a*b*d,a*c*d,b*c*d}

o8 = | bcd acd abd abc |

o8 : SimplicialComplex
i9 : I = monomialIdeal D

o9 = monomialIdeal(a*b*c*d)

o9 : MonomialIdeal of R
i10 : facets D

o10 = | bcd acd abd abc |

              1       4
o10 : Matrix R  <--- R
i11 : dim D

o11 = 2
i12 : dim I

o12 = 3
Note that the dimension of the zero ideal is -1.

See also