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ReesAlgebra :: analyticSpread

analyticSpread -- Compute the analytic spread of a module or ideal

Synopsis

Description

The analytic spread of a module is the dimension of its special fiber ring. When I is an ideal (and more generally, with the right definitions) the analytic spread of I is the smallest number of generators of an ideal J such that I is integral over J. See for example the book Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge, 2006, by Craig Huneke and Irena Swanson.

i1 : R=QQ[a..h]

o1 = R

o1 : PolynomialRing
i2 : M=matrix{{a,b,c,d},{e,f,g,h}}

o2 = | a b c d |
     | e f g h |

             2       4
o2 : Matrix R  <--- R
i3 : analyticSpread minors(2,M)

o3 = 5
i4 : specialFiberIdeal minors(2,M)

o4 = ideal(Z Z  - Z Z  + Z Z )
            2 3    1 4    0 5

o4 : Ideal of QQ[Z , Z , Z , Z , Z , Z ]
                  0   1   2   3   4   5
i5 : R=QQ[a,b,c,d]

o5 = R

o5 : PolynomialRing
i6 : M=matrix{{a,b,c,d},{b,c,d,a}}

o6 = | a b c d |
     | b c d a |

             2       4
o6 : Matrix R  <--- R
i7 : analyticSpread minors(2,M)

o7 = 4
i8 : specialFiberIdeal minors(2,M)

                                 2                  2
o8 = ideal (Z Z  - Z Z  + Z Z , Z  - Z Z  - Z Z  - Z  + Z Z  + Z Z )
             2 3    1 4    0 5   1    0 2    0 3    4    2 5    3 5

o8 : Ideal of QQ[Z , Z , Z , Z , Z , Z ]
                  0   1   2   3   4   5

See also

Ways to use analyticSpread :