composedFunctions -- composed functions

Synopsis

Usage:

cmds=composedFunctions()
Outputs:
- cmds, a string, the commands used in the testing/illustration of composed functions

Description

Prints the commands which illustrate / test various composition of functions.

i1 : n={1,1}, v=n+{1,1}

o1 = ({1, 1}, {2, 2})

o1 : Sequence

i2 : high=3*n, low=-high

o2 = ({3, 3}, {-3, -3})

o2 : Sequence

i3 : (S,E)=productOfProjectiveSpaces n

o3 = (S, E)

o3 : Sequence

We build the example from Section 4 of the paper Tate Resolutions on Products of Projective Spaces which corresponds to a rank 3 vector bundle on P^1xP^1.

i4 : P=(image transpose gens trim (ideal vars E)^2)**E^{n}

o4 = image {-1, -3} | -e_(1,0)e_(1,1) |
           {-2, -2} | -e_(0,1)e_(1,1) |
           {-2, -2} | -e_(0,0)e_(1,1) |
           {-2, -2} | -e_(0,1)e_(1,0) |
           {-2, -2} | -e_(0,0)e_(1,0) |
           {-3, -1} | -e_(0,0)e_(0,1) |

                             6
o4 : E-module, submodule of E

i5 : betti P

            0 1
o5 = total: 1 4
        -2: 1 .
        -1: . .
         0: . 4

o5 : BettiTally

i6 : LP=bgg P

      6      4      1
o6 = S  <-- S  <-- S
                    
     0      1      2

o6 : ChainComplex

i7 : M = (HH^0 LP)**S^{-n}

o7 = cokernel {2, 0} | x_(1,1)  -x_(1,0) 0       0        |
              {1, 1} | -x_(0,0) 0        x_(1,0) 0        |
              {1, 1} | 0        -x_(0,0) x_(1,1) 0        |
              {1, 1} | -x_(0,1) 0        0       x_(1,0)  |
              {1, 1} | 0        -x_(0,1) 0       x_(1,1)  |
              {0, 2} | 0        0        x_(0,1) -x_(0,0) |

                            6
o7 : S-module, quotient of S

i8 : betti res M

            0 1 2
o8 = total: 6 4 1
         2: 6 4 1

o8 : BettiTally

i9 : T = tateResolution(M,low,high)

      136      55      32      44      39      36      54      91      136      184      239      304      382      476      589
o9 = E    <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E
                                                                                                                             
     -8       -7      -6      -5      -4      -3      -2      -1      0        1        2        3        4        5        6

o9 : ChainComplex

i10 : cohomologyMatrix(T,low,high)

o10 = | 28h  18h  8h  2  12  22  32  |
      | 20h  13h  6h  1  8   15  22  |
      | 12h  8h   4h  0  4   8   12  |
      | 4h   3h   2h  h  0   1   2   |
      | 4h2  2h2  0   2h 4h  6h  8h  |
      | 12h2 7h2  2h2 3h 8h  13h 18h |
      | 20h2 12h2 4h2 4h 12h 20h 28h |

                       7               7
o10 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

T is the part of the Tate resolution, which is complete in the range low to high. (In a wider range some terms are missing or are incorrect)

i11 : cohomologyMatrix(T,2*low,2*high)

o11 = | 0    0    0    0    0    0    0  0   0   0   0   0   0 |
      | 0    0    0    0    0    0    0  0   0   0   68k 0   0 |
      | 0    0    0    0    0    0    0  0   0   0   55k 68k 0 |
      | 58h  48h  38h  28h  18h  8h   2  12  22  32  0   0   0 |
      | 41h  34h  27h  20h  13h  6h   1  8   15  22  0   0   0 |
      | 24h  20h  16h  12h  8h   4h   0  4   8   12  0   0   0 |
      | 7h   6h   5h   4h   3h   2h   h  0   1   2   0   0   0 |
      | 10h2 8h2  6h2  4h2  2h2  0    2h 4h  6h  8h  0   0   0 |
      | 27h2 22h2 17h2 12h2 7h2  2h2  3h 8h  13h 18h 0   0   0 |
      | 0    36h2 28h2 20h2 12h2 4h2  4h 12h 20h 28h 0   0   0 |
      | 0    0    39h2 28h2 17h2 6h2  5h 16h 27h 38h 0   0   0 |
      | 0    0    0    36h2 22h2 8h2  6h 20h 34h 48h 0   0   0 |
      | 0    0    0    0    27h2 10h2 7h 24h 41h 58h 0   0   0 |

                       13               13
o11 : Matrix (ZZ[h, k])   <-- (ZZ[h, k])

Alternatively we can recover M from its Beilinson monad derived from T.

i12 : B=beilinson T

       1      4
o12 = S  <-- S
              
      -1     0

o12 : ChainComplex

i13 : M'=prune HH^0 B

o13 = cokernel {2, 0} | -x_(1,0) 0        0        x_(1,1)  |
               {1, 1} | x_(0,0)  -x_(1,1) 0        0        |
               {1, 1} | -x_(0,1) 0        -x_(1,1) 0        |
               {0, 2} | 0        x_(0,1)  x_(0,0)  0        |
               {1, 1} | 0        -x_(1,0) 0        x_(0,0)  |
               {1, 1} | 0        0        -x_(1,0) -x_(0,1) |

                             6
o13 : S-module, quotient of S

i14 : prune HH^1 B

o14 = cokernel | x_(1,1) x_(1,0) x_(0,1) x_(0,0) |

                             1
o14 : S-module, quotient of S

i15 : isIsomorphic(M,M')

o15 = true

We study the corner complex of T at c=\{0,0\} .

i16 : C=cornerComplex(T,{0,0});

i17 : betti C

             -7 -6 -5 -4 -3 -2 -1 0  1  2  3   4
o17 = total: 55 32 44 39 20  6  1 4 15 36 70 120
         -1: 55  .  .  .  .  .  . .  .  .  .   .
          0:  . 32 44 39 20  6  . .  .  .  .   .
          1:  .  .  .  .  .  .  1 .  .  .  .   .
          2:  .  .  .  .  .  .  . .  .  .  .   .
          3:  .  .  .  .  .  .  . 4 15 36 70 120

o17 : BettiTally

i18 : cohomologyMatrix(C,low,high)

o18 = | 0    0    0   2 12 22 32 |
      | 0    0    0   1 8  15 22 |
      | 0    0    0   0 4  8  12 |
      | 0    0    0   h 0  1  2  |
      | 4h3  2h3  0   0 0  0  0  |
      | 12h3 7h3  2h3 0 0  0  0  |
      | 20h3 12h3 4h3 0 0  0  0  |

                       7               7
o18 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i19 : betti C.dd_0

             0 1
o19 = total: 1 4
          0: 1 .
          1: . .
          2: . 4

o19 : BettiTally

i20 : P=ker C.dd_0**E^{v}

o20 = image {-1, 0} | e_(1,1) e_(1,0) e_(0,0) e_(0,1)  0       0       0       0       0       0       0       0       0       0       0       |
            {0, -1} | 0       0       0       -e_(1,1) e_(1,0) e_(0,1) e_(0,0) 0       0       0       e_(1,1) 0       e_(1,1) 0       0       |
            {-1, 0} | 0       0       0       0        0       0       0       e_(1,1) e_(1,0) e_(0,1) e_(0,0) 0       0       0       0       |
            {0, -1} | 0       0       0       0        0       0       0       0       0       0       0       e_(1,1) e_(1,0) e_(0,1) e_(0,0) |

                              4
o20 : E-module, submodule of E

The tensor product with E^{\{v\}} is necessary because we work with E instead of $\omega_E$. M can be recovered by applying the bgg functor to P.

i21 : LP=bgg P;

i22 : betti LP

             -3 -2 -1  0
o22 = total:  4 16 24 15
          0:  4 16 24 15

o22 : BettiTally

i23 : coLP=apply(toList(min LP..max LP),i->prune HH^(-i) LP);

i24 : apply(coLP,h->dim h)

o24 = {0, -1, 0, 4}

o24 : List

i25 : M1=HH^0 LP

o25 = image {1, -1} | x_(0,0)x_(1,1) -x_(1,1)^2      0              0              x_(1,0)x_(1,1) 0               |
            {1, -1} | x_(0,0)x_(1,0) -x_(1,0)x_(1,1) 0              0              x_(1,0)^2      0               |
            {1, -1} | x_(0,1)x_(1,1) 0               x_(1,1)^2      0              0              -x_(1,0)x_(1,1) |
            {1, -1} | x_(0,1)x_(1,0) 0               x_(1,0)x_(1,1) 0              0              -x_(1,0)^2      |
            {0, 0}  | x_(0,0)^2      -x_(0,0)x_(1,1) 0              0              x_(0,0)x_(1,0) 0               |
            {0, 0}  | x_(0,0)x_(0,1) -x_(0,1)x_(1,1) 0              0              x_(0,1)x_(1,0) 0               |
            {0, 0}  | 0              x_(0,1)x_(1,0)  x_(0,0)x_(1,0) x_(1,0)^2      0              0               |
            {0, 0}  | x_(0,1)^2      0               x_(0,1)x_(1,1) 0              0              -x_(0,1)x_(1,0) |
            {0, 0}  | x_(0,0)x_(0,1) 0               x_(0,0)x_(1,1) 0              0              -x_(0,0)x_(1,0) |
            {0, 0}  | 0              0               0              x_(1,1)^2      x_(0,1)x_(1,1) x_(0,0)x_(1,1)  |
            {0, 0}  | 0              0               0              x_(1,0)x_(1,1) x_(0,1)x_(1,0) x_(0,0)x_(1,0)  |
            {-1, 1} | 0              x_(0,1)^2       x_(0,0)x_(0,1) x_(0,1)x_(1,0) 0              0               |
            {-1, 1} | 0              x_(0,0)x_(0,1)  x_(0,0)^2      x_(0,0)x_(1,0) 0              0               |
            {-1, 1} | 0              0               0              x_(0,1)x_(1,1) x_(0,1)^2      x_(0,0)x_(0,1)  |
            {-1, 1} | 0              0               0              x_(0,0)x_(1,1) x_(0,0)x_(0,1) x_(0,0)^2       |

                              15
o25 : S-module, submodule of S

i26 : betti M1,betti M

              0 1         0 1
o26 = (total: 6 4, total: 6 4)
           2: 6 4      2: 6 4

o26 : Sequence

i27 : isIsomorphic(M,M1)

o27 = true

It works also for different syzygy modules in the corner complex. It works for all P=ker C.dd_k in the range where C.dd_k is computed completely. We check the case k=1 and k=-2.

i28 : k=1

o28 = 1

i29 : P=ker C.dd_(-k)**E^{v}; betti P

             0  1
o30 = total: 4 15
         -1: 4 15

o30 : BettiTally

i31 : LP=bgg P

       1      4
o31 = S  <-- S
              
      0      1

o31 : ChainComplex

i32 : betti LP

             0 1
o32 = total: 1 4
          0: 1 4

o32 : BettiTally

i33 : coLP=apply(toList(min LP..max LP),i->prune HH^(-i) LP);

i34 : apply(coLP,h->dim h)

o34 = {0, 4}

o34 : List

i35 : M1=HH^(-k) LP

o35 = image {1, 0} | x_(0,1) 0       -x_(1,0) 0       0        x_(1,1) |
            {1, 0} | x_(0,0) x_(1,0) 0        0       -x_(1,1) 0       |
            {0, 1} | 0       x_(0,1) x_(0,0)  x_(1,1) 0        0       |
            {0, 1} | 0       0       0        x_(1,0) x_(0,1)  x_(0,0) |

                              4
o35 : S-module, submodule of S

i36 : betti M1, betti M

              0 1         0 1
o36 = (total: 6 4, total: 6 4)
           2: 6 4      2: 6 4

o36 : Sequence

i37 : isIsomorphic(M,M1)

o37 = true

Note that we have to take HH^{(-k)} == HH_k because of the homological position in which P sits.

i38 : k=-2

o38 = -2

i39 : P=ker C.dd_(-k)**E^{v}; betti P

              0   1
o40 = total: 70 120
          2: 70 120

o40 : BettiTally

i41 : LP=bgg P;

i42 : betti LP

             -5  -4  -3 -2
o42 = total: 36 129 160 70
          0: 36 129 160 70

o42 : BettiTally

i43 : coLP=apply(toList(min LP..max LP),i->prune HH^(-i) LP);

i44 : apply(coLP,h->dim h)

o44 = {0, -1, 0, 4}

o44 : List

i45 : M1=HH^(-k) LP

o45 = image {1, -3}  | x_(0,1)x_(1,1)^3             0                             -x_(1,0)x_(1,1)^3             0                            0                            x_(1,1)^4                    |
            {1, -3}  | x_(0,0)x_(1,1)^3             x_(1,0)x_(1,1)^3              0                             0                            -x_(1,1)^4                   0                            |
            {1, -3}  | x_(0,1)x_(1,0)x_(1,1)^2      0                             -x_(1,0)^2x_(1,1)^2           0                            0                            x_(1,0)x_(1,1)^3             |
            {1, -3}  | x_(0,0)x_(1,0)x_(1,1)^2      x_(1,0)^2x_(1,1)^2            0                             0                            -x_(1,0)x_(1,1)^3            0                            |
            {1, -3}  | x_(0,1)x_(1,0)^2x_(1,1)      0                             -x_(1,0)^3x_(1,1)             0                            0                            x_(1,0)^2x_(1,1)^2           |
            {1, -3}  | x_(0,1)x_(1,0)^3             0                             -x_(1,0)^4                    0                            0                            x_(1,0)^3x_(1,1)             |
            {1, -3}  | x_(0,0)x_(1,0)^2x_(1,1)      x_(1,0)^3x_(1,1)              0                             0                            -x_(1,0)^2x_(1,1)^2          0                            |
            {1, -3}  | x_(0,0)x_(1,0)^3             x_(1,0)^4                     0                             0                            -x_(1,0)^3x_(1,1)            0                            |
            {0, -2}  | 0                            x_(0,1)x_(1,1)^3              x_(0,0)x_(1,1)^3              x_(1,1)^4                    0                            0                            |
            {0, -2}  | x_(0,0)x_(0,1)x_(1,1)^2      x_(0,1)x_(1,0)x_(1,1)^2       0                             0                            -x_(0,1)x_(1,1)^3            0                            |
            {0, -2}  | 0                            x_(0,1)x_(1,0)x_(1,1)^2       x_(0,0)x_(1,0)x_(1,1)^2       x_(1,0)x_(1,1)^3             0                            0                            |
            {0, -2}  | x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,1)x_(1,0)^2x_(1,1)       0                             0                            -x_(0,1)x_(1,0)x_(1,1)^2     0                            |
            {0, -2}  | x_(0,1)^2x_(1,1)^2           0                             -x_(0,1)x_(1,0)x_(1,1)^2      0                            0                            x_(0,1)x_(1,1)^3             |
            {0, -2}  | x_(0,0)x_(0,1)x_(1,1)^2      0                             -x_(0,0)x_(1,0)x_(1,1)^2      0                            0                            x_(0,0)x_(1,1)^3             |
            {0, -2}  | x_(0,0)^2x_(1,1)^2           x_(0,0)x_(1,0)x_(1,1)^2       0                             0                            -x_(0,0)x_(1,1)^3            0                            |
            {0, -2}  | 0                            x_(0,1)x_(1,0)^2x_(1,1)       x_(0,0)x_(1,0)^2x_(1,1)       x_(1,0)^2x_(1,1)^2           0                            0                            |
            {0, -2}  | x_(0,0)x_(0,1)x_(1,0)^2      x_(0,1)x_(1,0)^3              0                             0                            -x_(0,1)x_(1,0)^2x_(1,1)     0                            |
            {0, -2}  | 0                            x_(0,1)x_(1,0)^3              x_(0,0)x_(1,0)^3              x_(1,0)^3x_(1,1)             0                            0                            |
            {0, -2}  | 0                            0                             0                             x_(1,0)^4                    x_(0,1)x_(1,0)^3             x_(0,0)x_(1,0)^3             |
            {0, -2}  | x_(0,1)^2x_(1,0)x_(1,1)      0                             -x_(0,1)x_(1,0)^2x_(1,1)      0                            0                            x_(0,1)x_(1,0)x_(1,1)^2      |
            {0, -2}  | x_(0,1)^2x_(1,0)^2           0                             -x_(0,1)x_(1,0)^3             0                            0                            x_(0,1)x_(1,0)^2x_(1,1)      |
            {0, -2}  | x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0                             -x_(0,0)x_(1,0)^2x_(1,1)      0                            0                            x_(0,0)x_(1,0)x_(1,1)^2      |
            {0, -2}  | x_(0,0)x_(0,1)x_(1,0)^2      0                             -x_(0,0)x_(1,0)^3             0                            0                            x_(0,0)x_(1,0)^2x_(1,1)      |
            {0, -2}  | x_(0,0)^2x_(1,0)x_(1,1)      x_(0,0)x_(1,0)^2x_(1,1)       0                             0                            -x_(0,0)x_(1,0)x_(1,1)^2     0                            |
            {0, -2}  | x_(0,0)^2x_(1,0)^2           x_(0,0)x_(1,0)^3              0                             0                            -x_(0,0)x_(1,0)^2x_(1,1)     0                            |
            {-1, -1} | 0                            -x_(0,1)^2x_(1,0)x_(1,1)      -x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0                            x_(0,1)^2x_(1,1)^2           x_(0,0)x_(0,1)x_(1,1)^2      |
            {-1, -1} | 0                            x_(0,1)^2x_(1,1)^2            x_(0,0)x_(0,1)x_(1,1)^2       x_(0,1)x_(1,1)^3             0                            0                            |
            {-1, -1} | 0                            -x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)^2x_(1,0)x_(1,1)      0                            x_(0,0)x_(0,1)x_(1,1)^2      x_(0,0)^2x_(1,1)^2           |
            {-1, -1} | 0                            x_(0,0)x_(0,1)x_(1,1)^2       x_(0,0)^2x_(1,1)^2            x_(0,0)x_(1,1)^3             0                            0                            |
            {-1, -1} | 0                            -x_(0,1)^2x_(1,0)^2           -x_(0,0)x_(0,1)x_(1,0)^2      0                            x_(0,1)^2x_(1,0)x_(1,1)      x_(0,0)x_(0,1)x_(1,0)x_(1,1) |
            {-1, -1} | 0                            0                             0                             x_(0,1)x_(1,0)^3             x_(0,1)^2x_(1,0)^2           x_(0,0)x_(0,1)x_(1,0)^2      |
            {-1, -1} | 0                            x_(0,1)^2x_(1,0)x_(1,1)       x_(0,0)x_(0,1)x_(1,0)x_(1,1)  x_(0,1)x_(1,0)x_(1,1)^2      0                            0                            |
            {-1, -1} | 0                            x_(0,1)^2x_(1,0)^2            x_(0,0)x_(0,1)x_(1,0)^2       x_(0,1)x_(1,0)^2x_(1,1)      0                            0                            |
            {-1, -1} | 0                            -x_(0,0)x_(0,1)x_(1,0)^2      -x_(0,0)^2x_(1,0)^2           0                            x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,0)^2x_(1,0)x_(1,1)      |
            {-1, -1} | 0                            0                             0                             x_(0,0)x_(1,0)^3             x_(0,0)x_(0,1)x_(1,0)^2      x_(0,0)^2x_(1,0)^2           |
            {-1, -1} | 0                            x_(0,0)x_(0,1)x_(1,0)x_(1,1)  x_(0,0)^2x_(1,0)x_(1,1)       x_(0,0)x_(1,0)x_(1,1)^2      0                            0                            |
            {-1, -1} | 0                            x_(0,0)x_(0,1)x_(1,0)^2       x_(0,0)^2x_(1,0)^2            x_(0,0)x_(1,0)^2x_(1,1)      0                            0                            |
            {-1, -1} | x_(0,1)^3x_(1,1)             0                             -x_(0,1)^2x_(1,0)x_(1,1)      0                            0                            x_(0,1)^2x_(1,1)^2           |
            {-1, -1} | x_(0,1)^3x_(1,0)             0                             -x_(0,1)^2x_(1,0)^2           0                            0                            x_(0,1)^2x_(1,0)x_(1,1)      |
            {-1, -1} | x_(0,0)x_(0,1)^2x_(1,1)      0                             -x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0                            0                            x_(0,0)x_(0,1)x_(1,1)^2      |
            {-1, -1} | x_(0,0)x_(0,1)^2x_(1,0)      0                             -x_(0,0)x_(0,1)x_(1,0)^2      0                            0                            x_(0,0)x_(0,1)x_(1,0)x_(1,1) |
            {-1, -1} | x_(0,0)^2x_(0,1)x_(1,1)      0                             -x_(0,0)^2x_(1,0)x_(1,1)      0                            0                            x_(0,0)^2x_(1,1)^2           |
            {-1, -1} | x_(0,0)^2x_(0,1)x_(1,0)      0                             -x_(0,0)^2x_(1,0)^2           0                            0                            x_(0,0)^2x_(1,0)x_(1,1)      |
            {-1, -1} | x_(0,0)^3x_(1,1)             x_(0,0)^2x_(1,0)x_(1,1)       0                             0                            -x_(0,0)^2x_(1,1)^2          0                            |
            {-1, -1} | x_(0,0)^3x_(1,0)             x_(0,0)^2x_(1,0)^2            0                             0                            -x_(0,0)^2x_(1,0)x_(1,1)     0                            |
            {-2, 0}  | x_(0,1)^4                    0                             -x_(0,1)^3x_(1,0)             0                            0                            x_(0,1)^3x_(1,1)             |
            {-2, 0}  | x_(0,0)x_(0,1)^3             0                             -x_(0,0)x_(0,1)^2x_(1,0)      0                            0                            x_(0,0)x_(0,1)^2x_(1,1)      |
            {-2, 0}  | x_(0,0)^2x_(0,1)^2           0                             -x_(0,0)^2x_(0,1)x_(1,0)      0                            0                            x_(0,0)^2x_(0,1)x_(1,1)      |
            {-2, 0}  | x_(0,0)^3x_(0,1)             0                             -x_(0,0)^3x_(1,0)             0                            0                            x_(0,0)^3x_(1,1)             |
            {-2, 0}  | x_(0,0)^4                    x_(0,0)^3x_(1,0)              0                             0                            -x_(0,0)^3x_(1,1)            0                            |
            {-2, 0}  | 0                            0                             0                             x_(0,0)^2x_(1,0)x_(1,1)      x_(0,0)^2x_(0,1)x_(1,1)      x_(0,0)^3x_(1,1)             |
            {-2, 0}  | 0                            0                             0                             x_(0,0)^2x_(1,0)^2           x_(0,0)^2x_(0,1)x_(1,0)      x_(0,0)^3x_(1,0)             |
            {-2, 0}  | 0                            0                             0                             x_(0,1)^2x_(1,0)x_(1,1)      x_(0,1)^3x_(1,1)             x_(0,0)x_(0,1)^2x_(1,1)      |
            {-2, 0}  | 0                            0                             0                             x_(0,1)^2x_(1,0)^2           x_(0,1)^3x_(1,0)             x_(0,0)x_(0,1)^2x_(1,0)      |
            {-2, 0}  | 0                            x_(0,1)^3x_(1,1)              x_(0,0)x_(0,1)^2x_(1,1)       x_(0,1)^2x_(1,1)^2           0                            0                            |
            {-2, 0}  | 0                            x_(0,1)^3x_(1,0)              x_(0,0)x_(0,1)^2x_(1,0)       x_(0,1)^2x_(1,0)x_(1,1)      0                            0                            |
            {-2, 0}  | 0                            x_(0,0)^2x_(0,1)x_(1,1)       x_(0,0)^3x_(1,1)              x_(0,0)^2x_(1,1)^2           0                            0                            |
            {-2, 0}  | 0                            x_(0,0)^2x_(0,1)x_(1,0)       x_(0,0)^3x_(1,0)              x_(0,0)^2x_(1,0)x_(1,1)      0                            0                            |
            {-2, 0}  | 0                            x_(0,0)x_(0,1)^2x_(1,1)       x_(0,0)^2x_(0,1)x_(1,1)       x_(0,0)x_(0,1)x_(1,1)^2      0                            0                            |
            {-2, 0}  | 0                            x_(0,0)x_(0,1)^2x_(1,0)       x_(0,0)^2x_(0,1)x_(1,0)       x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0                            0                            |
            {-2, 0}  | 0                            0                             0                             x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,0)x_(0,1)^2x_(1,1)      x_(0,0)^2x_(0,1)x_(1,1)      |
            {-2, 0}  | 0                            0                             0                             x_(0,0)x_(0,1)x_(1,0)^2      x_(0,0)x_(0,1)^2x_(1,0)      x_(0,0)^2x_(0,1)x_(1,0)      |
            {-3, 1}  | 0                            0                             0                             x_(0,0)^3x_(1,0)             x_(0,0)^3x_(0,1)             x_(0,0)^4                    |
            {-3, 1}  | 0                            0                             0                             x_(0,1)^3x_(1,0)             x_(0,1)^4                    x_(0,0)x_(0,1)^3             |
            {-3, 1}  | 0                            x_(0,1)^4                     x_(0,0)x_(0,1)^3              x_(0,1)^3x_(1,1)             0                            0                            |
            {-3, 1}  | 0                            x_(0,0)^3x_(0,1)              x_(0,0)^4                     x_(0,0)^3x_(1,1)             0                            0                            |
            {-3, 1}  | 0                            x_(0,0)x_(0,1)^3              x_(0,0)^2x_(0,1)^2            x_(0,0)x_(0,1)^2x_(1,1)      0                            0                            |
            {-3, 1}  | 0                            x_(0,0)^2x_(0,1)^2            x_(0,0)^3x_(0,1)              x_(0,0)^2x_(0,1)x_(1,1)      0                            0                            |
            {-3, 1}  | 0                            0                             0                             x_(0,0)x_(0,1)^2x_(1,0)      x_(0,0)x_(0,1)^3             x_(0,0)^2x_(0,1)^2           |
            {-3, 1}  | 0                            0                             0                             x_(0,0)^2x_(0,1)x_(1,0)      x_(0,0)^2x_(0,1)^2           x_(0,0)^3x_(0,1)             |

                              70
o45 : S-module, submodule of S

i46 : betti M1,betti M

              0 1         0 1
o46 = (total: 6 4, total: 6 4)
           2: 6 4      2: 6 4

o46 : Sequence

i47 : isIsomorphic(M,M1)

o47 = true

Next we check the functor bgg on S-modules.

i48 : RM=bgg M

       216      140      84      45      20      6
o48 = E    <-- E    <-- E   <-- E   <-- E   <-- E
                                                 
      -7       -6       -5      -4      -3      -2

o48 : ChainComplex

i49 : cohomologyMatrix(RM,low,high)

o49 = | 0 0 0 2 12 22 32 |
      | 0 0 0 1 8  15 22 |
      | 0 0 0 0 4  8  12 |
      | 0 0 0 0 0  1  2  |
      | 0 0 0 0 0  0  0  |
      | 0 0 0 0 0  0  0  |
      | 0 0 0 0 0  0  0  |

                       7               7
o49 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i50 : betti RM

              -7  -6 -5 -4 -3 -2
o50 = total: 216 140 84 45 20  6
          0: 216 140 84 45 20  6

o50 : BettiTally

i51 : uQ=firstQuadrantComplex(T,{0,0});

i52 : cohomologyMatrix(uQ,low,high)

o52 = | 0 0 0 2 12 22 32 |
      | 0 0 0 1 8  15 22 |
      | 0 0 0 0 4  8  12 |
      | 0 0 0 h 0  1  2  |
      | 0 0 0 0 0  0  0  |
      | 0 0 0 0 0  0  0  |
      | 0 0 0 0 0  0  0  |

                       7               7
o52 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

The additional entry h in the zero position of the cohomology matrix of uQ indicates that we can recover the original square of the maximal ideal of E from the differential of the first quadrant complex uQ in this specific case.

i53 : uQ.dd_(-1)

o53 = {-2, 0}  | e_(0,0)e_(0,1)  |
      {-1, -1} | e_(0,0)e_(1,1)  |
      {-1, -1} | e_(0,1)e_(1,0)  |
      {-1, -1} | -e_(0,1)e_(1,1) |
      {-1, -1} | -e_(0,0)e_(1,0) |
      {0, -2}  | e_(1,0)e_(1,1)  |

              6      1
o53 : Matrix E  <-- E

Next we test reciprocity.

i54 : T1=tateResolution(M,low,3*high);

i55 : c={2,2}

o55 = {2, 2}

o55 : List

i56 : CM=cornerComplex(T1,c);

i57 : RMc=firstQuadrantComplex(T1,c);

i58 : cohomologyMatrix(CM,low,3*high)

o58 = | 0    0    0   0   0    64 92 120 148 176 204 232 260 |
      | 0    0    0   0   0    57 82 107 132 157 182 207 232 |
      | 0    0    0   0   0    50 72 94  116 138 160 182 204 |
      | 0    0    0   0   0    43 62 81  100 119 138 157 176 |
      | 0    0    0   0   0    36 52 68  84  100 116 132 148 |
      | 0    0    0   0   0    29 42 55  68  81  94  107 120 |
      | 0    0    0   0   0    22 32 42  52  62  72  82  92  |
      | 0    0    0   0   0    15 22 29  36  43  50  57  64  |
      | 12h2 8h2  4h2 0   4h   0  0  0   0   0   0   0   0   |
      | 4h2  3h2  2h2 h2  0    0  0  0   0   0   0   0   0   |
      | 4h3  2h3  0   2h2 4h2  0  0  0   0   0   0   0   0   |
      | 12h3 7h3  2h3 3h2 8h2  0  0  0   0   0   0   0   0   |
      | 20h3 12h3 4h3 4h2 12h2 0  0  0   0   0   0   0   0   |

                       13               13
o58 : Matrix (ZZ[h, k])   <-- (ZZ[h, k])

i59 : coRMc=apply(toList(-10..-4),i-> HH^(-i) RMc==0)

o59 = {true, true, true, true, true, true, false}

o59 : List

i60 : P1=ker CM.dd_(-sum c)

o60 = image {-2, -2} | 0              0              e_(0,0)e_(1,0)  -e_(0,0)e_(1,1) |
            {-2, -2} | 0              e_(0,0)e_(1,1) 0               -e_(0,1)e_(1,1) |
            {-2, -2} | 0              e_(0,0)e_(1,0) 0               -e_(0,1)e_(1,0) |
            {-2, -2} | e_(0,1)e_(1,1) 0              0               0               |
            {-2, -2} | e_(0,0)e_(1,1) 0              -e_(0,1)e_(1,1) 0               |
            {-2, -2} | e_(0,1)e_(1,0) 0              0               0               |
            {-2, -2} | 0              0              e_(0,1)e_(1,0)  -e_(0,1)e_(1,1) |
            {-2, -2} | 0              0              e_(0,1)e_(1,1)  0               |
            {-2, -2} | 0              e_(0,1)e_(1,1) 0               0               |
            {-2, -2} | e_(0,0)e_(1,0) 0              -e_(0,1)e_(1,0) 0               |
            {-2, -2} | 0              e_(0,1)e_(1,0) 0               0               |
            {-2, -2} | 0              0              0               -e_(0,1)e_(1,0) |
            {-2, -2} | 0              0              e_(0,0)e_(1,1)  0               |
            {-2, -2} | 0              0              0               e_(0,0)e_(1,1)  |
            {-2, -2} | 0              0              0               e_(0,0)e_(1,0)  |

                              15
o60 : E-module, submodule of E

i61 : LP=bgg (P1**E^{-c+v})

       15      16      4
o61 = S   <-- S   <-- S
                       
      0       1       2

o61 : ChainComplex

i62 : betti LP

              0  1 2
o62 = total: 15 16 4
          0: 15 16 4

o62 : BettiTally

i63 : coLP=apply(toList(min LP..max LP),i->dim HH^(-i) LP)

o63 = {4, -1, -1}

o63 : List

Hence both Lp and RMc are acyclic.

i64 : Mc=prune truncate(c,M)**S^{c}

o64 = cokernel | x_(1,0)  0        0        0        0        0        0        0        0        0        -x_(0,0) 0        0        0       0        0       |
               | -x_(1,1) x_(1,0)  0        0        0        0        0        0        0        0        0        0        -x_(0,0) x_(0,0) 0        0       |
               | 0        -x_(1,1) 0        0        0        0        0        0        0        0        0        0        0        0       0        x_(0,0) |
               | 0        0        x_(0,0)  x_(1,0)  0        0        0        0        0        0        0        0        0        0       0        0       |
               | 0        0        -x_(0,1) 0        x_(1,0)  0        0        0        0        0        x_(0,0)  0        0        0       0        0       |
               | 0        0        0        -x_(1,1) 0        x_(0,0)  0        0        0        0        0        0        0        0       0        0       |
               | 0        0        0        0        -x_(1,1) -x_(0,1) 0        0        0        0        0        0        x_(0,0)  0       0        0       |
               | 0        0        0        0        0        0        x_(0,0)  x_(1,0)  0        0        0        0        0        0       0        0       |
               | 0        0        0        0        0        0        -x_(0,1) 0        x_(1,0)  0        0        0        0        x_(0,0) 0        0       |
               | 0        0        0        0        0        0        0        -x_(1,1) 0        x_(0,0)  0        0        0        0       0        0       |
               | 0        0        0        0        0        0        0        0        -x_(1,1) -x_(0,1) 0        0        0        0       0        x_(0,0) |
               | 0        0        0        0        0        0        0        0        0        0        x_(0,1)  x_(1,0)  0        0       0        0       |
               | 0        0        0        0        0        0        0        0        0        0        0        -x_(1,1) x_(0,1)  0       0        0       |
               | 0        0        0        0        0        0        0        0        0        0        0        0        0        x_(0,1) x_(1,0)  0       |
               | 0        0        0        0        0        0        0        0        0        0        0        0        0        0       -x_(1,1) x_(0,1) |

                             15
o64 : S-module, quotient of S

i65 : betti (Mc'=HH^0 LP), betti Mc

               0  1          0  1
o65 = (total: 15 16, total: 15 16)
           0: 15 16      0: 15 16

o65 : Sequence

i66 : isIsomorphic(Mc',Mc)

o66 = true

i67 : c={3,1}

o67 = {3, 1}

o67 : List

i68 : cohomologyMatrix(T1,low,2*high)

o68 = | 52h  33h  14h 5  24  43  62  81  100 119 |
      | 44h  28h  12h 4  20  36  52  68  84  100 |
      | 36h  23h  10h 3  16  29  42  55  68  81  |
      | 28h  18h  8h  2  12  22  32  42  52  62  |
      | 20h  13h  6h  1  8   15  22  29  36  43  |
      | 12h  8h   4h  0  4   8   12  16  20  24  |
      | 4h   3h   2h  h  0   1   2   3   4   5   |
      | 4h2  2h2  0   2h 4h  6h  8h  10h 12h 14h |
      | 12h2 7h2  2h2 3h 8h  13h 18h 23h 28h 33h |
      | 20h2 12h2 4h2 4h 12h 20h 28h 36h 44h 52h |

                       10               10
o68 : Matrix (ZZ[h, k])   <-- (ZZ[h, k])

i69 : CM=cornerComplex(T1,c);

i70 : cohomologyMatrix(CM,low,3*high)

o70 = | 0    0    0   0   0    0    92 120 148 176 204 232 260 |
      | 0    0    0   0   0    0    82 107 132 157 182 207 232 |
      | 0    0    0   0   0    0    72 94  116 138 160 182 204 |
      | 0    0    0   0   0    0    62 81  100 119 138 157 176 |
      | 0    0    0   0   0    0    52 68  84  100 116 132 148 |
      | 0    0    0   0   0    0    42 55  68  81  94  107 120 |
      | 0    0    0   0   0    0    32 42  52  62  72  82  92  |
      | 0    0    0   0   0    0    22 29  36  43  50  57  64  |
      | 0    0    0   0   0    0    12 16  20  24  28  32  36  |
      | 4h2  3h2  2h2 h2  0    h    0  0   0   0   0   0   0   |
      | 4h3  2h3  0   2h2 4h2  6h2  0  0   0   0   0   0   0   |
      | 12h3 7h3  2h3 3h2 8h2  13h2 0  0   0   0   0   0   0   |
      | 20h3 12h3 4h3 4h2 12h2 20h2 0  0   0   0   0   0   0   |

                       13               13
o70 : Matrix (ZZ[h, k])   <-- (ZZ[h, k])

i71 : RMc=firstQuadrantComplex(T1,c);

i72 : coRMc=apply(toList(-9..-4),i-> HH^(-i) RMc==0)

o72 = {true, true, true, true, true, false}

o72 : List

i73 : P1=ker CM.dd_(-sum c)

o73 = image {-3, -1} | 0              0                     e_(0,0)e_(1,0)e_(1,1)  0                     0                     0                     0                      |
            {-3, -1} | e_(0,1)e_(1,1) 0                     0                      0                     0                     0                     0                      |
            {-3, -1} | e_(0,1)e_(1,0) 0                     0                      0                     0                     0                     0                      |
            {-3, -1} | 0              0                     0                      0                     e_(0,0)e_(1,0)e_(1,1) 0                     -e_(0,1)e_(1,0)e_(1,1) |
            {-3, -1} | 0              e_(0,1)e_(1,0)e_(1,1) 0                      0                     0                     0                     0                      |
            {-3, -1} | 0              e_(0,0)e_(1,0)e_(1,1) 0                      e_(0,1)e_(1,0)e_(1,1) 0                     0                     0                      |
            {-3, -1} | e_(0,0)e_(1,1) 0                     e_(0,1)e_(1,0)e_(1,1)  0                     0                     0                     0                      |
            {-3, -1} | 0              0                     -e_(0,1)e_(1,0)e_(1,1) e_(0,0)e_(1,0)e_(1,1) 0                     0                     0                      |
            {-3, -1} | 0              0                     0                      0                     0                     e_(0,1)e_(1,0)e_(1,1) 0                      |
            {-3, -1} | 0              0                     0                      0                     e_(0,1)e_(1,0)e_(1,1) e_(0,0)e_(1,0)e_(1,1) 0                      |
            {-3, -1} | e_(0,0)e_(1,0) 0                     0                      0                     0                     0                     -e_(0,1)e_(1,0)e_(1,1) |
            {-3, -1} | 0              0                     0                      0                     0                     0                     e_(0,0)e_(1,0)e_(1,1)  |

                              12
o73 : E-module, submodule of E

i74 : LP=bgg (P1**E^{-c+v})

       12      10      1
o74 = S   <-- S   <-- S
                       
      0       1       2

o74 : ChainComplex

i75 : betti LP

              0  1 2
o75 = total: 12 10 1
          0: 12 10 1

o75 : BettiTally

i76 : coLP=apply(toList(min LP..max LP),i->dim HH^(-i) LP)

o76 = {4, -1, -1}

o76 : List

i77 : Mc=prune truncate(c,M)**S^{c}

o77 = cokernel | x_(0,0)  x_(1,0)  0        0        0        0        0        0        0        0       |
               | -x_(0,1) 0        x_(1,0)  0        0        0        0        0        -x_(0,0) 0       |
               | 0        -x_(1,1) 0        x_(0,0)  0        0        0        0        0        0       |
               | 0        0        -x_(1,1) -x_(0,1) 0        0        0        0        0        x_(0,0) |
               | 0        0        0        0        x_(0,0)  0        0        0        0        0       |
               | 0        0        0        0        -x_(0,1) x_(0,0)  0        0        0        0       |
               | 0        0        0        0        0        -x_(0,1) 0        0        x_(0,0)  0       |
               | 0        0        0        0        0        0        x_(0,0)  0        0        0       |
               | 0        0        0        0        0        0        -x_(0,1) x_(0,0)  0        0       |
               | 0        0        0        0        0        0        0        -x_(0,1) 0        x_(0,0) |
               | 0        0        0        0        0        0        0        0        x_(0,1)  0       |
               | 0        0        0        0        0        0        0        0        0        x_(0,1) |

                             12
o77 : S-module, quotient of S

i78 : betti (Mc'=HH^0 LP), betti Mc

               0  1          0  1
o78 = (total: 12 10, total: 12 10)
           0: 12 10      0: 12 10

o78 : Sequence

i79 : isIsomorphic(Mc',Mc)

o79 = true

Now we test tateExtension.

i80 : W=beilinsonWindow T

       1      4
o80 = E  <-- E
              
      -1     0

o80 : ChainComplex

i81 : T'=tateExtension W

       1250      1044      867      716      588      480      389      312      246      189      136      88      55      46      54
o81 = E     <-- E     <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E   <-- E   <-- E   <-- E
                                                                                                                                    
      -10       -9        -8       -7       -6       -5       -4       -3       -2       -1       0        1       2       3       4

o81 : ChainComplex

i82 : comT'=cohomologyMatrix(T',low,high)

o82 = | 28h  18h  8h  2  12  22  32  |
      | 20h  13h  6h  1  8   15  22  |
      | 12h  8h   4h  0  4   8   12  |
      | 4h   3h   2h  h  0   1   2   |
      | 4h2  2h2  0   2h 4h  6h  8h  |
      | 12h2 7h2  2h2 3h 8h  13h 18h |
      | 20h2 12h2 4h2 4h 12h 20h 28h |

                       7               7
o82 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i83 : comT=cohomologyMatrix(T,low,high)

o83 = | 28h  18h  8h  2  12  22  32  |
      | 20h  13h  6h  1  8   15  22  |
      | 12h  8h   4h  0  4   8   12  |
      | 4h   3h   2h  h  0   1   2   |
      | 4h2  2h2  0   2h 4h  6h  8h  |
      | 12h2 7h2  2h2 3h 8h  13h 18h |
      | 20h2 12h2 4h2 4h 12h 20h 28h |

                       7               7
o83 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i84 : assert(sub(comT',vars ring comT)==comT)

Finally we illustrate how shifting the Beilinson window works.

i85 : cohomologyMatrix(T,low,high)

o85 = | 28h  18h  8h  2  12  22  32  |
      | 20h  13h  6h  1  8   15  22  |
      | 12h  8h   4h  0  4   8   12  |
      | 4h   3h   2h  h  0   1   2   |
      | 4h2  2h2  0   2h 4h  6h  8h  |
      | 12h2 7h2  2h2 3h 8h  13h 18h |
      | 20h2 12h2 4h2 4h 12h 20h 28h |

                       7               7
o85 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i86 : cohomologyMatrix(beilinsonWindow T,low, high)

o86 = | 0 0 0  0  0 0 0 |
      | 0 0 0  0  0 0 0 |
      | 0 0 0  0  0 0 0 |
      | 0 0 2h h  0 0 0 |
      | 0 0 0  2h 0 0 0 |
      | 0 0 0  0  0 0 0 |
      | 0 0 0  0  0 0 0 |

                       7               7
o86 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i87 : B = beilinson T

       1      4
o87 = S  <-- S
              
      -1     0

o87 : ChainComplex

i88 : d={2,2}

o88 = {2, 2}

o88 : List

i89 : T1=T**E^{d}[sum d]

       136      55      32      44      39      36      54      91      136      184      239      304      382      476      589
o89 = E    <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E
                                                                                                                              
      -12      -11     -10     -9      -8      -7      -6      -5      -4       -3       -2       -1       0        1        2

o89 : ChainComplex

i90 : cohomologyMatrix(beilinsonWindow T1,low,high)

o90 = | 0 0 0    0    0 0 0 |
      | 0 0 0    0    0 0 0 |
      | 0 0 0    0    0 0 0 |
      | 0 0 12h2 7h2  0 0 0 |
      | 0 0 20h2 12h2 0 0 0 |
      | 0 0 0    0    0 0 0 |
      | 0 0 0    0    0 0 0 |

                       7               7
o90 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i91 : B1 =beilinson T1

       7      24      20
o91 = S  <-- S   <-- S
                      
      -2     -1      0

o91 : ChainComplex

i92 : decompose annihilator HH^1 B1

o92 = {ideal (x   , x   ), ideal (x   , x   )}
               0,1   0,0           1,1   1,0

o92 : List

i93 : decompose annihilator HH^2 B1

o93 = {ideal (x   , x   , x   , x   )}
               1,1   1,0   0,1   0,0

o93 : List

i94 : M1=HH^0 B1

o94 = image {1, 1} | x_(0,1)x_(1,0)x_(1,1)^2 x_(0,0)x_(0,1)x_(1,1)^2      0                             x_(0,1)^2x_(1,1)^2           0                             0                        |
            {1, 1} | x_(0,1)x_(1,1)^3        0                            x_(0,0)x_(0,1)x_(1,1)^2       0                            x_(0,1)^2x_(1,1)^2            0                        |
            {1, 1} | 0                       x_(0,0)^2x_(1,1)^2           -x_(0,0)^2x_(1,0)x_(1,1)      0                            0                             x_(0,0)^2x_(0,1)x_(1,1)  |
            {1, 1} | x_(0,0)x_(1,1)^3        0                            x_(0,0)^2x_(1,1)^2            0                            x_(0,0)x_(0,1)x_(1,1)^2       0                        |
            {1, 1} | x_(0,1)x_(1,0)^2x_(1,1) x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0                             x_(0,1)^2x_(1,0)x_(1,1)      0                             0                        |
            {1, 1} | x_(0,1)x_(1,0)x_(1,1)^2 0                            x_(0,0)x_(0,1)x_(1,0)x_(1,1)  0                            x_(0,1)^2x_(1,0)x_(1,1)       0                        |
            {1, 1} | 0                       x_(0,0)^2x_(1,0)x_(1,1)      -x_(0,0)^2x_(1,0)^2           0                            0                             x_(0,0)^2x_(0,1)x_(1,0)  |
            {1, 1} | x_(0,0)x_(1,0)x_(1,1)^2 0                            x_(0,0)^2x_(1,0)x_(1,1)       0                            x_(0,0)x_(0,1)x_(1,0)x_(1,1)  0                        |
            {1, 1} | 0                       x_(0,1)^2x_(1,1)^2           -x_(0,1)^2x_(1,0)x_(1,1)      0                            0                             x_(0,1)^3x_(1,1)         |
            {1, 1} | 0                       x_(0,0)x_(0,1)x_(1,1)^2      -x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0                            0                             x_(0,0)x_(0,1)^2x_(1,1)  |
            {1, 1} | 0                       0                            0                             x_(0,0)x_(0,1)x_(1,1)^2      -x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)^2x_(0,1)x_(1,1) |
            {1, 1} | 0                       0                            0                             -x_(0,0)^2x_(1,1)^2          x_(0,0)^2x_(1,0)x_(1,1)       x_(0,0)^3x_(1,1)         |
            {1, 1} | x_(0,1)x_(1,0)^3        x_(0,0)x_(0,1)x_(1,0)^2      0                             x_(0,1)^2x_(1,0)^2           0                             0                        |
            {1, 1} | x_(0,1)x_(1,0)^2x_(1,1) 0                            x_(0,0)x_(0,1)x_(1,0)^2       0                            x_(0,1)^2x_(1,0)^2            0                        |
            {1, 1} | x_(0,0)x_(1,0)^3        x_(0,0)^2x_(1,0)^2           0                             x_(0,0)x_(0,1)x_(1,0)^2      0                             0                        |
            {1, 1} | x_(0,0)x_(1,0)^2x_(1,1) 0                            x_(0,0)^2x_(1,0)^2            0                            x_(0,0)x_(0,1)x_(1,0)^2       0                        |
            {1, 1} | 0                       x_(0,1)^2x_(1,0)x_(1,1)      -x_(0,1)^2x_(1,0)^2           0                            0                             x_(0,1)^3x_(1,0)         |
            {1, 1} | 0                       x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)x_(0,1)x_(1,0)^2      0                            0                             x_(0,0)x_(0,1)^2x_(1,0)  |
            {1, 1} | 0                       0                            0                             x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)x_(0,1)x_(1,0)^2      -x_(0,0)^2x_(0,1)x_(1,0) |
            {1, 1} | 0                       0                            0                             -x_(0,0)^2x_(1,0)x_(1,1)     x_(0,0)^2x_(1,0)^2            x_(0,0)^3x_(1,0)         |

                              20
o94 : S-module, submodule of S

i95 : dim M1

o95 = 4

i96 : betti M1, betti M

              0 1         0 1
o96 = (total: 6 4, total: 6 4)
           6: 6 4      2: 6 4

o96 : Sequence

i97 : isIsomorphic(M1,M**S^{-d})

o97 = true

Another shift:

i98 : d={-1,-2}

o98 = {-1, -2}

o98 : List

i99 : T2=T**E^{d}[sum d]

       136      55      32      44      39      36      54      91      136      184      239      304      382      476      589
o99 = E    <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E
                                                                                                                              
      -5       -4      -3      -2      -1      0       1       2       3        4        5        6        7        8        9

o99 : ChainComplex

i100 : cohomologyMatrix(beilinsonWindow T2,low,high)

o100 = | 0 0 0 0 0 0 0 |
       | 0 0 0 0 0 0 0 |
       | 0 0 0 0 0 0 0 |
       | 0 0 1 8 0 0 0 |
       | 0 0 0 4 0 0 0 |
       | 0 0 0 0 0 0 0 |
       | 0 0 0 0 0 0 0 |

                        7               7
o100 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i101 : cohomologyMatrix(T,low,high)

o101 = | 28h  18h  8h  2  12  22  32  |
       | 20h  13h  6h  1  8   15  22  |
       | 12h  8h   4h  0  4   8   12  |
       | 4h   3h   2h  h  0   1   2   |
       | 4h2  2h2  0   2h 4h  6h  8h  |
       | 12h2 7h2  2h2 3h 8h  13h 18h |
       | 20h2 12h2 4h2 4h 12h 20h 28h |

                        7               7
o101 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

i102 : B2 =beilinson T2

        8      5
o102 = S  <-- S
               
       0      1

o102 : ChainComplex

i103 : HH^(-1) B2 == 0

o103 = true

i104 : M2=HH^0 B2

o104 = cokernel | -x_(1,1) 0        0        -x_(1,0) 0       |
                | 0        x_(1,0)  0        0        x_(0,0) |
                | 0        -x_(1,1) 0        0        0       |
                | 0        0        x_(1,0)  0        0       |
                | 0        0        -x_(1,1) 0        x_(0,0) |
                | x_(1,0)  0        0        0        0       |
                | 0        0        0        x_(1,0)  x_(0,1) |
                | 0        0        0        -x_(1,1) 0       |

                              8
o104 : S-module, quotient of S

i105 : dim M2

o105 = 4

i106 : betti M2, betti M, betti truncate(-d,M)

               0 1         0 1         0 1
o106 = (total: 8 5, total: 6 4, total: 8 5)
            0: 8 5      2: 6 4      3: 8 5

o106 : Sequence

i107 : isIsomorphic(M2,truncate(-d,M)**S^{-d})

o107 = true

For the programmer

The object composedFunctions is a function closure.

composedFunctions -- composed functions

Synopsis

Description

See also

For the programmer