-- -*- M2-comint -*- {* hash: 68326226 *}

i1 : S = ZZ/32003[x_0..x_2]; 

i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];

i3 : alphad = map(E^1,E^{2:-1},{{e_1,e_2}});

             1       2
o3 : Matrix E  <--- E

i4 : alpha = map(E^{2:-1},E^{1:-2},{{e_1},{e_2}});

             2       1
o4 : Matrix E  <--- E

i5 : alphad' = beilinson(alphad,S)

o5 = | x_0 0 -x_2 0 x_0 x_1 |

o5 : Matrix

i6 : alpha' = beilinson(alpha,S)

o6 = {1} | 0  |
     {1} | -1 |
     {1} | 0  |
     {1} | 1  |
     {1} | 0  |
     {1} | 0  |

o6 : Matrix

i7 : F = prune homology(alphad',alpha')

o7 = cokernel {1} | x_1^2-x_2^2 |
              {1} | x_1x_2      |
              {2} | -x_0        |

                            3
o7 : S-module, quotient of S

i8 : betti F

            0 1
o8 = total: 3 1
         1: 2 .
         2: 1 1

o8 : BettiTally

i9 : cohomologyTable(presentation F,E,-2,3)

        -2 -1 0 1 2  3  4
o9 = 2:  7  2 . . .  .  .
     1:  .  1 2 1 .  .  .
     0:  .  . . 2 7 14 23

o9 : CohomologyTally

i10 : S = ZZ/32003[x_0..x_4]; 

i11 : E = ZZ/32003[e_0..e_4, SkewCommutative=>true];

i12 : alphad = map(E^5,E^{2:-2},{{e_4*e_1,e_2*e_3},{e_0*e_2,e_3*e_4},{e_1*e_3,e_4*e_0},{e_2*e_4,e_0*e_1},{e_3*e_0,e_1*e_2}})

o12 = | -e_1e_4 e_2e_3  |
      | e_0e_2  e_3e_4  |
      | e_1e_3  -e_0e_4 |
      | e_2e_4  e_0e_1  |
      | -e_0e_3 e_1e_2  |

              5       2
o12 : Matrix E  <--- E

i13 : alpha = syz alphad

o13 = {2} | e_0e_1  e_2e_3 e_0e_4 e_1e_2 -e_3e_4 |
      {2} | -e_2e_4 e_1e_4 e_1e_3 e_0e_3 e_0e_2  |

              2       5
o13 : Matrix E  <--- E

i14 : alphad' = beilinson(alphad,S)

o14 = | 0   0    0    0   x_0 0    -x_2 0   -x_3 0   0    0    -x_0 -x_1 0   
      | x_1 0    -x_3 0   0   -x_4 0    0   0    0   0    0    0    0    0   
      | 0   -x_0 0    x_2 0   0    0    0   -x_4 0   0    0    0    0    -x_1
      | 0   0    0    0   0   -x_0 -x_1 0   0    x_3 -x_2 -x_3 0    0    -x_4
      | 0   -x_1 -x_2 0   0   0    0    x_4 0    0   -x_0 0    0    -x_3 0   
      -----------------------------------------------------------------------
      0    0    0    0    -x_4 |
      0    0    -x_0 -x_1 -x_2 |
      -x_2 0    -x_3 0    0    |
      0    0    0    0    0    |
      0    -x_4 0    0    0    |

o14 : Matrix

i15 : alpha' = beilinson(alpha,S)

o15 = {1} | 0  0  0 0  1 |
      {1} | 0  0  0 0  0 |
      {1} | 0  0  0 0  0 |
      {1} | 0  0  1 0  0 |
      {1} | 0  -1 0 0  0 |
      {1} | 0  0  0 0  0 |
      {1} | 0  0  0 0  0 |
      {1} | 0  0  0 -1 0 |
      {1} | 0  0  0 0  0 |
      {1} | -1 0  0 0  0 |
      {1} | 0  0  0 0  0 |
      {1} | -1 0  0 0  0 |
      {1} | 0  -1 0 0  0 |
      {1} | 0  0  0 0  0 |
      {1} | 0  0  0 0  0 |
      {1} | 0  0  1 0  0 |
      {1} | 0  0  0 -1 0 |
      {1} | 0  0  0 0  0 |
      {1} | 0  0  0 0  1 |
      {1} | 0  0  0 0  0 |

o15 : Matrix

i16 : F = prune homology(alphad',alpha');

i17 : betti res F

              0  1  2 3
o17 = total: 19 35 20 2
          3:  4  .  . .
          4: 15 35 20 .
          5:  .  .  . 2

o17 : BettiTally

i18 : regularity F

o18 = 5

i19 : cohomologyTable(presentation F,E,-6,6)

          -6  -5 -4 -3 -2 -1 0 1  2  3  4   5   6   7
o19 = 4: 210 100 35  4  .  . . .  .  .  .   .   .   .
      3:   .   .  2 10 10  5 . .  .  .  .   .   .   .
      2:   .   .  .  .  .  . 2 .  .  .  .   .   .   .
      1:   .   .  .  .  .  . . 5 10 10  2   .   .   .
      0:   .   .  .  .  .  . . .  .  4 35 100 210 380

o19 : CohomologyTally

i20 :