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

Synopsis

Usage:

symmetricKernel(...,Variable=>w)

reesIdeal(...,Variable=>w)

reesAlgebra(...,Variable=>w)

specialFiberIdeal(...,Variable=>w)

specialFiber(...,Variable=>w)

distinguished(...,Variable=>w)

isReduction(...,Variable=>w)

jacobianDual(...,Variable=>w)

Description

Each of these functions creates a new ring of the form R[w_0,\ldots, w_r] or R[w_0,\ldots, w_r]/J, where R is the ring of the input ideal or module (except for specialFiber, which creates a ring $K[w_0,\ldots, w_r]$, 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     7 2  
o3 = ideal (x*p  - y*p , y*p  - z*p , x*p  - z*p , z p  - y p , z p  -
               1      2     0      1     0      2     0      2     0  
     ------------------------------------------------------------------------
              6 3    2
     y*p p , z p  - p p )
        1 2     0    1 2

o3 : Ideal of R[p ..p ]
                 0   2

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

i4 : describe ring I

o4 = R[p ..p , Degrees => {3:{1}}, Heft => {1, 0}, Join => false]
        0   2                {1}

i5 : I1 = first flattenRing I

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

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

i6 : describe ring oo

o6 = QQ[p ..p , x..z, Degrees => {3:{1}, 3:{0}}, Heft => {0..1}, Join => false, MonomialOrder => {MonomialSize => 32}]
         0   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 , x..z, Degrees => {6:1}, Heft => {1}, Join => false, MonomialOrder => {MonomialSize => 32}]
         0   2                                                                       {GRevLex => {3:1}  }
                                                                                     {Position => Up    }
                                                                                     {GRevLex => {3:1}  }

i9 : I2 = sub(I1,vars S)

               9      2                                      8      2   2 7  
o9 = ideal (- z  + x*y , p x - p y, p y - p z, p x - p z, p z  - p y , p z  -
                          1     2    0     1    0     2    0      2     0    
     ------------------------------------------------------------------------
             3 6    2
     p p y, p z  - p p )
      1 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
Function: symmetricKernel -- Compute the Rees ring of the image of a matrix
Option key: Variable -- specify a name for a variable

Functions with optional argument named Variable:

conwayPolynomial(...,Variable=>...) (missing documentation)
GF(...,Variable=>...) -- see GF -- make a finite field
Grassmannian(...,Variable=>...) -- see Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space
idealizer(...,Variable=>...) -- Sets the name of the indexed variables introduced in computing the endomorphism ring Hom(J,J).
integralClosure(...,Variable=>...) -- set the base letter for the indexed variables introduced while computing the integral closure
intersectInP(...,Variable=>...) -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
multiplicity(...,Variable=>...) -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
makeS2(...,Variable=>...) -- Sets the name of the indexed variables introduced in computing the S2-ification.
ringFromFractions(...,Variable=>...) -- see ringFromFractions -- find presentation for f.g. ring
Schubert(...,Variable=>...) -- see Schubert(ZZ,ZZ,VisibleList) -- find the Plücker ideal of a Schubert variety
associatedGradedRing(...,Variable=>...)
distinguished(...,Variable=>...)
isReduction(...,Variable=>...)
jacobianDual(...,Variable=>...)
normalCone(Ideal,RingElement,Variable=>...)
normalCone(Ideal,Variable=>...)
reesAlgebra(...,Variable=>...)
reesIdeal(...,Variable=>...)
specialFiber(...,Variable=>...)
specialFiberIdeal(...,Variable=>...)
symmetricKernel(...,Variable=>...) -- Choose name for variables in the created ring

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

Synopsis

Description

Further information

See also

Functions with optional argument named Variable: