Prints the commands which illustrate / test various composition of functions.
i1 : n={1,1}, v=n+{1,1}
o1 = ({1, 1}, {2, 2})
o1 : Sequence
|
i2 : high=3*n, low=-high
o2 = ({3, 3}, {-3, -3})
o2 : Sequence
|
i3 : (S,E)=productOfProjectiveSpaces n o3 = (S, E) o3 : Sequence |
We build the example from Section 4 of the paper Tate Resolutions on Products of Projective Spaces which corresponds to a rank 3 vector bundle on P1xP1.
i4 : P=(image transpose gens trim (ideal vars E)^2)**E^{n}
o4 = image {-1, -3} | -e_(1,0)e_(1,1) |
{-2, -2} | -e_(0,1)e_(1,1) |
{-2, -2} | -e_(0,0)e_(1,1) |
{-2, -2} | -e_(0,1)e_(1,0) |
{-2, -2} | -e_(0,0)e_(1,0) |
{-3, -1} | -e_(0,0)e_(0,1) |
6
o4 : E-module, submodule of E
|
i5 : betti P
0 1
o5 = total: 1 4
-2: 1 .
-1: . .
0: . 4
o5 : BettiTally
|
i6 : LP=bgg P
6 4 1
o6 = S <-- S <-- S
0 1 2
o6 : ChainComplex
|
i7 : M = (HH^0 LP)**S^{-n}
o7 = cokernel {2, 0} | x_(1,1) -x_(1,0) 0 0 |
{1, 1} | -x_(0,0) 0 x_(1,0) 0 |
{1, 1} | 0 -x_(0,0) x_(1,1) 0 |
{1, 1} | -x_(0,1) 0 0 x_(1,0) |
{1, 1} | 0 -x_(0,1) 0 x_(1,1) |
{0, 2} | 0 0 x_(0,1) -x_(0,0) |
6
o7 : S-module, quotient of S
|
i8 : betti res M
0 1 2
o8 = total: 6 4 1
2: 6 4 1
o8 : BettiTally
|
i9 : T = tateResolution(M,low,high)
136 55 32 44 39 36 54 91 136 184 239 304 382 476 589
o9 = E <-- 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 6
o9 : ChainComplex
|
i10 : cohomologyMatrix(T,low,high)
o10 = | 28h 18h 8h 2 12 22 32 |
| 20h 13h 6h 1 8 15 22 |
| 12h 8h 4h 0 4 8 12 |
| 4h 3h 2h h 0 1 2 |
| 4h2 2h2 0 2h 4h 6h 8h |
| 12h2 7h2 2h2 3h 8h 13h 18h |
| 20h2 12h2 4h2 4h 12h 20h 28h |
7 7
o10 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
T is the part of the Tate resolution, which is complete in the range low to high. (In a wider range some terms are missing or are incorrect)
i11 : cohomologyMatrix(T,2*low,2*high)
o11 = | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 68k 0 0 |
| 0 0 0 0 0 0 0 0 0 0 55k 68k 0 |
| 58h 48h 38h 28h 18h 8h 2 12 22 32 0 0 0 |
| 41h 34h 27h 20h 13h 6h 1 8 15 22 0 0 0 |
| 24h 20h 16h 12h 8h 4h 0 4 8 12 0 0 0 |
| 7h 6h 5h 4h 3h 2h h 0 1 2 0 0 0 |
| 10h2 8h2 6h2 4h2 2h2 0 2h 4h 6h 8h 0 0 0 |
| 27h2 22h2 17h2 12h2 7h2 2h2 3h 8h 13h 18h 0 0 0 |
| 0 36h2 28h2 20h2 12h2 4h2 4h 12h 20h 28h 0 0 0 |
| 0 0 39h2 28h2 17h2 6h2 5h 16h 27h 38h 0 0 0 |
| 0 0 0 36h2 22h2 8h2 6h 20h 34h 48h 0 0 0 |
| 0 0 0 0 27h2 10h2 7h 24h 41h 58h 0 0 0 |
13 13
o11 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
Alternatively we can recover M from its Beilinson monad derived from T.
i12 : B=beilinson T
1 4
o12 = S <-- S
-1 0
o12 : ChainComplex
|
i13 : M'=prune HH^0 B
o13 = cokernel {2, 0} | -x_(1,0) 0 0 x_(1,1) |
{1, 1} | x_(0,0) -x_(1,1) 0 0 |
{1, 1} | -x_(0,1) 0 -x_(1,1) 0 |
{0, 2} | 0 x_(0,1) x_(0,0) 0 |
{1, 1} | 0 -x_(1,0) 0 x_(0,0) |
{1, 1} | 0 0 -x_(1,0) -x_(0,1) |
6
o13 : S-module, quotient of S
|
i14 : prune HH^1 B
o14 = cokernel | x_(1,1) x_(1,0) x_(0,1) x_(0,0) |
1
o14 : S-module, quotient of S
|
i15 : isIsomorphic(M,M') o15 = true |
We study the corner complex of T at c={0,0}.
i16 : C=cornerComplex(T,{0,0});
|
i17 : betti C
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
o17 = total: 55 32 44 39 20 6 1 4 15 36 70 120
-1: 55 . . . . . . . . . . .
0: . 32 44 39 20 6 . . . . . .
1: . . . . . . 1 . . . . .
2: . . . . . . . . . . . .
3: . . . . . . . 4 15 36 70 120
o17 : BettiTally
|
i18 : cohomologyMatrix(C,low,high)
o18 = | 0 0 0 2 12 22 32 |
| 0 0 0 1 8 15 22 |
| 0 0 0 0 4 8 12 |
| 0 0 0 h 0 1 2 |
| 4h3 2h3 0 0 0 0 0 |
| 12h3 7h3 2h3 0 0 0 0 |
| 20h3 12h3 4h3 0 0 0 0 |
7 7
o18 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i19 : betti C.dd_0
0 1
o19 = total: 1 4
0: 1 .
1: . .
2: . 4
o19 : BettiTally
|
i20 : P=ker C.dd_0**E^{v}
o20 = image {-1, 0} | e_(1,1) e_(1,0) e_(0,0) 0 0 0 e_(0,1) 0 0 0 0 0 0 0 0 |
{-1, 0} | 0 0 0 e_(1,1) e_(1,0) e_(0,1) 0 e_(0,0) 0 0 0 0 0 0 0 |
{0, -1} | 0 0 0 0 0 0 -e_(1,1) e_(1,1) e_(1,0) e_(0,1) e_(0,0) 0 e_(1,1) 0 0 |
{0, -1} | 0 0 0 0 0 0 0 0 0 0 0 e_(1,1) e_(1,0) e_(0,1) e_(0,0) |
4
o20 : E-module, submodule of E
|
The tensor product with E{v} is necessary because we work with E instead of ωE. M can be recovered by applying the bgg functor to P.
i21 : LP=bgg P; |
i22 : betti LP
-3 -2 -1 0
o22 = total: 4 16 24 15
0: 4 16 24 15
o22 : BettiTally
|
i23 : coLP=apply(toList(min LP..max LP),i->prune HH^(-i) LP); |
i24 : apply(coLP,h->dim h)
o24 = {0, -1, 0, 4}
o24 : List
|
i25 : M1=HH^0 LP
o25 = image {1, -1} | x_(0,0)x_(1,1) -x_(1,1)^2 0 0 x_(1,0)x_(1,1) 0 |
{1, -1} | x_(0,0)x_(1,0) -x_(1,0)x_(1,1) 0 0 x_(1,0)^2 0 |
{1, -1} | x_(0,1)x_(1,1) 0 x_(1,1)^2 0 0 -x_(1,0)x_(1,1) |
{1, -1} | x_(0,1)x_(1,0) 0 x_(1,0)x_(1,1) 0 0 -x_(1,0)^2 |
{0, 0} | x_(0,0)^2 -x_(0,0)x_(1,1) 0 0 x_(0,0)x_(1,0) 0 |
{0, 0} | x_(0,1)^2 0 x_(0,1)x_(1,1) 0 0 -x_(0,1)x_(1,0) |
{0, 0} | x_(0,0)x_(0,1) -x_(0,1)x_(1,1) 0 0 x_(0,1)x_(1,0) 0 |
{0, 0} | x_(0,0)x_(0,1) 0 x_(0,0)x_(1,1) 0 0 -x_(0,0)x_(1,0) |
{0, 0} | 0 x_(0,1)x_(1,0) x_(0,0)x_(1,0) x_(1,0)^2 0 0 |
{0, 0} | 0 0 0 x_(1,1)^2 x_(0,1)x_(1,1) x_(0,0)x_(1,1) |
{0, 0} | 0 0 0 x_(1,0)x_(1,1) x_(0,1)x_(1,0) x_(0,0)x_(1,0) |
{-1, 1} | 0 x_(0,1)^2 x_(0,0)x_(0,1) x_(0,1)x_(1,0) 0 0 |
{-1, 1} | 0 x_(0,0)x_(0,1) x_(0,0)^2 x_(0,0)x_(1,0) 0 0 |
{-1, 1} | 0 0 0 x_(0,1)x_(1,1) x_(0,1)^2 x_(0,0)x_(0,1) |
{-1, 1} | 0 0 0 x_(0,0)x_(1,1) x_(0,0)x_(0,1) x_(0,0)^2 |
15
o25 : S-module, submodule of S
|
i26 : betti M1,betti M
0 1 0 1
o26 = (total: 6 4, total: 6 4)
2: 6 4 2: 6 4
o26 : Sequence
|
i27 : isIsomorphic(M,M1) o27 = true |
It works also for different syzygy modules in the corner complex. It works for all P=ker C.ddk in the range where C.ddk is computed completely. We check the case k=1 and k=-2.
i28 : k=1 o28 = 1 |
i29 : P=ker C.dd_(-k)**E^{v}; betti P
0 1
o30 = total: 4 15
-1: 4 15
o30 : BettiTally
|
i31 : LP=bgg P
1 4
o31 = S <-- S
0 1
o31 : ChainComplex
|
i32 : betti LP
0 1
o32 = total: 1 4
0: 1 4
o32 : BettiTally
|
i33 : coLP=apply(toList(min LP..max LP),i->prune HH^(-i) LP); |
i34 : apply(coLP,h->dim h)
o34 = {0, 4}
o34 : List
|
i35 : M1=HH^(-k) LP
o35 = image {1, 0} | x_(0,1) 0 -x_(1,0) 0 0 x_(1,1) |
{1, 0} | x_(0,0) x_(1,0) 0 0 -x_(1,1) 0 |
{0, 1} | 0 x_(0,1) x_(0,0) x_(1,1) 0 0 |
{0, 1} | 0 0 0 x_(1,0) x_(0,1) x_(0,0) |
4
o35 : S-module, submodule of S
|
i36 : betti M1, betti M
0 1 0 1
o36 = (total: 6 4, total: 6 4)
2: 6 4 2: 6 4
o36 : Sequence
|
i37 : isIsomorphic(M,M1) o37 = true |
Note that we have to take HH(-k) == HHk because of the homological position in which P sits.
i38 : k=-2 o38 = -2 |
i39 : P=ker C.dd_(-k)**E^{v}; betti P
0 1
o40 = total: 70 120
2: 70 120
o40 : BettiTally
|
i41 : LP=bgg P; |
i42 : betti LP
-5 -4 -3 -2
o42 = total: 36 129 160 70
0: 36 129 160 70
o42 : BettiTally
|
i43 : coLP=apply(toList(min LP..max LP),i->prune HH^(-i) LP); |
i44 : apply(coLP,h->dim h)
o44 = {0, -1, 0, 4}
o44 : List
|
i45 : M1=HH^(-k) LP
o45 = image {1, -3} | x_(0,1)x_(1,1)^3 0 -x_(1,0)x_(1,1)^3 0 0 x_(1,1)^4 |
{1, -3} | x_(0,0)x_(1,1)^3 x_(1,0)x_(1,1)^3 0 0 -x_(1,1)^4 0 |
{1, -3} | x_(0,1)x_(1,0)x_(1,1)^2 0 -x_(1,0)^2x_(1,1)^2 0 0 x_(1,0)x_(1,1)^3 |
{1, -3} | x_(0,0)x_(1,0)x_(1,1)^2 x_(1,0)^2x_(1,1)^2 0 0 -x_(1,0)x_(1,1)^3 0 |
{1, -3} | x_(0,1)x_(1,0)^2x_(1,1) 0 -x_(1,0)^3x_(1,1) 0 0 x_(1,0)^2x_(1,1)^2 |
{1, -3} | x_(0,1)x_(1,0)^3 0 -x_(1,0)^4 0 0 x_(1,0)^3x_(1,1) |
{1, -3} | x_(0,0)x_(1,0)^2x_(1,1) x_(1,0)^3x_(1,1) 0 0 -x_(1,0)^2x_(1,1)^2 0 |
{1, -3} | x_(0,0)x_(1,0)^3 x_(1,0)^4 0 0 -x_(1,0)^3x_(1,1) 0 |
{0, -2} | 0 x_(0,1)x_(1,1)^3 x_(0,0)x_(1,1)^3 x_(1,1)^4 0 0 |
{0, -2} | x_(0,0)x_(0,1)x_(1,1)^2 x_(0,1)x_(1,0)x_(1,1)^2 0 0 -x_(0,1)x_(1,1)^3 0 |
{0, -2} | 0 x_(0,1)x_(1,0)x_(1,1)^2 x_(0,0)x_(1,0)x_(1,1)^2 x_(1,0)x_(1,1)^3 0 0 |
{0, -2} | x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,1)x_(1,0)^2x_(1,1) 0 0 -x_(0,1)x_(1,0)x_(1,1)^2 0 |
{0, -2} | x_(0,1)^2x_(1,1)^2 0 -x_(0,1)x_(1,0)x_(1,1)^2 0 0 x_(0,1)x_(1,1)^3 |
{0, -2} | x_(0,0)x_(0,1)x_(1,1)^2 0 -x_(0,0)x_(1,0)x_(1,1)^2 0 0 x_(0,0)x_(1,1)^3 |
{0, -2} | x_(0,0)^2x_(1,1)^2 x_(0,0)x_(1,0)x_(1,1)^2 0 0 -x_(0,0)x_(1,1)^3 0 |
{0, -2} | 0 x_(0,1)x_(1,0)^2x_(1,1) x_(0,0)x_(1,0)^2x_(1,1) x_(1,0)^2x_(1,1)^2 0 0 |
{0, -2} | x_(0,0)x_(0,1)x_(1,0)^2 x_(0,1)x_(1,0)^3 0 0 -x_(0,1)x_(1,0)^2x_(1,1) 0 |
{0, -2} | 0 x_(0,1)x_(1,0)^3 x_(0,0)x_(1,0)^3 x_(1,0)^3x_(1,1) 0 0 |
{0, -2} | 0 0 0 x_(1,0)^4 x_(0,1)x_(1,0)^3 x_(0,0)x_(1,0)^3 |
{0, -2} | x_(0,1)^2x_(1,0)x_(1,1) 0 -x_(0,1)x_(1,0)^2x_(1,1) 0 0 x_(0,1)x_(1,0)x_(1,1)^2 |
{0, -2} | x_(0,1)^2x_(1,0)^2 0 -x_(0,1)x_(1,0)^3 0 0 x_(0,1)x_(1,0)^2x_(1,1) |
{0, -2} | x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 -x_(0,0)x_(1,0)^2x_(1,1) 0 0 x_(0,0)x_(1,0)x_(1,1)^2 |
{0, -2} | x_(0,0)x_(0,1)x_(1,0)^2 0 -x_(0,0)x_(1,0)^3 0 0 x_(0,0)x_(1,0)^2x_(1,1) |
{0, -2} | x_(0,0)^2x_(1,0)x_(1,1) x_(0,0)x_(1,0)^2x_(1,1) 0 0 -x_(0,0)x_(1,0)x_(1,1)^2 0 |
{0, -2} | x_(0,0)^2x_(1,0)^2 x_(0,0)x_(1,0)^3 0 0 -x_(0,0)x_(1,0)^2x_(1,1) 0 |
{-1, -1} | 0 -x_(0,1)^2x_(1,0)x_(1,1) -x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 x_(0,1)^2x_(1,1)^2 x_(0,0)x_(0,1)x_(1,1)^2 |
{-1, -1} | 0 x_(0,1)^2x_(1,1)^2 x_(0,0)x_(0,1)x_(1,1)^2 x_(0,1)x_(1,1)^3 0 0 |
{-1, -1} | 0 -x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)^2x_(1,0)x_(1,1) 0 x_(0,0)x_(0,1)x_(1,1)^2 x_(0,0)^2x_(1,1)^2 |
{-1, -1} | 0 x_(0,0)x_(0,1)x_(1,1)^2 x_(0,0)^2x_(1,1)^2 x_(0,0)x_(1,1)^3 0 0 |
{-1, -1} | 0 -x_(0,1)^2x_(1,0)^2 -x_(0,0)x_(0,1)x_(1,0)^2 0 x_(0,1)^2x_(1,0)x_(1,1) x_(0,0)x_(0,1)x_(1,0)x_(1,1) |
{-1, -1} | 0 0 0 x_(0,1)x_(1,0)^3 x_(0,1)^2x_(1,0)^2 x_(0,0)x_(0,1)x_(1,0)^2 |
{-1, -1} | 0 x_(0,1)^2x_(1,0)x_(1,1) x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,1)x_(1,0)x_(1,1)^2 0 0 |
{-1, -1} | 0 x_(0,1)^2x_(1,0)^2 x_(0,0)x_(0,1)x_(1,0)^2 x_(0,1)x_(1,0)^2x_(1,1) 0 0 |
{-1, -1} | 0 -x_(0,0)x_(0,1)x_(1,0)^2 -x_(0,0)^2x_(1,0)^2 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,0)^2x_(1,0)x_(1,1) |
{-1, -1} | 0 0 0 x_(0,0)x_(1,0)^3 x_(0,0)x_(0,1)x_(1,0)^2 x_(0,0)^2x_(1,0)^2 |
{-1, -1} | 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,0)^2x_(1,0)x_(1,1) x_(0,0)x_(1,0)x_(1,1)^2 0 0 |
{-1, -1} | 0 x_(0,0)x_(0,1)x_(1,0)^2 x_(0,0)^2x_(1,0)^2 x_(0,0)x_(1,0)^2x_(1,1) 0 0 |
{-1, -1} | x_(0,1)^3x_(1,1) 0 -x_(0,1)^2x_(1,0)x_(1,1) 0 0 x_(0,1)^2x_(1,1)^2 |
{-1, -1} | x_(0,1)^3x_(1,0) 0 -x_(0,1)^2x_(1,0)^2 0 0 x_(0,1)^2x_(1,0)x_(1,1) |
{-1, -1} | x_(0,0)x_(0,1)^2x_(1,1) 0 -x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 0 x_(0,0)x_(0,1)x_(1,1)^2 |
{-1, -1} | x_(0,0)x_(0,1)^2x_(1,0) 0 -x_(0,0)x_(0,1)x_(1,0)^2 0 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) |
{-1, -1} | x_(0,0)^2x_(0,1)x_(1,1) 0 -x_(0,0)^2x_(1,0)x_(1,1) 0 0 x_(0,0)^2x_(1,1)^2 |
{-1, -1} | x_(0,0)^2x_(0,1)x_(1,0) 0 -x_(0,0)^2x_(1,0)^2 0 0 x_(0,0)^2x_(1,0)x_(1,1) |
{-1, -1} | x_(0,0)^3x_(1,1) x_(0,0)^2x_(1,0)x_(1,1) 0 0 -x_(0,0)^2x_(1,1)^2 0 |
{-1, -1} | x_(0,0)^3x_(1,0) x_(0,0)^2x_(1,0)^2 0 0 -x_(0,0)^2x_(1,0)x_(1,1) 0 |
{-2, 0} | x_(0,1)^4 0 -x_(0,1)^3x_(1,0) 0 0 x_(0,1)^3x_(1,1) |
{-2, 0} | 0 0 0 x_(0,1)^2x_(1,0)x_(1,1) x_(0,1)^3x_(1,1) x_(0,0)x_(0,1)^2x_(1,1) |
{-2, 0} | 0 0 0 x_(0,1)^2x_(1,0)^2 x_(0,1)^3x_(1,0) x_(0,0)x_(0,1)^2x_(1,0) |
{-2, 0} | x_(0,0)x_(0,1)^3 0 -x_(0,0)x_(0,1)^2x_(1,0) 0 0 x_(0,0)x_(0,1)^2x_(1,1) |
{-2, 0} | x_(0,0)^2x_(0,1)^2 0 -x_(0,0)^2x_(0,1)x_(1,0) 0 0 x_(0,0)^2x_(0,1)x_(1,1) |
{-2, 0} | x_(0,0)^3x_(0,1) 0 -x_(0,0)^3x_(1,0) 0 0 x_(0,0)^3x_(1,1) |
{-2, 0} | x_(0,0)^4 x_(0,0)^3x_(1,0) 0 0 -x_(0,0)^3x_(1,1) 0 |
{-2, 0} | 0 0 0 x_(0,0)^2x_(1,0)x_(1,1) x_(0,0)^2x_(0,1)x_(1,1) x_(0,0)^3x_(1,1) |
{-2, 0} | 0 0 0 x_(0,0)^2x_(1,0)^2 x_(0,0)^2x_(0,1)x_(1,0) x_(0,0)^3x_(1,0) |
{-2, 0} | 0 x_(0,1)^3x_(1,1) x_(0,0)x_(0,1)^2x_(1,1) x_(0,1)^2x_(1,1)^2 0 0 |
{-2, 0} | 0 x_(0,1)^3x_(1,0) x_(0,0)x_(0,1)^2x_(1,0) x_(0,1)^2x_(1,0)x_(1,1) 0 0 |
{-2, 0} | 0 x_(0,0)^2x_(0,1)x_(1,1) x_(0,0)^3x_(1,1) x_(0,0)^2x_(1,1)^2 0 0 |
{-2, 0} | 0 x_(0,0)^2x_(0,1)x_(1,0) x_(0,0)^3x_(1,0) x_(0,0)^2x_(1,0)x_(1,1) 0 0 |
{-2, 0} | 0 x_(0,0)x_(0,1)^2x_(1,1) x_(0,0)^2x_(0,1)x_(1,1) x_(0,0)x_(0,1)x_(1,1)^2 0 0 |
{-2, 0} | 0 x_(0,0)x_(0,1)^2x_(1,0) x_(0,0)^2x_(0,1)x_(1,0) x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 0 |
{-2, 0} | 0 0 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) x_(0,0)x_(0,1)^2x_(1,1) x_(0,0)^2x_(0,1)x_(1,1) |
{-2, 0} | 0 0 0 x_(0,0)x_(0,1)x_(1,0)^2 x_(0,0)x_(0,1)^2x_(1,0) x_(0,0)^2x_(0,1)x_(1,0) |
{-3, 1} | 0 0 0 x_(0,1)^3x_(1,0) x_(0,1)^4 x_(0,0)x_(0,1)^3 |
{-3, 1} | 0 0 0 x_(0,0)^3x_(1,0) x_(0,0)^3x_(0,1) x_(0,0)^4 |
{-3, 1} | 0 x_(0,1)^4 x_(0,0)x_(0,1)^3 x_(0,1)^3x_(1,1) 0 0 |
{-3, 1} | 0 x_(0,0)^3x_(0,1) x_(0,0)^4 x_(0,0)^3x_(1,1) 0 0 |
{-3, 1} | 0 x_(0,0)x_(0,1)^3 x_(0,0)^2x_(0,1)^2 x_(0,0)x_(0,1)^2x_(1,1) 0 0 |
{-3, 1} | 0 x_(0,0)^2x_(0,1)^2 x_(0,0)^3x_(0,1) x_(0,0)^2x_(0,1)x_(1,1) 0 0 |
{-3, 1} | 0 0 0 x_(0,0)x_(0,1)^2x_(1,0) x_(0,0)x_(0,1)^3 x_(0,0)^2x_(0,1)^2 |
{-3, 1} | 0 0 0 x_(0,0)^2x_(0,1)x_(1,0) x_(0,0)^2x_(0,1)^2 x_(0,0)^3x_(0,1) |
70
o45 : S-module, submodule of S
|
i46 : betti M1,betti M
0 1 0 1
o46 = (total: 6 4, total: 6 4)
2: 6 4 2: 6 4
o46 : Sequence
|
i47 : isIsomorphic(M,M1) o47 = true |
Next we check the functor bgg on S-modules.
i48 : RM=bgg M
216 140 84 45 20 6
o48 = E <-- E <-- E <-- E <-- E <-- E
-7 -6 -5 -4 -3 -2
o48 : ChainComplex
|
i49 : cohomologyMatrix(RM,low,high)
o49 = | 0 0 0 2 12 22 32 |
| 0 0 0 1 8 15 22 |
| 0 0 0 0 4 8 12 |
| 0 0 0 0 0 1 2 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
7 7
o49 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i50 : betti RM
-7 -6 -5 -4 -3 -2
o50 = total: 216 140 84 45 20 6
0: 216 140 84 45 20 6
o50 : BettiTally
|
i51 : uQ=firstQuadrantComplex(T,{0,0});
|
i52 : cohomologyMatrix(uQ,low,high)
o52 = | 0 0 0 2 12 22 32 |
| 0 0 0 1 8 15 22 |
| 0 0 0 0 4 8 12 |
| 0 0 0 h 0 1 2 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
7 7
o52 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
The additional entry h in the zero position of the cohomology matrix of uQ indicates that we can recover the original square of the maximal ideal of E from the differential of of the first quadrant complex uQ in this specific case.
i53 : uQ.dd_(-1)
o53 = {-2, 0} | e_(0,0)e_(0,1) |
{-1, -1} | -e_(0,1)e_(1,1) |
{-1, -1} | e_(0,0)e_(1,1) |
{-1, -1} | -e_(0,1)e_(1,0) |
{-1, -1} | -e_(0,0)e_(1,0) |
{0, -2} | e_(1,0)e_(1,1) |
6 1
o53 : Matrix E <--- E
|
Next we test reciprocity.
i54 : T1=tateResolution(M,low,3*high); |
i55 : c={2,2}
o55 = {2, 2}
o55 : List
|
i56 : CM=cornerComplex(T1,c); |
i57 : RMc=firstQuadrantComplex(T1,c); |
i58 : cohomologyMatrix(CM,low,3*high)
o58 = | 0 0 0 0 0 64 92 120 148 176 204 232 260 |
| 0 0 0 0 0 57 82 107 132 157 182 207 232 |
| 0 0 0 0 0 50 72 94 116 138 160 182 204 |
| 0 0 0 0 0 43 62 81 100 119 138 157 176 |
| 0 0 0 0 0 36 52 68 84 100 116 132 148 |
| 0 0 0 0 0 29 42 55 68 81 94 107 120 |
| 0 0 0 0 0 22 32 42 52 62 72 82 92 |
| 0 0 0 0 0 15 22 29 36 43 50 57 64 |
| 12h2 8h2 4h2 0 4h 0 0 0 0 0 0 0 0 |
| 4h2 3h2 2h2 h2 0 0 0 0 0 0 0 0 0 |
| 4h3 2h3 0 2h2 4h2 0 0 0 0 0 0 0 0 |
| 12h3 7h3 2h3 3h2 8h2 0 0 0 0 0 0 0 0 |
| 20h3 12h3 4h3 4h2 12h2 0 0 0 0 0 0 0 0 |
13 13
o58 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i59 : coRMc=apply(toList(-10..-4),i-> HH^(-i) RMc==0)
o59 = {true, true, true, true, true, true, false}
o59 : List
|
i60 : P1=ker CM.dd_(-sum c)
o60 = image {-2, -2} | -e_(0,0)e_(1,1) 0 0 0 |
{-2, -2} | -e_(0,0)e_(1,0) 0 e_(0,0)e_(1,1) 0 |
{-2, -2} | 0 0 e_(0,0)e_(1,0) 0 |
{-2, -2} | e_(0,1)e_(1,1) 0 0 0 |
{-2, -2} | e_(0,0)e_(1,1) e_(0,1)e_(1,1) 0 0 |
{-2, -2} | 0 e_(0,0)e_(1,1) 0 0 |
{-2, -2} | -e_(0,1)e_(1,0) 0 e_(0,1)e_(1,1) 0 |
{-2, -2} | e_(0,0)e_(1,0) e_(0,1)e_(1,0) 0 0 |
{-2, -2} | 0 e_(0,0)e_(1,0) 0 0 |
{-2, -2} | 0 0 e_(0,1)e_(1,1) 0 |
{-2, -2} | 0 0 e_(0,0)e_(1,1) e_(0,1)e_(1,1) |
{-2, -2} | 0 0 0 e_(0,0)e_(1,1) |
{-2, -2} | 0 0 e_(0,1)e_(1,0) 0 |
{-2, -2} | 0 0 e_(0,0)e_(1,0) e_(0,1)e_(1,0) |
{-2, -2} | 0 0 0 e_(0,0)e_(1,0) |
15
o60 : E-module, submodule of E
|
i61 : LP=bgg (P1**E^{-c+v})
15 16 4
o61 = S <-- S <-- S
0 1 2
o61 : ChainComplex
|
i62 : betti LP
0 1 2
o62 = total: 15 16 4
0: 15 16 4
o62 : BettiTally
|
i63 : coLP=apply(toList(min LP..max LP),i->dim HH^(-i) LP)
o63 = {4, -1, -1}
o63 : List
|
Hence both Lp and RMc are azyclic.
i64 : Mc=prune truncate(c,M)**S^{c}
o64 = cokernel | x_(1,0) 0 0 0 -x_(0,1) 0 0 0 0 0 0 0 0 0 0 0 |
| -x_(1,1) x_(1,0) 0 0 0 0 0 -x_(0,1) 0 0 x_(0,1) 0 0 0 0 0 |
| 0 -x_(1,1) 0 0 0 0 0 0 0 0 0 0 0 x_(0,1) 0 0 |
| 0 0 x_(1,0) 0 x_(0,0) 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 -x_(1,1) 0 0 0 0 x_(0,0) 0 0 0 0 0 0 0 0 |
| 0 0 0 x_(1,0) 0 0 0 0 0 0 x_(0,0) 0 0 0 0 0 |
| 0 0 0 -x_(1,1) 0 0 0 0 0 0 0 0 0 x_(0,0) 0 0 |
| 0 0 0 0 x_(0,1) x_(0,0) x_(1,0) 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 -x_(0,1) 0 0 x_(1,0) 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 -x_(1,1) x_(0,1) 0 x_(0,0) 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 -x_(1,1) -x_(0,1) 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 x_(0,1) x_(0,0) x_(1,0) 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 -x_(0,1) 0 0 x_(1,0) 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 -x_(1,1) x_(0,1) 0 x_(0,0) |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_(1,1) -x_(0,1) |
15
o64 : S-module, quotient of S
|
i65 : betti (Mc'=HH^0 LP), betti Mc
0 1 0 1
o65 = (total: 15 16, total: 15 16)
0: 15 16 0: 15 16
o65 : Sequence
|
i66 : isIsomorphic(Mc',Mc) o66 = true |
i67 : c={3,1}
o67 = {3, 1}
o67 : List
|
i68 : cohomologyMatrix(T1,low,2*high)
o68 = | 52h 33h 14h 5 24 43 62 81 100 119 |
| 44h 28h 12h 4 20 36 52 68 84 100 |
| 36h 23h 10h 3 16 29 42 55 68 81 |
| 28h 18h 8h 2 12 22 32 42 52 62 |
| 20h 13h 6h 1 8 15 22 29 36 43 |
| 12h 8h 4h 0 4 8 12 16 20 24 |
| 4h 3h 2h h 0 1 2 3 4 5 |
| 4h2 2h2 0 2h 4h 6h 8h 10h 12h 14h |
| 12h2 7h2 2h2 3h 8h 13h 18h 23h 28h 33h |
| 20h2 12h2 4h2 4h 12h 20h 28h 36h 44h 52h |
10 10
o68 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i69 : CM=cornerComplex(T1,c); |
i70 : cohomologyMatrix(CM,low,3*high)
o70 = | 0 0 0 0 0 0 92 120 148 176 204 232 260 |
| 0 0 0 0 0 0 82 107 132 157 182 207 232 |
| 0 0 0 0 0 0 72 94 116 138 160 182 204 |
| 0 0 0 0 0 0 62 81 100 119 138 157 176 |
| 0 0 0 0 0 0 52 68 84 100 116 132 148 |
| 0 0 0 0 0 0 42 55 68 81 94 107 120 |
| 0 0 0 0 0 0 32 42 52 62 72 82 92 |
| 0 0 0 0 0 0 22 29 36 43 50 57 64 |
| 0 0 0 0 0 0 12 16 20 24 28 32 36 |
| 4h2 3h2 2h2 h2 0 h 0 0 0 0 0 0 0 |
| 4h3 2h3 0 2h2 4h2 6h2 0 0 0 0 0 0 0 |
| 12h3 7h3 2h3 3h2 8h2 13h2 0 0 0 0 0 0 0 |
| 20h3 12h3 4h3 4h2 12h2 20h2 0 0 0 0 0 0 0 |
13 13
o70 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i71 : RMc=firstQuadrantComplex(T1,c); |
i72 : coRMc=apply(toList(-9..-4),i-> HH^(-i) RMc==0)
o72 = {true, true, true, true, true, false}
o72 : List
|
i73 : P1=ker CM.dd_(-sum c)
o73 = image {-3, -1} | e_(0,1)e_(1,1) -e_(0,0)e_(1,0)e_(1,1) 0 0 0 0 0 |
{-3, -1} | e_(0,0)e_(1,1) 0 0 0 0 0 0 |
{-3, -1} | e_(0,1)e_(1,0) 0 0 0 e_(0,0)e_(1,0)e_(1,1) 0 0 |
{-3, -1} | e_(0,0)e_(1,0) 0 0 0 0 0 0 |
{-3, -1} | 0 -e_(0,1)e_(1,0)e_(1,1) 0 0 0 0 0 |
{-3, -1} | 0 e_(0,0)e_(1,0)e_(1,1) e_(0,1)e_(1,0)e_(1,1) 0 0 0 0 |
{-3, -1} | 0 0 e_(0,0)e_(1,0)e_(1,1) e_(0,1)e_(1,0)e_(1,1) 0 0 0 |
{-3, -1} | 0 0 0 e_(0,0)e_(1,0)e_(1,1) 0 0 0 |
{-3, -1} | 0 0 0 0 e_(0,1)e_(1,0)e_(1,1) 0 0 |
{-3, -1} | 0 0 0 0 e_(0,0)e_(1,0)e_(1,1) e_(0,1)e_(1,0)e_(1,1) 0 |
{-3, -1} | 0 0 0 0 0 e_(0,0)e_(1,0)e_(1,1) e_(0,1)e_(1,0)e_(1,1) |
{-3, -1} | 0 0 0 0 0 0 e_(0,0)e_(1,0)e_(1,1) |
12
o73 : E-module, submodule of E
|
i74 : LP=bgg (P1**E^{-c+v})
12 10 1
o74 = S <-- S <-- S
0 1 2
o74 : ChainComplex
|
i75 : betti LP
0 1 2
o75 = total: 12 10 1
0: 12 10 1
o75 : BettiTally
|
i76 : coLP=apply(toList(min LP..max LP),i->dim HH^(-i) LP)
o76 = {4, -1, -1}
o76 : List
|
i77 : Mc=prune truncate(c,M)**S^{c}
o77 = cokernel | x_(0,0) x_(1,0) 0 0 -x_(0,1) 0 0 0 0 0 |
| -x_(0,1) 0 x_(1,0) 0 0 0 0 0 0 0 |
| 0 -x_(1,1) 0 x_(0,0) 0 0 0 x_(0,1) 0 0 |
| 0 0 -x_(1,1) -x_(0,1) 0 0 0 0 0 0 |
| 0 0 0 0 x_(0,0) 0 0 0 0 0 |
| 0 0 0 0 0 0 0 x_(0,0) 0 0 |
| 0 0 0 0 x_(0,1) x_(0,0) 0 0 0 0 |
| 0 0 0 0 0 -x_(0,1) x_(0,0) 0 0 0 |
| 0 0 0 0 0 0 -x_(0,1) 0 0 0 |
| 0 0 0 0 0 0 0 x_(0,1) x_(0,0) 0 |
| 0 0 0 0 0 0 0 0 -x_(0,1) x_(0,0) |
| 0 0 0 0 0 0 0 0 0 -x_(0,1) |
12
o77 : S-module, quotient of S
|
i78 : betti (Mc'=HH^0 LP), betti Mc
0 1 0 1
o78 = (total: 12 10, total: 12 10)
0: 12 10 0: 12 10
o78 : Sequence
|
i79 : isIsomorphic(Mc',Mc) o79 = true |
Now we test tateExtension.
i80 : W=beilinsonWindow T
1 4
o80 = E <-- E
-1 0
o80 : ChainComplex
|
i81 : T'=tateExtension W
1250 1044 867 716 588 480 389 312 246 189 136 88 55 46 54
o81 = E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
o81 : ChainComplex
|
i82 : comT'=cohomologyMatrix(T',low,high)
o82 = | 28h 18h 8h 2 12 22 32 |
| 20h 13h 6h 1 8 15 22 |
| 12h 8h 4h 0 4 8 12 |
| 4h 3h 2h h 0 1 2 |
| 4h2 2h2 0 2h 4h 6h 8h |
| 12h2 7h2 2h2 3h 8h 13h 18h |
| 20h2 12h2 4h2 4h 12h 20h 28h |
7 7
o82 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i83 : comT=cohomologyMatrix(T,low,high)
o83 = | 28h 18h 8h 2 12 22 32 |
| 20h 13h 6h 1 8 15 22 |
| 12h 8h 4h 0 4 8 12 |
| 4h 3h 2h h 0 1 2 |
| 4h2 2h2 0 2h 4h 6h 8h |
| 12h2 7h2 2h2 3h 8h 13h 18h |
| 20h2 12h2 4h2 4h 12h 20h 28h |
7 7
o83 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i84 : assert(sub(comT',vars ring comT)==comT) |
Finally we illustrate how shifting the Beilinson window works.
i85 : cohomologyMatrix(T,low,high)
o85 = | 28h 18h 8h 2 12 22 32 |
| 20h 13h 6h 1 8 15 22 |
| 12h 8h 4h 0 4 8 12 |
| 4h 3h 2h h 0 1 2 |
| 4h2 2h2 0 2h 4h 6h 8h |
| 12h2 7h2 2h2 3h 8h 13h 18h |
| 20h2 12h2 4h2 4h 12h 20h 28h |
7 7
o85 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i86 : cohomologyMatrix(beilinsonWindow T,low, high)
o86 = | 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 2h h 0 0 0 |
| 0 0 0 2h 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
7 7
o86 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i87 : B = beilinson T
1 4
o87 = S <-- S
-1 0
o87 : ChainComplex
|
i88 : d={2,2}
o88 = {2, 2}
o88 : List
|
i89 : T1=T**E^{d}[sum d]
136 55 32 44 39 36 54 91 136 184 239 304 382 476 589
o89 = E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
o89 : ChainComplex
|
i90 : cohomologyMatrix(beilinsonWindow T1,low,high)
o90 = | 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 12h2 7h2 0 0 0 |
| 0 0 20h2 12h2 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
7 7
o90 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i91 : B1 =beilinson T1
7 24 20
o91 = S <-- S <-- S
-2 -1 0
o91 : ChainComplex
|
i92 : decompose annihilator HH^1 B1
o92 = {ideal (x , x ), ideal (x , x )}
0,1 0,0 1,1 1,0
o92 : List
|
i93 : decompose annihilator HH^2 B1
o93 = {ideal (x , x , x , x )}
1,1 1,0 0,1 0,0
o93 : List
|
i94 : M1=HH^0 B1
o94 = image {1, 1} | x_(0,1)x_(1,0)x_(1,1)^2 x_(0,0)x_(0,1)x_(1,1)^2 0 x_(0,1)^2x_(1,1)^2 0 0 |
{1, 1} | x_(0,1)x_(1,1)^3 0 x_(0,0)x_(0,1)x_(1,1)^2 0 x_(0,1)^2x_(1,1)^2 0 |
{1, 1} | 0 x_(0,0)^2x_(1,1)^2 -x_(0,0)^2x_(1,0)x_(1,1) 0 0 x_(0,0)^2x_(0,1)x_(1,1) |
{1, 1} | x_(0,0)x_(1,1)^3 0 x_(0,0)^2x_(1,1)^2 0 x_(0,0)x_(0,1)x_(1,1)^2 0 |
{1, 1} | x_(0,1)x_(1,0)^2x_(1,1) x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 x_(0,1)^2x_(1,0)x_(1,1) 0 0 |
{1, 1} | x_(0,1)x_(1,0)x_(1,1)^2 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 x_(0,1)^2x_(1,0)x_(1,1) 0 |
{1, 1} | 0 x_(0,0)^2x_(1,0)x_(1,1) -x_(0,0)^2x_(1,0)^2 0 0 x_(0,0)^2x_(0,1)x_(1,0) |
{1, 1} | x_(0,0)x_(1,0)x_(1,1)^2 0 x_(0,0)^2x_(1,0)x_(1,1) 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 |
{1, 1} | 0 x_(0,1)^2x_(1,1)^2 -x_(0,1)^2x_(1,0)x_(1,1) 0 0 x_(0,1)^3x_(1,1) |
{1, 1} | 0 x_(0,0)x_(0,1)x_(1,1)^2 -x_(0,0)x_(0,1)x_(1,0)x_(1,1) 0 0 x_(0,0)x_(0,1)^2x_(1,1) |
{1, 1} | 0 0 0 x_(0,0)x_(0,1)x_(1,1)^2 -x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)^2x_(0,1)x_(1,1) |
{1, 1} | 0 0 0 -x_(0,0)^2x_(1,1)^2 x_(0,0)^2x_(1,0)x_(1,1) x_(0,0)^3x_(1,1) |
{1, 1} | x_(0,1)x_(1,0)^3 x_(0,0)x_(0,1)x_(1,0)^2 0 x_(0,1)^2x_(1,0)^2 0 0 |
{1, 1} | x_(0,1)x_(1,0)^2x_(1,1) 0 x_(0,0)x_(0,1)x_(1,0)^2 0 x_(0,1)^2x_(1,0)^2 0 |
{1, 1} | x_(0,0)x_(1,0)^3 x_(0,0)^2x_(1,0)^2 0 x_(0,0)x_(0,1)x_(1,0)^2 0 0 |
{1, 1} | x_(0,0)x_(1,0)^2x_(1,1) 0 x_(0,0)^2x_(1,0)^2 0 x_(0,0)x_(0,1)x_(1,0)^2 0 |
{1, 1} | 0 x_(0,1)^2x_(1,0)x_(1,1) -x_(0,1)^2x_(1,0)^2 0 0 x_(0,1)^3x_(1,0) |
{1, 1} | 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)x_(0,1)x_(1,0)^2 0 0 x_(0,0)x_(0,1)^2x_(1,0) |
{1, 1} | 0 0 0 x_(0,0)x_(0,1)x_(1,0)x_(1,1) -x_(0,0)x_(0,1)x_(1,0)^2 -x_(0,0)^2x_(0,1)x_(1,0) |
{1, 1} | 0 0 0 -x_(0,0)^2x_(1,0)x_(1,1) x_(0,0)^2x_(1,0)^2 x_(0,0)^3x_(1,0) |
20
o94 : S-module, submodule of S
|
i95 : dim M1 o95 = 4 |
i96 : betti M1, betti M
0 1 0 1
o96 = (total: 6 4, total: 6 4)
6: 6 4 2: 6 4
o96 : Sequence
|
i97 : isIsomorphic(M1,M**S^{-d})
o97 = true
|
Another shift:
i98 : d={-1,-2}
o98 = {-1, -2}
o98 : List
|
i99 : T2=T**E^{d}[sum d]
136 55 32 44 39 36 54 91 136 184 239 304 382 476 589
o99 = E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E <-- E
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
o99 : ChainComplex
|
i100 : cohomologyMatrix(beilinsonWindow T2,low,high)
o100 = | 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 1 8 0 0 0 |
| 0 0 0 4 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
7 7
o100 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i101 : cohomologyMatrix(T,low,high)
o101 = | 28h 18h 8h 2 12 22 32 |
| 20h 13h 6h 1 8 15 22 |
| 12h 8h 4h 0 4 8 12 |
| 4h 3h 2h h 0 1 2 |
| 4h2 2h2 0 2h 4h 6h 8h |
| 12h2 7h2 2h2 3h 8h 13h 18h |
| 20h2 12h2 4h2 4h 12h 20h 28h |
7 7
o101 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i102 : B2 =beilinson T2
8 5
o102 = S <-- S
0 1
o102 : ChainComplex
|
i103 : HH^(-1) B2 == 0 o103 = true |
i104 : M2=HH^0 B2
o104 = cokernel | -x_(1,0) 0 0 0 0 |
| 0 x_(1,0) 0 0 0 |
| -x_(1,1) 0 x_(1,0) 0 0 |
| 0 -x_(1,1) 0 0 x_(0,1) |
| 0 0 x_(1,0) 0 -x_(0,0) |
| 0 0 0 x_(1,0) x_(0,1) |
| 0 0 -x_(1,1) 0 0 |
| 0 0 0 -x_(1,1) 0 |
8
o104 : S-module, quotient of S
|
i105 : dim M2 o105 = 4 |
i106 : betti M2, betti M, betti truncate(-d,M)
0 1 0 1 0 1
o106 = (total: 8 5, total: 6 4, total: 8 5)
0: 8 5 2: 6 4 3: 8 5
o106 : Sequence
|
i107 : isIsomorphic(M2,truncate(-d,M)**S^{-d})
o107 = true
|