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TateOnProducts :: sloppyTateExtension

sloppyTateExtension -- extend the terms in the Beilinson window to a part of a corner complex of the corresponding Tate resolution

Synopsis

Description

Every object F in in the derived category Dd(P) of coherent sheaves on a product P=Pn1x..xPnt of t projective space is of the form U(W) with W a complex with terms in the Beilinson range only. The function computes with the algorithm (not!) described in section 4 of Tate Resolutions on Products of Projective Spaces computes part of a suitable choosen corner complex of the Tate resolution T(F).

The phrase sloppy refers to the fact that the Beilinson window of T is not equal, but only isomorphic to W. Moreover the bounds in the computation are only a guess and certainly not optimal.

i1 : n={1,1};(S,E)=setupRings(ZZ/101,n);
i3 : T1 = (dual res trim (ideal vars E)^2)[1];
i4 : isChainComplex T1

o4 = true
i5 : a=-{2,2};
i6 : T2=T1**E^{a}[sum a];
i7 : W=beilinsonWindow T2

                    15      16      4
o7 = 0  <-- 0  <-- E   <-- E   <-- E  <-- 0
                                           
     -2     -1     0       1       2      3

o7 : ChainComplex
i8 : cohomologyTable(W,-2*n,2*n)

o8 = | 0 0 0  0 0 |
     | 0 0 0  0 0 |
     | 0 8 15 0 0 |
     | 0 4 8  0 0 |
     | 0 0 0  0 0 |

                      5                5
o8 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i9 : T=sloppyTateExtension W

      1462      1189      954      754      586      447      334      244      174      121      82      54      35      20      10      7
o9 = E     <-- E     <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E    <-- E   <-- E   <-- E   <-- E   <-- E   <-- E
                                                                                                                                          
     -9        -8        -7       -6       -5       -4       -3       -2       -1       0        1       2       3       4       5       6

o9 : ChainComplex
i10 : cohomologyTable(T,-3*n,4*n)

o10 = | 14h 5  24 43 62 81  100 119 |
      | 12h 4  20 36 52 68  84  100 |
      | 10h 3  16 29 42 55  68  81  |
      | 8h  2  12 22 32 42  52  62  |
      | 6h  1  8  15 22 29  36  43  |
      | 4h  0  4  8  12 16  20  24  |
      | 2h  h  0  1  2  3   4   5   |
      | 0   2h 4h 6h 8h 10h 12h 14h |

                       8                8
o10 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i11 : cohomologyTable(beilinsonWindow T,-n,n)

o11 = | 0 0  0 |
      | 8 15 0 |
      | 4 8  0 |

                       3                3
o11 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i12 : cohomologyTable(T,-5*n,4*n) -- the view including the corner

o12 = | 0 33h 14h 5  24 43  62  81  100 119 |
      | 0 28h 12h 4  20 36  52  68  84  100 |
      | 0 23h 10h 3  16 29  42  55  68  81  |
      | 0 18h 8h  2  12 22  32  42  52  62  |
      | 0 13h 6h  1  8  15  22  29  36  43  |
      | 0 8h  4h  0  4  8   12  16  20  24  |
      | 0 3h  2h  h  0  1   2   3   4   5   |
      | 0 2h2 0   2h 4h 6h  8h  10h 12h 14h |
      | 0 7h2 2h2 3h 8h 13h 18h 23h 28h 33h |
      | 0 0   0   0  0  0   0   0   0   0   |

                       10                10
o12 : Matrix (ZZ[h, k])   <--- (ZZ[h, k])

Ways to use sloppyTateExtension :