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

## Synopsis

• Usage:
T=tateExtension W
• Inputs:
• W, , terms in the Beilinson window of a Tate resolution
• Outputs:
• T, , a corner complex of the corresponding Tate resolution

## Description

Every object F in the derived category D^d(P) of coherent sheaves on a product P=P^{n_1}x..xP^{n_t} 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 chosen corner complex of the Tate resolution T(F).

 i1 : n={1,1}; i2 : (S,E) = productOfProjectiveSpaces n; i3 : T1 = (dual res trim (ideal vars E)^2)[1]; i4 : a=-{2,2}; i5 : T2=T1**E^{a}[sum a]; i6 : W=beilinsonWindow T2 15 16 4 o6 = E <-- E <-- E 0 1 2 o6 : ChainComplex 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]) 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 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]) 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]) 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])

## Caveat

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.