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ReesAlgebra :: isLinearType

isLinearType -- Determine whether module has linear type

Synopsis

Description

A module or ideal M is said to be “of linear type” if the natural map from the symmetric algebra of M to the Rees algebra of M is an isomorphism. It is known, for example, that any complete intersection ideal is of linear type.

This routine computes the reesIdeal of M. Giving the element f computes the reesIdeal in a different manner, which is sometimes faster, sometimes slower.

i1 : S = QQ[x_0..x_4]

o1 = S

o1 : PolynomialRing
i2 : i = monomialCurveIdeal(S,{2,3,5,6})

                          2                       3      2     2      2     2
o2 = ideal (x x  - x x , x  - x x , x x  - x x , x  - x x , x x  - x x , x x 
             2 3    1 4   2    0 4   1 2    0 3   3    2 4   1 3    0 4   0 3
     ------------------------------------------------------------------------
        2     2              3    2
     - x x , x x  - x x x , x  - x x )
        1 4   1 3    0 2 4   1    0 4

o2 : Ideal of S
i3 : isLinearType i

o3 = false
i4 : isLinearType(i, i_0)

o4 = false
i5 : I = reesIdeal i

                                                     2                 
o5 = ideal (x w  - x w  + x w , x w  - x w  + x w , x w  + x w  - x w ,
             2 0    3 1    4 2   0 0    1 1    2 2   4 2    1 3    3 4 
     ------------------------------------------------------------------------
                            2                   2                          
     x x w  + x w  - x w , x w  + x w  - x w , x w  + x w  - x w , x x w  -
      0 4 2    1 5    3 6   3 2    0 3    2 4   1 2    0 5    2 6   1 4 1  
     ------------------------------------------------------------------------
                                                           2      2         
     x x w  - x w  + x w , x x w  - x x w  - x w  + x w , x w  - x w  - x w 
      2 4 2    1 4    3 5   0 4 1    1 3 2    0 4    2 5   3 0    4 1    2 3
     ------------------------------------------------------------------------
                                    2                                2    2  
     + x w , x x w  - x w  + x w , x w  - x x w  - x w  + x w , x x w  - w  -
        4 4   1 3 0    2 4    4 5   1 0    1 3 2    0 4    4 6   1 4 2    5  
     ------------------------------------------------------------------------
                                         2                         2
     x w w  + w w , x x w w  - x w w  + w  - w w , x x w w  - x x w  + w w  -
      4 1 6    4 6   3 4 0 2    4 1 4    4    3 5   1 4 0 2    3 4 2    4 5  
     ------------------------------------------------------------------------
     w w )
      3 6

o5 : Ideal of S[w , w , w , w , w , w , w ]
                 0   1   2   3   4   5   6
i6 : select(I_*, f -> first degree f > 1)

           2    2                                       2                   
o6 = {x x w  - w  - x w w  + w w , x x w w  - x w w  + w  - w w , x x w w  -
       1 4 2    5    4 1 6    4 6   3 4 0 2    4 1 4    4    3 5   1 4 0 2  
     ------------------------------------------------------------------------
          2
     x x w  + w w  - w w }
      3 4 2    4 5    3 6

o6 : List
i7 : S = ZZ/101[x,y,z]

o7 = S

o7 : PolynomialRing
i8 : for p from 1 to 5 do print isLinearType (ideal vars S)^p
true
false
false
false
false

See also

Ways to use isLinearType :