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
|