If P is an E-module, then LP becomes a linear complex of free S-modules, where (S,E) is the Koszul pair corresponding to a product of projective spaces. Similarly, if M is an S-module, them RM becomes a linear free complex over the exterior algebra E of length bounded by the LengthLimit.
The complex LP is that produced from P by the Bernstein-Gel’fand-Gel’fand functor called L in our paper Tate Resolutions on Products of Projective Spaces. Similarly, the complex RM produced from M is a bounded piece of the infinite complex of the Bernstein-Gel’fand-Gel’fand functor called R in loc.cit. L and R form a pair of adjoint functors.
i1 : (S,E) = productOfProjectiveSpaces{1,2}
o1 = (S, E)
o1 : Sequence
|
i2 : P = prune truncate({1,2},E^1)**E^{{1,2}};
|
i3 : LP = bgg P
1 5 6
o3 = S <-- S <-- S
-2 -1 0
o3 : ChainComplex
|
i4 : netList apply(toList(min LP..max LP), i-> decompose ann HH_i LP)
+------------------------------------+------------------+
o4 = |ideal (x , x , x , x , x )| |
| 1,2 1,1 1,0 0,1 0,0 | |
+------------------------------------+------------------+
|ideal (x , x , x ) |ideal (x , x )|
| 1,2 1,1 1,0 | 0,1 0,0 |
+------------------------------------+------------------+
|ideal () | |
+------------------------------------+------------------+
|
i5 : M = prune HH_0 LP
o5 = cokernel {1, 1} | -x_(1,2) |
{1, 1} | x_(1,0) |
{1, 1} | -x_(1,1) |
3
o5 : S-module, quotient of S
|
i6 : betti res M
0 1
o6 = total: 3 1
2: 3 1
o6 : BettiTally
|
i7 : high = {3,3}
o7 = {3, 3}
o7 : List
|
i8 : cohomologyMatrix(M, -high, high)
o8 = | 45h 30h 15h 0 15 30 45 |
| 24h 16h 8h 0 8 16 24 |
| 9h 6h 3h 0 3 6 9 |
| 0 0 0 0 0 0 0 |
| 3h2 2h2 h2 0 h 2h 3h |
| 0 0 0 0 0 0 0 |
| 9h3 6h3 3h3 0 3h2 6h2 9h2 |
7 7
o8 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
|
i9 : M=module ideal vars S
o9 = image | x_(0,0) x_(0,1) x_(1,0) x_(1,1) x_(1,2) |
1
o9 : S-module, submodule of S
|
i10 : RM = bgg(M,LengthLimit=>3)
70 35 15 5
o10 = E <-- E <-- E <-- E
-4 -3 -2 -1
o10 : ChainComplex
|
i11 : betti RM
-4 -3 -2 -1
o11 = total: 70 35 15 5
0: 70 35 15 5
o11 : BettiTally
|
i12 : tallyDegrees RM
o12 = (Tally{{-1, -3} => 20}, Tally{{-1, -2} => 12}, Tally{{-1, -1} => 6},
{-2, -2} => 18 {-2, -1} => 9 {-2, 0} => 3
{-3, -1} => 12 {-3, 0} => 4 {0, -2} => 6
{-4, 0} => 5 {0, -3} => 10
{0, -4} => 15
-----------------------------------------------------------------------
Tally{{-1, 0} => 2})
{0, -1} => 3
o12 : Sequence
|