-- -*- M2-comint -*- hash: 15349168943331426539

i1 : R = ZZ/32003[a..f, SkewCommutative => true]

o1 = R

o1 : PolynomialRing, 6 skew commutative variable(s)

i2 : F = 1 + a + b + a*b + b*c*d + a*b*c*d*e

o2 = a*b*c*d*e + b*c*d + a*b + a + b + 1

o2 : R

i3 : antipode F

o3 = a*b*c*d*e - b*c*d - a*b + a + b + 1

o3 : R

i4 : S = ZZ/32003[a..f, SkewCommutative => {0,1,2}]

o4 = S

o4 : PolynomialRing, 3 skew commutative variable(s)

i5 : F = 1 + a + b + a*b + b*c*d + a*b*c*d*e

o5 = a*b*c*d*e + b*c*d + a*b + a + b + 1

o5 : S

i6 : antipode F

o6 = - a*b*c*d*e - b*c*d - a*b + a + b + 1

o6 : S

i7 : use R

o7 = R

o7 : PolynomialRing, 6 skew commutative variable(s)

i8 : M = matrix{{a*b-1, a*c-d, e-a*b*c}}

o8 = | ab-1 ac-d -abc+e |

             1      3
o8 : Matrix R  <-- R

i9 : N = syz M

o9 = {2} | -ad 0  cd   be ae -abd+ac-d -abc-bd -bcd 2abd-ac+d abc-abe-e
     {2} | a   cd bd-c 0  0  1         b       bc   -ab-1     0        
     {3} | 0   0  0    b  a  0         0       0    0         -1       
     ------------------------------------------------------------------------
     bde+ce abce |
     be     0    |
     c      e    |

             3      12
o9 : Matrix R  <-- R

i10 : M*N

o10 = 0

              1      12
o10 : Matrix R  <-- R

i11 : (transpose N) * (transpose M)

o11 = 0

              12      1
o11 : Matrix R   <-- R

i12 : Mt = matrix transpose entries M -- this does not use antipode

o12 = | ab-1   |
      | ac-d   |
      | -abc+e |

              3      1
o12 : Matrix R  <-- R

i13 : Nt = matrix transpose entries N -- this does not use antipode

o13 = | -ad       a     0  |
      | 0         cd    0  |
      | cd        bd-c  0  |
      | be        0     b  |
      | ae        0     a  |
      | -abd+ac-d 1     0  |
      | -abc-bd   b     0  |
      | -bcd      bc    0  |
      | 2abd-ac+d -ab-1 0  |
      | abc-abe-e 0     -1 |
      | bde+ce    be    c  |
      | abce      0     e  |

              12      3
o13 : Matrix R   <-- R

i14 : Nt*Mt -- is not zero!

o14 = | 2ad    |
      | 0      |
      | -2cd   |
      | -2be   |
      | -2ae   |
      | 0      |
      | 2bd    |
      | 0      |
      | 0      |
      | 0      |
      | -2ce   |
      | -2abce |

              12      1
o14 : Matrix R   <-- R

i15 : 
