symmetricKernel(..., Variable => ...) -- Choose name for variables in the created ring

Synopsis

Usage:
symmetricKernel(...,Variable=>w)
reesIdeal(...,Variable=>w)
reesAlgebra(...,Variable=>w)
normalCone(...,Variable=>w)
associatedGradedRing(...,Variable=>w)
specialFiberIdeal(...,Variable=>w)
specialFiber(...,Variable=>w)
distinguished(...,Variable=>w)
distinguishedAndMult(...,Variable=>w)

Description

Each of these functions creates a new ring of the form R[w₀, ..., w_r] or R[w₀, ..., w_r]/J, where R is the ring of the input ideal or module (except for specialFiber, which creates a ring K[w₀, ..., w_r], where K is the ultimate coefficient ring of the input ideal or module.) This option allows the user to change the names of the new variables in this ring. The default variable is w.

i1 : R = QQ[x,y,z]/ideal(x*y^2-z^9)

o1 = R

o1 : QuotientRing

i2 : J = ideal(x,y,z)

o2 = ideal (x, y, z)

o2 : Ideal of R

i3 : I = reesIdeal(J, Variable => p)

                                                             8       2  
o3 = ideal (z*p  - y*p , z*p  - x*p , y*p  - x*p , x*y*p  - z p , x*p  -
               1      2     0      2     0      1       1      2     1  
     ------------------------------------------------------------------------
      7 2     2    6 3
     z p , p p  - z p )
        2   0 1      2

o3 : Ideal of R[p , p , p ]
                 0   1   2

To lift the result to an ideal in a flattened ring, use flattenRing:

i4 : describe ring I

o4 = R[p , p , p , Degrees => {3:{1}}, Heft => {1, 0}, MonomialOrder =>
        0   1   2                {1}                                   
                                                                       
     ------------------------------------------------------------------------
     {MonomialSize => 32}, DegreeRank => 2]
     {GRevLex => {3:1}  }
     {Position => Up    }

i5 : I1 = first flattenRing I

               9      2                                              8   2   
o5 = ideal (- z  + x*y , p z - p y, p z - p x, p y - p x, p x*y - p z , p x -
                          1     2    0     2    0     1    1       2     1   
     ------------------------------------------------------------------------
      2 7     2    3 6
     p z , p p  - p z )
      2     0 1    2

o5 : Ideal of QQ[p , p , p , x, y, z]
                  0   1   2

i6 : describe ring oo

o6 = QQ[p , p , p , x..z, Degrees => {3:{1}, 3:{0}}, Heft => {0..1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 2]
         0   1   2                      {1}    {1}                                    {GRevLex => {3:1}  }
                                                                                      {Position => Up    }
                                                                                      {GRevLex => {3:1}  }

Note that the rings of I and I1 both have bigradings. Use newRing to make a new ring with different degrees.

i7 : S = newRing(ring I1, Degrees=>{numgens ring I1:1})

o7 = S

o7 : PolynomialRing

i8 : describe S

o8 = QQ[p , p , p , x..z, Degrees => {6:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1]
         0   1   2                                                        {GRevLex => {3:1}  }
                                                                          {Position => Up    }
                                                                          {GRevLex => {3:1}  }

i9 : I2 = sub(I1,vars S)

               9      2                                              8   2   
o9 = ideal (- z  + x*y , p z - p y, p z - p x, p y - p x, p x*y - p z , p x -
                          1     2    0     2    0     1    1       2     1   
     ------------------------------------------------------------------------
      2 7     2    3 6
     p z , p p  - p z )
      2     0 1    2

o9 : Ideal of S

i10 : res I2

       1      7      11      6      1
o10 = S  <-- S  <-- S   <-- S  <-- S  <-- 0
                                           
      0      1      2       3      4      5

o10 : ChainComplex

Further information

Default value: w (missing documentation)
Function: symmetricKernel -- Compute the Rees ring of the image of a matrix
Option name: Variable -- specify a name for a variable

symmetricKernel(..., Variable => ...) -- Choose name for variables in the created ring

Synopsis

Description

Further information

See also