-- -*- M2-comint -*- hash: 17784646901002943296

i1 : A = QQ{x,y,z}

o1 = A

o1 : NCPolynomialRing

i2 : f = y*z + z*y - x^2

            2
o2 = zy+yz-x

o2 : A

i3 : g = x*z + z*x - y^2

         2
o3 = zx-y +xz

o3 : A

i4 : h = z^2 - x*y - y*x

      2
o4 = z -yx-xy

o4 : A

i5 : B = A/ncIdeal{f,g,h}
--Calling Bergman for NCGB calculation.
Complete!

o5 = B

o5 : NCQuotientRing

i6 : centralElements(B,3)

o6 = | y^3-y*x*z+x*y*z-x^3 |

o6 : NCMatrix

i7 : j = y^3+x*y*z-y*x*z-x^3

      3          3
o7 = y -yxz+xyz-x

o7 : B

i8 : isCentral j

o8 = true

i9 : apply(5,i->isLeftRegular(j,i+1))

o9 = {true, true, true, true, true}

o9 : List

i10 : normalElements(B,3,n,o)
Normal monomials of degree 3:
none
Components of the normal variety, excluding normal monomials:

o10 = {| n_(y^2*z) n_(y*x*z)+n_(x*y*z) n_(x^2*z) n_(y^3)-n_(x*y*z) n_(x*y^2)
      -----------------------------------------------------------------------
      n_(y*x*y) n_(x^2*y) n_(x*y*x) n_(x^3)+n_(x*y*z) |}

o10 : List

i11 : basis(3,B)

o11 = | x^3 x*y*x x^2*y y*x*y x*y^2 y^3 x^2*z y*x*z x*y*z y^2*z |

o11 : NCMatrix

i12 : use A

o12 = A

o12 : NCPolynomialRing

i13 : I = B.ideal

                              2      2      2
o13 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}

o13 : NCIdeal

i14 : J = ncIdeal promote(j,A)

                        3          3
o14 = Two-sided ideal {y -yxz+xyz-x }

o14 : NCIdeal

i15 : B' = A/(I+J)
--Calling Bergman for NCGB calculation.
Complete!

o15 = B'

o15 : NCQuotientRing

i16 : k = ncMatrix {gens B'}

o16 = | x y z |

o16 : NCMatrix

i17 : M = rightKernelBergman rightKernelBergman k
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!

o17 = | -z y*z     -y^2    -y*x-2*x*y |
      | x  y*x+x*y y*z+x^2 -y^2+3*x*z |
      | y  -x*z    -x*y    x^2        |
      | 0  -2*y    -2*x    -2*z       |

o17 : NCMatrix

i18 : N = rightKernelBergman M
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!

o18 = | 0  2*y*x 2*y^2-2*x*z -y*x*z-x^3 |
      | -z x     y           -y^2+x*z   |
      | x  y     -z          -y*z+x^2   |
      | -y -z    -x          0          |

o18 : NCMatrix

i19 : BprimeToB = ncMap(B,B',gens B)

o19 = NCRingMap B <--- B'

o19 : NCRingMap

i20 : liftM = BprimeToB M

o20 = | -z y*z     -y^2    -y*x-2*x*y |
      | x  y*x+x*y y*z+x^2 -y^2+3*x*z |
      | y  -x*z    -x*y    x^2        |
      | 0  -2*y    -2*x    -2*z       |

o20 : NCMatrix

i21 : liftN = BprimeToB N

o21 = | 0  2*y*x 2*y^2-2*x*z -y*x*z-x^3 |
      | -z x     y           -y^2+x*z   |
      | x  y     -z          -y*z+x^2   |
      | -y -z    -x          0          |

o21 : NCMatrix

i22 : liftM*liftN

o22 = | 0                           2*y^3-2*y*x*z+2*x*y*z-2*x^3 0                           0                           |
      | 2*y^3-2*y*x*z+2*x*y*z-2*x^3 0                           0                           0                           |
      | 0                           0                           2*y^3-2*y*x*z+2*x*y*z-2*x^3 0                           |
      | 0                           0                           0                           2*y^3-2*y*x*z+2*x*y*z-2*x^3 |

o22 : NCMatrix

i23 : jId = promote(j,B)*(ncMatrix applyTable(entries id_(ZZ^4), i -> promote(i,B)))

o23 = | y^3-y*x*z+x*y*z-x^3 0                   0                   0                   |
      | 0                   y^3-y*x*z+x*y*z-x^3 0                   0                   |
      | 0                   0                   y^3-y*x*z+x*y*z-x^3 0                   |
      | 0                   0                   0                   y^3-y*x*z+x*y*z-x^3 |

o23 : NCMatrix

i24 : assignDegrees(jId,{2,2,2,3},{5,5,5,6});

i25 : M' = jId // liftM
--Calling Bergman for NCGB calculation.
Complete!

o25 = | y*x    0      y^2-x*z -1/2*y*x*z-1/2*x^3 |
      | 1/2*x  -1/2*z 1/2*y   -1/2*y^2+1/2*x*z   |
      | 1/2*y  1/2*x  -1/2*z  -1/2*y*z+1/2*x^2   |
      | -1/2*z -1/2*y -1/2*x  0                  |

o25 : NCMatrix

i26 : N

o26 = | 0  2*y*x 2*y^2-2*x*z -y*x*z-x^3 |
      | -z x     y           -y^2+x*z   |
      | x  y     -z          -y*z+x^2   |
      | -y -z    -x          0          |

o26 : NCMatrix

i27 : liftM*M'

o27 = | y^3-y*x*z+x*y*z-x^3 0                   0                   0                   |
      | 0                   y^3-y*x*z+x*y*z-x^3 0                   0                   |
      | 0                   0                   y^3-y*x*z+x*y*z-x^3 0                   |
      | 0                   0                   0                   y^3-y*x*z+x*y*z-x^3 |

o27 : NCMatrix

i28 : M'*liftM

o28 = | y^3-y*x*z+x*y*z-x^3 0                   0                   0                   |
      | 0                   y^3-y*x*z+x*y*z-x^3 0                   0                   |
      | 0                   0                   y^3-y*x*z+x*y*z-x^3 0                   |
      | 0                   0                   0                   y^3-y*x*z+x*y*z-x^3 |

o28 : NCMatrix

i29 : C = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
--Calling Bergman for NCGB calculation.
Complete!

o29 = C

o29 : NCQuotientRing

i30 : R = QQ[q]/ideal{q^4+q^3+q^2+q+1}

o30 = R

o30 : QuotientRing

i31 : B = skewPolynomialRing(R,q,{x,y,z,w})

o31 = B

o31 : NCQuotientRing

i32 : x*y == q*y*x

o32 = true

i33 : use C

o33 = C

o33 : NCQuotientRing

i34 : sigma = ncMap(C,C,{y,z,w,x})

o34 = NCRingMap C <--- C

o34 : NCRingMap

i35 : isWellDefined sigma

o35 = true

i36 : D = oreExtension(C,sigma,a)
--Calling Bergman for NCGB calculation.
Complete!

o36 = D

o36 : NCQuotientRing

i37 : isNormal a

o37 = true

i38 : apply(5,i-> isLeftRegular(a,i+1))

o38 = {true, true, true, true, true}

o38 : List

i39 : sigmaD = normalAutomorphism a

o39 = NCRingMap D <--- D

o39 : NCRingMap

i40 : normalElements(sigmaD,1)

o40 = | a |

o40 : NCMatrix

i41 : normalElements(sigmaD,2)

o41 = 0

               1
o41 : Matrix QQ  <-- 0

i42 : normalElements(D,2,P,Q)
Normal monomials of degree 2:
 2
a
Components of the normal variety, excluding normal monomials:

o42 = {| P_(w*a) P_(z*a) P_(y*a) P_(x*a) P_(z*w) P_(y*w) P_(x*w) P_(y*z)
      -----------------------------------------------------------------------
      P_(x*z) P_(x*y) P_(z^2)^2-P_(y^2)P_(w^2) P_(y^2)P_(z^2)-P_(x^2)P_(w^2)
      -----------------------------------------------------------------------
      P_(x^2)P_(z^2)-P_(w^2)^2 P_(y^2)^2-P_(w^2)^2
      -----------------------------------------------------------------------
      P_(x^2)P_(y^2)-P_(z^2)P_(w^2) P_(x^2)^2-P_(y^2)P_(w^2) |}

o42 : List

i43 : basis(2,D)

o43 = | x^2 x*y y^2 x*z y*z z^2 x*w y*w z*w w^2 x*a y*a z*a w*a a^2 |

o43 : NCMatrix

i44 : isNormal (x^2+z^2-y^2-w^2)

o44 = true

i45 : E' = QQ[x,y,z,w,SkewCommutative=>true]

o45 = E'

o45 : PolynomialRing, 4 skew commutative variable(s)

i46 : E = toNCRing E'
--Calling Bergman for NCGB calculation.
Complete!

o46 = E

o46 : NCQuotientRing

i47 : f = ncMap(E,C,gens E)

o47 = NCRingMap E <--- C

o47 : NCRingMap

i48 : f x^2

o48 = 0

o48 : E

i49 : use C

o49 = C

o49 : NCQuotientRing

i50 : x^2 == 0

o50 = false

i51 : C' = toM2Ring C

o51 = C'

o51 : QuotientRing

i52 : x*y

o52 = 0

o52 : C'

i53 : x*x

       2
o53 = x

o53 : C'

i54 : C'' = toM2Ring(C,SkewCommutative=>true)

o54 = C''

o54 : PolynomialRing, 4 skew commutative variable(s)

i55 : y*x

o55 = -x*y

o55 : C''

i56 : y*y

o56 = 0

o56 : C''

i57 : Dop = oppositeRing D
--Calling Bergman for NCGB calculation.
Complete!

o57 = Dop

o57 : NCQuotientRing

i58 : a*x == w*a

o58 = true

i59 : use D

o59 = D

o59 : NCQuotientRing

i60 : a*w == x*a

o60 = true

i61 : Q = QQ[a,b,c]

o61 = Q

o61 : PolynomialRing

i62 : R = Q/ideal{a*b-c^2}

o62 = R

o62 : QuotientRing

i63 : kRes = res(coker vars R, LengthLimit=>7);

i64 : M = coker kRes.dd_5

o64 = cokernel {4} | 0  -a 0  -c |
               {4} | -b 0  -c 0  |
               {4} | c  0  a  0  |
               {4} | 0  c  0  b  |

                             4
o64 : R-module, quotient of R

i65 : B = endomorphismRing(M,X);

i66 : gensI = gens ideal B

                                                                          
o66 = {X X +-X , -X +X X , -X +X X , X X , X X , X X +-X , X X , X X +-X ,
        0 1   0    3  0 2    2  1 2   0 3   1 3   2 3   2   1 0   2 0   1 
      -----------------------------------------------------------------------
                                    2    2        2    2
      X X , X X +-X , X X , X X , X  , X  +-X , X  , X  +-X }
       2 1   3 0   0   3 1   3 2   0    1    1   2    3    3

o66 : List

i67 : gensIMin = minimizeRelations(gensI, Verbosity=>1)
Eliminating variable X
                      1
Eliminating variable X
                      3
Reducing X_0*X_2*X_0+-X_0
  (1 of 12)
  (Pass 1)
Reducing X_2*X_0*X_2+-X_2
  (2 of 12)
  (Pass 1)
Reducing X_0^2*X_2
  (3 of 12)
  (Pass 1)
Reducing X_2*X_0^2*X_2
  (4 of 12)
  (Pass 1)
Reducing X_2*X_0^2
  (5 of 12)
  (Pass 1)
Reducing X_2^2*X_0
  (6 of 12)
  (Pass 1)
Reducing X_0*X_2^2*X_0
  (7 of 12)
  (Pass 1)
Reducing X_0*X_2^2
  (8 of 12)
  (Pass 1)
Reducing X_0^2
  (9 of 12)
  (Pass 1)
Reducing X_2*X_0*X_2*X_0+-X_2*X_0
  (10 of 12)
  (Pass 1)
Reducing X_2^2
  (11 of 12)
  (Pass 1)
Reducing X_0*X_2*X_0*X_2+-X_0*X_2
  (12 of 12)
  (Pass 1)
Reducing X_0*X_2*X_0+-X_0
  (1 of 4)
  (Pass 2)
Reducing X_2*X_0*X_2+-X_2
  (2 of 4)
  (Pass 2)
Reducing X_0^2
  (3 of 4)
  (Pass 2)
Reducing X_2^2
  (4 of 4)
  (Pass 2)
Reducing X_0*X_2*X_0+-X_0
  (1 of 4)
  (Pass 1)
Reducing X_2*X_0*X_2+-X_2
  (2 of 4)
  (Pass 1)
Reducing X_0^2
  (3 of 4)
  (Pass 1)
Reducing X_2^2
  (4 of 4)
  (Pass 1)

                                 2    2
o67 = {X X X +-X , X X X +-X , X  , X  }
        0 2 0   0   2 0 2   2   0    2

o67 : List

i68 : maps = B.cache.endomorphismRingGens

o68 = {{4} | 0 0 1 0 |, {4} | 0 0 0 0 |, {4} | 0 0 0 0 |, {4} | 1 0 0 0 |}
       {4} | 0 0 0 0 |  {4} | 0 1 0 0 |  {4} | 0 0 0 1 |  {4} | 0 0 0 0 |
       {4} | 0 0 0 0 |  {4} | 0 0 1 0 |  {4} | 1 0 0 0 |  {4} | 0 0 0 0 |
       {4} | 0 1 0 0 |  {4} | 0 0 0 0 |  {4} | 0 0 0 0 |  {4} | 0 0 0 1 |

o68 : List

i69 : maps#3 == maps#0*maps#2

o69 = true

i70 : 
