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

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

Functions with optional argument named Variable:

GF(...,Variable=>...) -- see GF -- make a finite field
Grassmannian(...,Variable=>...) -- see Grassmannian -- compute the ideal of the Grassmannian of linear subspaces of a vector space
intersectInP(...,Variable=>...) -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
multiplicity(...,Variable=>...) -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
Schubert(...,Variable=>...) -- see Schubert -- compute 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

Further information

Default value: w
Function: symmetricKernel -- Compute the Rees ring of the image of a matrix
Option key: Variable -- an optional argument

The source of this document is in ReesAlgebra.m2:1978:0.

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

Description

See also

Functions with optional argument named Variable:

Further information