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TateOnProducts :: pushAboveWindow

pushAboveWindow -- push a projective resolution of the Beilinson complex out of the window

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. This function is the first step in our computation of the algorithm (not!) described in section 4 of Tate Resolutions on Products of Projective Spaces that computes part of a suitable choosen corner complex of the Tate resolution T(F).

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;
i10 : cohomologyTable(T,-5*n,4*n) -- a view with the corner

o10 = | 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
o10 : Matrix (ZZ[h, k])   <--- (ZZ[h, k])
i11 : puT=trivialHomologicalTruncation(pushAboveWindow W,-1, 6)

                     15      16      19      36      60
o11 = 0  <-- 0  <-- E   <-- E   <-- E   <-- E   <-- E   <-- 0 <-- 0 <-- 0
                                                                         
      -2     -1     0       1       2       3       4       5     6     7

o11 : ChainComplex
i12 : cohomologyTable(puT,-3*n,{1,1})

o12 = | 0  0  0   0  0 |
      | 6h 1  8   15 0 |
      | 4h 0  4   8  0 |
      | 2h h  h+1 1  0 |
      | 0  2h 4h  6h 0 |

                       5                5
o12 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i13 : betti W

              0  1 2
o13 = total: 15 16 4
          0: 15 16 4

o13 : BettiTally
i14 : qT=trivialHomologicalTruncation(lastQuadrantComplex(T,{0,0}),-1,6)

                                 4      9      20      10
o14 = 0  <-- 0  <-- 0 <-- 0 <-- E  <-- E  <-- E   <-- E   <-- 0 <-- 0
                                                                     
      -2     -1     0     1     2      3      4       5       6     7

o14 : ChainComplex
i15 : cohomologyTable(qT,-3*n,{1,1})

o15 = | 0  0  0  0 0 |
      | 0  0  0  0 0 |
      | 4h 0  4  0 0 |
      | 2h h  0  0 0 |
      | 0  2h 4h 0 0 |

                       5                5
o15 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])
i16 : betti puT

              0  1  2  3  4
o16 = total: 15 16 19 36 60
          0: 15 16  6  1  .
          1:  .  . 13 35 60

o16 : BettiTally
i17 : betti qT

             2 3  4  5
o17 = total: 4 9 20 10
          0: 4 .  .  .
          1: . 9 20  6
          2: . .  .  4

o17 : BettiTally
i18 : betti T

               -9   -8  -7  -6  -5  -4  -3  -2  -1   0  1  2  3  4  5 6
o18 = total: 1462 1189 954 754 586 447 334 244 174 121 82 54 35 20 10 7
          0: 1260 1001 780 594 440 315 216 140  84  45 20  6  .  .  . .
          1:  202  188 174 160 146 132 118 104  90  76 62 48 35 20  6 .
          2:    .    .   .   .   .   .   .   .   .   .  .  .  .  .  4 7

o18 : BettiTally
i19 : puT.dd_3_{0}

o19 = {1, 1} | -e_(1,1) |
      {1, 1} | 0        |
      {1, 1} | -e_(1,0) |
      {1, 1} | 0        |
      {2, 0} | 0        |
      {0, 2} | e_(0,0)  |
      {3, 0} | 0        |
      {3, 0} | 0        |
      {3, 0} | 0        |
      {3, 0} | 0        |
      {0, 3} | 0        |
      {0, 3} | 0        |
      {0, 3} | 0        |
      {3, 0} | 0        |
      {3, 0} | 0        |
      {0, 3} | 0        |
      {0, 3} | 0        |
      {1, 2} | -1       |
      {0, 3} | 0        |

              19       1
o19 : Matrix E   <--- E

Ways to use pushAboveWindow :