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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 $c$ corresponds to the Tate resolution $R{\pi_J}_*(F(c))$ where $J =\{0,\ldots,t-1\} - I$ is the complement and $\pi_J: \mathbb PP \to \prod_{j \in J} \mathbb P^{n_j}$ denotes the projection.

i1 : n={1,1};
 
i2 : (S,E) = productOfProjectiveSpaces 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,cohomologyMatrix(W,-2*n,2*n)

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

o6 : Sequence
 
i7 : T=tateExtension W;
 
i8 : low = -{2,2};high = {2,2};
 
i10 : cohomologyMatrix(T,low,high)

o10 = | 3 16 29 42 55 |
      | 2 12 22 32 42 |
      | 1 8  15 22 29 |
      | 0 4  8  12 16 |
      | h 0  1  2  3  |

                       5               5
o10 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])
 
i11 : sT1=strand(T,{-1,0},{1});
 
i12 : cohomologyMatrix(sT1,low,high)

o12 = | 0 0 0  0  0  |
      | 0 0 0  0  0  |
      | 1 8 15 22 29 |
      | 0 0 0  0  0  |
      | 0 0 0  0  0  |

                       5               5
o12 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])
 
i13 : sT2=strand(T,{-1,0},{0});
 
i14 : cohomologyMatrix(sT2,low,high)

o14 = | 0 16 0 0 0 |
      | 0 12 0 0 0 |
      | 0 8  0 0 0 |
      | 0 4  0 0 0 |
      | 0 0  0 0 0 |

                       5               5
o14 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])
 
i15 : sT3=strand(T,{-1,0},{0,1});
 
i16 : cohomologyMatrix(sT3, low,high)

o16 = | 0 0 0 0 0 |
      | 0 0 0 0 0 |
      | 0 8 0 0 0 |
      | 0 0 0 0 0 |
      | 0 0 0 0 0 |

                       5               5
o16 : Matrix (ZZ[h, k])  <-- (ZZ[h, k])

See also

Ways to use strand :

For the programmer

The object strand is a method function.