-- -*- M2-comint -*- {* hash: 570227892 *}

i1 : R = QQ[a..d];

i2 : M = image matrix{{a,b,c}}

o2 = image | a b c |

                             1
o2 : R-module, submodule of R

i3 : symmetricAlgebra M

                     R[p , p , p ]
                        0   1   2
o3 = ---------------------------------------------
     (- b*p  + a*p , - c*p  + b*p , - c*p  + a*p )
           0      1       1      2       0      2

o3 : QuotientRing

i4 : symmetricAlgebra(R^{1,2,3})

o4 = R[p , p , p ]
        0   1   2

o4 : PolynomialRing

i5 : A = symmetricAlgebra(M, Variables=>{x,y,z})

o5 = A

o5 : QuotientRing

i6 : describe A

                    R[x, y, z]
o6 = ---------------------------------------
     (- b*x + a*y, - c*y + b*z, - c*x + a*z)

i7 : B = symmetricAlgebra(M, VariableBaseName=>G, MonomialSize=>16)

o7 = B

o7 : QuotientRing

i8 : describe B

                     R[G , G , G ]
                        0   1   2
o8 = ---------------------------------------------
     (- b*G  + a*G , - c*G  + b*G , - c*G  + a*G )
           0      1       1      2       0      2

i9 : symmetricAlgebra(M, Degrees=> {3:1})

                     R[p , p , p ]
                        0   1   2
o9 = ---------------------------------------------
     (- b*p  + a*p , - c*p  + b*p , - c*p  + a*p )
           0      1       1      2       0      2

o9 : QuotientRing

i10 : symmetricAlgebra vars R

o10 = map(R[p ],R[p , p , p , p ],{a*p , b*p , c*p , d*p , a, b, c, d})
             0     0   1   2   3      0     0     0     0

o10 : RingMap R[p ] <--- R[p , p , p , p ]
                 0          0   1   2   3

i11 : symmetricAlgebra vars R

o11 = map(R[p ],R[p , p , p , p ],{a*p , b*p , c*p , d*p , a, b, c, d})
             0     0   1   2   3      0     0     0     0

o11 : RingMap R[p ] <--- R[p , p , p , p ]
                 0          0   1   2   3

i12 : p = symmetricAlgebra(A,B,id_M)

o12 = map(A,B,{x, y, z, a, b, c, d})

o12 : RingMap A <--- B

i13 : p^-1

o13 = map(B,A,{G , G , G , a, b, c, d})
                0   1   2

o13 : RingMap B <--- A

i14 : p * p^-1 === id_A

o14 = true

i15 : p^-1 * p === id_B

o15 = true

i16 :