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TateOnProducts :: tateResolution

tateResolution -- compute the Tate resolution

Synopsis

Description

The call

tateResolution(M,low,high)

forms a free subquotient complex the Tate resolution of the sheaf F represented by M in a range that covers all generators corresponding to cohomology groups of twists F(a) of F in the range low <= a <= high, see Tate Resolutions on Products of Projective Spaces.

i1 : (S,E) = productOfProjectiveSpaces{1,1}

o1 = (S, E)

o1 : Sequence
i2 : low = {-3,-3};high = {3,3};
i4 : T=tateResolution( S^{{1,1}},low, high);
i5 : cohomologyMatrix(T,low,high)

o5 = | 5h 0 5 10 15 20 25 |
     | 4h 0 4 8  12 16 20 |
     | 3h 0 3 6  9  12 15 |
     | 2h 0 2 4  6  8  10 |
     | h  0 1 2  3  4  5  |
     | 0  0 0 0  0  0  0  |
     | h2 0 h 2h 3h 4h 5h |

                      7                7
o5 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])

The complex contains some trailing terms and superfluous terms in a wider range, which can be removed using trivial homological truncation.

i6 : cohomologyMatrix(T,2*low,2*high)

o6 = | 0   0   0   0   0 0  0  0   0   0   0   0   0 |
     | 0   0   0   0   0 0  0  0   0   0   42k 0   0 |
     | 0   0   0   0   0 0  0  0   0   0   36k 42k 0 |
     | 20h 15h 10h 5h  0 5  10 15  20  25  0   0   0 |
     | 16h 12h 8h  4h  0 4  8  12  16  20  0   0   0 |
     | 12h 9h  6h  3h  0 3  6  9   12  15  0   0   0 |
     | 8h  6h  4h  2h  0 2  4  6   8   10  0   0   0 |
     | 4h  3h  2h  h   0 1  2  3   4   5   0   0   0 |
     | 0   0   0   0   0 0  0  0   0   0   0   0   0 |
     | 0   3h2 2h2 h2  0 h  2h 3h  4h  5h  0   0   0 |
     | 0   0   4h2 2h2 0 2h 4h 6h  8h  10h 0   0   0 |
     | 0   0   0   3h2 0 3h 6h 9h  12h 15h 0   0   0 |
     | 0   0   0   0   0 4h 8h 12h 16h 20h 0   0   0 |

                      13                13
o6 : Matrix (ZZ[h, k])   <--- (ZZ[h, k])
i7 : betti T

            -8 -7 -6 -5 -4 -3 -2 -1  0  1  2   3   4   5   6
o7 = total: 84 36 25 40 46 44 35 30 38 56 81 110 141 174 210
        -1: 84 36  .  .  .  .  .  .  .  .  .   .   .   .   .
         0:  .  . 25 40 46 44 35 20 10  4  1   .   .   .   .
         1:  .  .  .  .  .  .  . 10 28 52 80 110 140 170 200
         2:  .  .  .  .  .  .  .  .  .  .  .   .   1   4  10

o7 : BettiTally
i8 : T'=trivialHomologicalTruncation(T, -sum high,-sum low)

             25      40      46      44      35      30      38      56      81      110      141      174      210
o8 = 0  <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E   <-- E    <-- E    <-- E    <-- E    <-- 0
                                                                                                                         
     -7     -6      -5      -4      -3      -2      -1      0       1       2       3        4        5        6        7

o8 : ChainComplex
i9 : betti T'

            -6 -5 -4 -3 -2 -1  0  1  2   3   4   5   6
o9 = total: 25 40 46 44 35 30 38 56 81 110 141 174 210
         0: 25 40 46 44 35 20 10  4  1   .   .   .   .
         1:  .  .  .  .  . 10 28 52 80 110 140 170 200
         2:  .  .  .  .  .  .  .  .  .   .   1   4  10

o9 : BettiTally
i10 : cohomologyMatrix(T',2*low,2*high)

o10 = | 0   0   0   0   0 0  0  0   0   0   0 0 0 |
      | 0   0   0   0   0 0  0  0   0   0   0 0 0 |
      | 0   0   0   0   0 0  0  0   0   0   0 0 0 |
      | 20h 15h 10h 5h  0 5  10 15  20  25  0 0 0 |
      | 16h 12h 8h  4h  0 4  8  12  16  20  0 0 0 |
      | 12h 9h  6h  3h  0 3  6  9   12  15  0 0 0 |
      | 8h  6h  4h  2h  0 2  4  6   8   10  0 0 0 |
      | 4h  3h  2h  h   0 1  2  3   4   5   0 0 0 |
      | 0   0   0   0   0 0  0  0   0   0   0 0 0 |
      | 0   3h2 2h2 h2  0 h  2h 3h  4h  5h  0 0 0 |
      | 0   0   4h2 2h2 0 2h 4h 6h  8h  10h 0 0 0 |
      | 0   0   0   3h2 0 3h 6h 9h  12h 15h 0 0 0 |
      | 0   0   0   0   0 4h 8h 12h 16h 20h 0 0 0 |

                       13                13
o10 : Matrix (ZZ[h, k])   <--- (ZZ[h, k])

The call

tateResolution(A,low,high)

where A is a matrix representing the multi-graded module homomorphism from M to N computes the induced map between two free subquotients of Tate resolutions of M and N in the given range.

i11 : (S,E)=productOfProjectiveSpaces {2,1}

o11 = (S, E)

o11 : Sequence
i12 : low=-{2,1}; high={2,1};
i14 : A=map(S^1, S^{1:{-1,0}}, {{S_0}})

o14 = | x_(0,0) |

              1       1
o14 : Matrix S  <--- S
i15 : M=source A; N=target A;
i17 : TA = tateResolution(A, low, high);
i18 : TM = tateResolution(M, low, high);
i19 : TN = tateResolution(N, low, high);
i20 : (source TA == TM, target TA == TN)

o20 = (true, true)

o20 : Sequence

See also

Ways to use tateResolution :

For the programmer

The object tateResolution is a method function.