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TateOnProducts :: strand

strand -- take the strand

Synopsis

Description

We compute the strand of T as defined in Tate Resolutions on Products of Projective Spaces Theorem 0.4. If T is (part of) the Tate resolution of a sheaf F, then the I-strand of T through c correponds to the Tate resolution J*(F(c)) where J ={0,...,t-1}- I is the complement and πJ: ℙP →∏j ∈Jnj denotes the projection.

i1 : n={1,1};(S,E)=setupRings(ZZ/101,n);
i3 : T1 = (dual res trim (ideal vars E)^2)[1];
i4 : a=-{2,2};T2=T1**E^{a}[sum a];
i6 : W=beilinsonWindow T2,cohomologyTable(W,-2*n,2*n)

                     15      16      4
o6 = (0  <-- 0  <-- E   <-- E   <-- E  <-- 0, | 0 0 0  0 0 |)
                                              | 0 0 0  0 0 |
      -2     -1     0       1       2      3  | 0 8 15 0 0 |
                                              | 0 4 8  0 0 |
                                              | 0 0 0  0 0 |

o6 : Sequence
i7 : T=sloppyTateExtension W;
i8 : cohomologyTable(T,-{3,3},{3,3})

o8 = | 12h 4  20 36 52 68  84  |
     | 10h 3  16 29 42 55  68  |
     | 8h  2  12 22 32 42  52  |
     | 6h  1  8  15 22 29  36  |
     | 4h  0  4  8  12 16  20  |
     | 2h  h  0  1  2  3   4   |
     | 0   2h 4h 6h 8h 10h 12h |

                      7                7
o8 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i9 : sT1=strand(T,-{1,1},{1});
i10 : cohomologyTable(sT1,-{3,3},{3,3})

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  |
      | 4h 0 4 8 12 16 20 |
      | 0  0 0 0 0  0  0  |
      | 0  0 0 0 0  0  0  |

                       7                7
o10 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i11 : sT2=strand(T,{1,1},{0});
i12 : cohomologyTable(sT2,-{3,3},{3,3})

o12 = | 0 0 0 0 52 0 0 |
      | 0 0 0 0 42 0 0 |
      | 0 0 0 0 32 0 0 |
      | 0 0 0 0 22 0 0 |
      | 0 0 0 0 12 0 0 |
      | 0 0 0 0 2  0 0 |
      | 0 0 0 0 8h 0 0 |

                       7                7
o12 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i13 : sT3=removeZeroTrailingTerms strand(T,{1,-1},{0,1})

              12
o13 = 0  <-- E   <-- 0
                      
      -1     0       1

o13 : ChainComplex
i14 : cohomologyTable(sT3,-{3,3},{3,3})

o14 = | 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 12 0 0 |
      | 0 0 0 0 0  0 0 |
      | 0 0 0 0 0  0 0 |

                       7                7
o14 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])

Ways to use strand :