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Macaulay2Doc :: dim(Module)

dim(Module) -- compute the Krull dimension

Synopsis

Description

Computes the Krull dimension of the module M
i1 : R = ZZ/31991[a,b,c,d]

o1 = R

o1 : PolynomialRing
i2 : I = monomialCurveIdeal(R,{1,2,3})

             2                    2
o2 = ideal (c  - b*d, b*c - a*d, b  - a*c)

o2 : Ideal of R
i3 : M = Ext^1(I,R)
-- ker (41) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (41) returned CacheFunction: -*a cache function*-
-- ker (41) called with Matrix: 0
--                            2
-- ker (41) returned Module: R
assert( ker(map(R^0,R^{{3}, {3}},0)) === (R^{{3}, {3}}))
-- ker (42) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (42) returned CacheFunction: -*a cache function*-
-- ker (42) called with Matrix: 0
--                            3
-- ker (42) returned Module: R
assert( ker(map(R^0,R^{{2}, {2}, {2}},0)) === (R^{{2}, {2}, {2}}))

o3 = cokernel {-3} | c b a |
              {-3} | d c b |

                            2
o3 : R-module, quotient of R
i4 : dim M

o4 = 2
i5 : N = Ext^0(I,R)
-- ker (43) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (43) returned CacheFunction: -*a cache function*-
-- ker (43) called with Matrix: {-3} | b -c d |
--                              {-3} | a -b c |
-- ker (43) returned Module: image {-2} | c2-bd |
--                                 {-2} | bc-ad |
--                                 {-2} | b2-ac |
assert( ker(map(R^{{3}, {3}},R^{{2}, {2}, {2}},{{b,-c,d}, {a,-b,c}})) === (image(map(R^{{2}, {2}, {2}},R^1,{{c^2-b*d}, {b*c-a*d}, {b^2-a*c}}))))

o5 = image {-2} | c2-bd |
           {-2} | bc-ad |
           {-2} | b2-ac |

                             3
o5 : R-module, submodule of R
i6 : dim N

o6 = 4
Note that the dimension of the zero module is -1.

See also