The call
tateResolution(M,low,high)
forms the 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
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i2 : low = {-3,-3};high = {3,3};
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i4 : T=tateResolution( S^{{1,1}},low, high);
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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])
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The complex contains some trailing terms and superflous 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])
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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
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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
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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
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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])
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