The call
cornerComplex(T,c)
forms the corner complex with corner c of a (part of a) Tate resolution T as defined in Tate Resolutions on Products of Projective Spaces. The call
cornerComplex(M,c,low,high)
first computes the Tate resolution T of the sheaf F represented by M in the range covering low to high and then takes the corner complex of T.
i1 : (S,E) = productOfProjectiveSpaces{1,1}
o1 = (S, E)
o1 : Sequence
|
i2 : low = {-4,-4};high = {3,2};
|
i4 : T1= (dual res( trim (ideal vars E)^2,LengthLimit=>8))[1]; |
i5 : T2=res(coker upperCorner(T1,{4,3}),LengthLimit=>13)[7];
|
Finally, we can define T, the sufficient part of the Tate resolution:
i6 : T=trivialHomologicalTruncation (T2,-5,6); |
i7 : cohomologyMatrix(T,low,high)
o7 = | 27h 20h 13h 6h 1 8 15 22 |
| 16h 12h 8h 4h 0 4 8 12 |
| 5h 4h 3h 2h h 0 1 2 |
| 6h2 4h2 2h2 0 2h 4h 6h 8h |
| 17h2 12h2 7h2 2h2 3h 8h 13h 18h |
| 28h2 20h2 12h2 4h2 4h 12h 20h 28h |
| 39h2 28h2 17h2 6h2 5h 16h 27h 38h |
7 8
o7 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
In the following we will produce a corner complex cT with corner at c ={-2,-1}. To do this we need a big enough part T of a Tate resolution so that all the strands around the corner are exact. This example corresponds to the Example of Section 4 of our paper referenced above. The Tate resolution in question is that corresponding to a rank 3 natural sheaf on P1xP1.
i8 : c = -{2,1};
|
i9 : cT=cornerComplex(T,c); |
i10 : betti cT
-5 -4 -3 -2 -1 0 1 2 3 4 5
o10 = total: 22 27 26 18 22 12 5 12 37 78 138
0: 22 27 18 6 . . . . . . .
1: . . 8 12 22 12 3 . . . .
2: . . . . . . 2 . . . .
3: . . . . . . . 12 37 78 138
o10 : BettiTally
|
i11 : cohomologyMatrix(cT,low,high)
o11 = | 0 0 13h 6h 1 8 15 22 |
| 0 0 8h 4h 0 4 8 12 |
| 0 0 3h 2h h 0 1 2 |
| 0 0 2h2 0 2h 4h 6h 8h |
| 17h3 12h3 0 0 0 0 0 0 |
| 28h3 20h3 0 0 0 0 0 0 |
| 39h3 28h3 0 0 0 0 0 0 |
7 8
o11 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
The corner complex is built from a first quadrant complex fqT and a last quadrant complex lqT connected by the corner map between these complexes.
i12 : fqT=firstQuadrantComplex(T,c); |
i13 : lqT=lastQuadrantComplex(T,c); |
i14 : cohomologyMatrix(fqT,low,high)
o14 = | 0 0 13h 6h 1 8 15 22 |
| 0 0 8h 4h 0 4 8 12 |
| 0 0 3h 2h h 0 1 2 |
| 0 0 2h2 0 2h 4h 6h 8h |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
7 8
o14 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i15 : cohomologyMatrix(lqT,low,high)
o15 = | 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 |
| 17h2 12h2 0 0 0 0 0 0 |
| 28h2 20h2 0 0 0 0 0 0 |
| 39h2 28h2 0 0 0 0 0 0 |
7 8
o15 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i16 : betti fqT
-5 -4 -3 -2 -1 0 1
o16 = total: 22 27 26 18 22 12 5
0: 22 27 18 6 . . .
1: . . 8 12 22 12 3
2: . . . . . . 2
o16 : BettiTally
|
i17 : betti lqT
3 4 5 6
o17 = total: 12 37 78 138
2: 12 37 78 138
o17 : BettiTally
|
i18 : betti cT
-5 -4 -3 -2 -1 0 1 2 3 4 5
o18 = total: 22 27 26 18 22 12 5 12 37 78 138
0: 22 27 18 6 . . . . . . .
1: . . 8 12 22 12 3 . . . .
2: . . . . . . 2 . . . .
3: . . . . . . . 12 37 78 138
o18 : BettiTally
|
Here the corner map is cT.dd2
i19 : betti (cT.dd_(-sum c-1))
0 1
o19 = total: 5 12
2: 3 .
3: 2 .
4: . 12
o19 : BettiTally
|
In general the corner map is a chain complex map from lqT to fqT spread over several homological degrees.
Putting the corner in c = {-1,-1 } we get a different picture:
i20 : c = {-1,-1}
o20 = {-1, -1}
o20 : List
|
i21 : cT=cornerComplex(T,c); |
i22 : betti cT
-5 -4 -3 -2 -1 0 1 2 3 4 5
o22 = total: 22 27 26 18 9 4 7 24 54 100 165
0: 22 27 18 6 . . . . . . .
1: . . 8 12 9 4 . . . . .
2: . . . . . . . . . . .
3: . . . . . . 7 24 54 100 165
o22 : BettiTally
|
i23 : cohomologyMatrix(cT,low,high)
o23 = | 0 0 0 6h 1 8 15 22 |
| 0 0 0 4h 0 4 8 12 |
| 0 0 0 2h h 0 1 2 |
| 0 0 0 0 2h 4h 6h 8h |
| 17h3 12h3 7h3 0 0 0 0 0 |
| 28h3 20h3 12h3 0 0 0 0 0 |
| 39h3 28h3 17h3 0 0 0 0 0 |
7 8
o23 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
The corner complex is built from a first quadrant complex fqT and a last quadrant complex lqT connected by the corner map between these complexes.
i24 : fqT=firstQuadrantComplex(T,c); |
i25 : lqT=lastQuadrantComplex(T,c); |
i26 : cohomologyMatrix(fqT,low,high)
o26 = | 0 0 0 6h 1 8 15 22 |
| 0 0 0 4h 0 4 8 12 |
| 0 0 0 2h h 0 1 2 |
| 0 0 0 0 2h 4h 6h 8h |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
7 8
o26 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i27 : cohomologyMatrix(lqT,low,high)
o27 = | 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 |
| 17h2 12h2 7h2 0 0 0 0 0 |
| 28h2 20h2 12h2 0 0 0 0 0 |
| 39h2 28h2 17h2 0 0 0 0 0 |
7 8
o27 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i28 : betti fqT
-5 -4 -3 -2 -1 0
o28 = total: 22 27 26 18 9 4
0: 22 27 18 6 . .
1: . . 8 12 9 4
o28 : BettiTally
|
i29 : betti lqT
2 3 4 5 6
o29 = total: 7 24 54 100 165
2: 7 24 54 100 165
o29 : BettiTally
|
i30 : betti cT
-5 -4 -3 -2 -1 0 1 2 3 4 5
o30 = total: 22 27 26 18 9 4 7 24 54 100 165
0: 22 27 18 6 . . . . . . .
1: . . 8 12 9 4 . . . . .
2: . . . . . . . . . . .
3: . . . . . . 7 24 54 100 165
o30 : BettiTally
|
Here the corner map is cT.dd1
i31 : betti (cT.dd_1)
0 1
o31 = total: 4 7
1: 4 .
2: . .
3: . 7
o31 : BettiTally
|
In general the corner map is a chain complex map from lqT to fqT spread over several homological degrees.
Next we give an example obtained from a module
i32 : (S,E)=productOfProjectiveSpaces{2,1}
o32 = (S, E)
o32 : Sequence
|
i33 : M=beilinson(E^{-{1,1}})
o33 = cokernel {1, 1} | x_(0,2) |
{1, 1} | -x_(0,1) |
{1, 1} | x_(0,0) |
3
o33 : S-module, quotient of S
|
i34 : c={1,1}
o34 = {1, 1}
o34 : List
|
i35 : low={-3,-3},high={4,4}
o35 = ({-3, -3}, {4, 4})
o35 : Sequence
|
i36 : cohomologyMatrix(M,low,high)
o36 = | 12h2 0 4h 0 12 32 60 96 |
| 9h2 0 3h 0 9 24 45 72 |
| 6h2 0 2h 0 6 16 30 48 |
| 3h2 0 h 0 3 8 15 24 |
| 0 0 0 0 0 0 0 0 |
| 3h3 0 h2 0 3h 8h 15h 24h |
| 6h3 0 2h2 0 6h 16h 30h 48h |
| 9h3 0 3h2 0 9h 24h 45h 72h |
8 8
o36 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i37 : C=cornerComplex(M,c,low,high)
96 132 125 90 40 14 3 1 5 17 44 95 181 315
o37 = E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
o37 : ChainComplex
|
i38 : cohomologyMatrix(C,low,high)
o38 = | 0 0 0 0 12 32 60 96 |
| 0 0 0 0 9 24 45 72 |
| 0 0 0 0 6 16 30 48 |
| 0 0 0 0 3 8 15 24 |
| 0 0 0 0 0 0 0 0 |
| 3h4 0 h3 0 0 0 0 0 |
| 6h4 0 2h3 0 0 0 0 0 |
| 9h4 0 3h3 0 0 0 0 0 |
8 8
o38 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i39 : cohomologyMatrix(C,2*low,2*high)
o39 = | 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 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 12 32 60 96 0 0 0 0 |
| 0 0 0 0 0 0 0 9 24 45 72 0 0 0 0 |
| 0 0 0 0 0 0 0 6 16 30 48 0 0 0 0 |
| 0 0 0 0 0 0 0 3 8 15 24 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 24h4 15h4 8h4 3h4 0 h3 0 0 0 0 0 0 0 0 0 |
| 48h4 30h4 16h4 6h4 0 2h3 0 0 0 0 0 0 0 0 0 |
| 72h4 45h4 24h4 9h4 0 3h3 0 0 0 0 0 0 0 0 0 |
| 0 60h4 32h4 12h4 0 4h3 0 0 0 0 0 0 0 0 0 |
| 0 0 40h4 15h4 0 5h3 0 0 0 0 0 0 0 0 0 |
| 0 0 0 18h4 0 6h3 0 0 0 0 0 0 0 0 0 |
15 15
o39 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i40 : betti C
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
o40 = total: 96 132 125 90 40 14 3 1 5 17 44 95 181 315
0: 96 132 125 90 40 14 3 . . . . . . .
1: . . . . . . . . . . . . . .
2: . . . . . . . . . . . . . .
3: . . . . . . . 1 2 3 4 5 6 7
4: . . . . . . . . 3 14 40 90 175 308
o40 : BettiTally
|
i41 : C.dd_(-sum c +1)
o41 = {-1, -1} | e_(0,0)e_(0,1)e_(1,0)e_(1,1) |
{-1, -1} | e_(0,1)e_(0,2)e_(1,0)e_(1,1) |
{-1, -1} | e_(0,0)e_(0,2)e_(1,0)e_(1,1) |
3 1
o41 : Matrix E <--- E
|