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).
i1 : n={1,1};
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i2 : (S,E) = productOfProjectiveSpaces n; |
i3 : T1 = (dual res trim (ideal vars E)^2)[1]; |
i4 : a=-{2,2};
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i5 : T2=T1**E^{a}[sum a];
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i6 : W=beilinsonWindow T2
15 16 4
o6 = E <-- E <-- E
0 1 2
o6 : ChainComplex
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i7 : cohomologyMatrix(W,-2*n,2*n)
o7 = | 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
o7 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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i8 : T=tateExtension W
1462 1189 954 754 586 447 334 244 174 121 82 54 35 20 10 7
o8 = 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
o8 : ChainComplex
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i9 : cohomologyMatrix(T,-3*n,4*n)
o9 = | 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
o9 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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i10 : cohomologyMatrix(beilinsonWindow T,-n,n)
o10 = | 0 0 0 |
| 8 15 0 |
| 4 8 0 |
3 3
o10 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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i11 : cohomologyMatrix(T,-5*n,4*n) -- the view including the corner
o11 = | 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
o11 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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Note that the Beilinson window of tateExtension of the beilinson window W is not equal but just isomorphic to the original W.
The implicit bounds in the computation are only a guess and certainly not optimal. This should be improved.