Under the assumption that T is part of a Tate resolution of a sheaf F on a product of two projective space Pn1 x Pn2, the function returns a matrix of cohomology polynomials
for every c=(c1,c2) with a1 ≤c1 ≤b1 and a2 ≤c2 ≤b2. In case T corresponds to an object in the derived category Db(Pn1x Pn2), then hypercohomology polynomials are returned, with the convention that k stands for k=h -1.
If T is not a large enough part of the Tate resolution, such as W below, then the function collects only the contribution of T to the cohomology table of the Tate resolution, according to the formula in Corollary 0.2 of Tate Resolutions on Products of Projective Spaces.
The polynomial for (b1,b2) sits in the north-east corner, the one corresponding to (a1,a2) in the south-west corner.
i1 : n={1,2};kk=ZZ/101;
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i3 : (S,E)=setupRings(ZZ/101,n); |
i4 : a={1,1}; U=E^{ -a};
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i6 : W=(chainComplex {map(E^0,U,0),map(U,E^0,0)})[1]
1
o6 = 0 <-- E <-- 0
-1 0 1
o6 : ChainComplex
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i7 : cohomologyTable(W,-{3,3},{3,3})
o7 = | 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 h2 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
7 7
o7 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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i8 : time T=sloppyTateExtension W
-- used 6.00033 seconds
4331 3515 2839 2282 1825 1451 1145 894 687 515 371 251 156 92 68 90 125 132
o8 = E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
o8 : ChainComplex
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i9 : cohomologyTable(T,-{3,3},{3,3})
o9 = | 45h 30h 15h 0 15 30 45 |
| 24h 16h 8h 0 8 16 24 |
| 9h 6h 3h 0 3 6 9 |
| 0 0 0 0 0 0 0 |
| 3h2 2h2 h2 0 h 2h 3h |
| 0 0 0 0 0 0 0 |
| 9h3 6h3 3h3 0 3h2 6h2 9h2 |
7 7
o9 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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i10 : cohomologyTable(T,-{3,4},{3,3})
o10 = | 45h 30h 15h 0 15 30 45 |
| 24h 16h 8h 0 8 16 24 |
| 9h 6h 3h 0 3 6 9 |
| 0 0 0 0 0 0 0 |
| 3h2 2h2 h2 0 h 2h 3h |
| 0 0 0 0 0 0 0 |
| 9h3 6h3 3h3 0 3h2 6h2 9h2 |
| 24h3 16h3 8h3 0 8h2 16h2 24h2 |
8 7
o10 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|