If M is a multi-graded module representing a coherent sheaf F on Pn := Pn0 x .. x Pnt-1, the script returns a hash table with entries a,i => hi(F(a)) where a is a multi-index, low<=a<=high in the partial order (thus the value is 0 when i is not in the range 0..sum n.) In case T is a Tate resolution corresponding to an object F in Db(Pn), then the values returned are the dimensions of the hypercohomology groups of twists of F, and the values can be nonzero in a wider range.
In case the number of factors t is 2, the output of cohomologyMatrix is easier to parse.
The script computes a sufficient part of the Tate resolution for F, and then calls itself in the version for a Tate resolution.
If T is not a large enough part of the Tate resolution, such as W below, then the function collects only the contribution of T to the cohomology table of the Tate resolution, according to the formula in Corollary 0.2 of Tate Resolutions on Products of Projective Spaces.
i1 : (S,E) = productOfProjectiveSpaces{1,2}
o1 = (S, E)
o1 : Sequence
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i2 : M = S^1
1
o2 = S
o2 : S-module, free
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i3 : low = {-3,-3};high = {3,3};
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i5 : H = cohomologyHashTable(M, low,high); |
We can print just the entries representing nonzero cohomology groups:
i6 : H' = hashTable(select(pairs H, p-> p_1!=0))
o6 = HashTable{{{-2, -3}, 3} => 1}
{{-2, 0}, 1} => 1
{{-2, 1}, 1} => 3
{{-2, 2}, 1} => 6
{{-2, 3}, 1} => 10
{{-3, -3}, 3} => 2
{{-3, 0}, 1} => 2
{{-3, 1}, 1} => 6
{{-3, 2}, 1} => 12
{{-3, 3}, 1} => 20
{{0, -3}, 2} => 1
{{0, 0}, 0} => 1
{{0, 1}, 0} => 3
{{0, 2}, 0} => 6
{{0, 3}, 0} => 10
{{1, -3}, 2} => 2
{{1, 0}, 0} => 2
{{1, 1}, 0} => 6
{{1, 2}, 0} => 12
{{1, 3}, 0} => 20
{{2, -3}, 2} => 3
{{2, 0}, 0} => 3
{{2, 1}, 0} => 9
{{2, 2}, 0} => 18
{{2, 3}, 0} => 30
{{3, -3}, 2} => 4
{{3, 0}, 0} => 4
{{3, 1}, 0} => 12
{{3, 2}, 0} => 24
{{3, 3}, 0} => 40
o6 : HashTable
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In the case of two factors (t=2), the same information can be read conveniently from a matrix
i7 : cohomologyMatrix(M, low, high)
o7 = | 20h 10h 0 10 20 30 40 |
| 12h 6h 0 6 12 18 24 |
| 6h 3h 0 3 6 9 12 |
| 2h h 0 1 2 3 4 |
| 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 2h3 h3 0 h2 2h2 3h2 4h2 |
7 7
o7 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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where the entry in the a= {a0,a1}place is sumi hi(F(a)*hi ∈ZZ[h]. In the case of more factors, the same format is available through the command
i8 : eulerPolynomialTable H'
3
o8 = HashTable{{-2, -3} => h }
{-2, 0} => h
{-2, 1} => 3h
{-2, 2} => 6h
{-2, 3} => 10h
3
{-3, -3} => 2h
{-3, 0} => 2h
{-3, 1} => 6h
{-3, 2} => 12h
{-3, 3} => 20h
2
{0, -3} => h
{0, 0} => 1
{0, 1} => 3
{0, 2} => 6
{0, 3} => 10
2
{1, -3} => 2h
{1, 0} => 2
{1, 1} => 6
{1, 2} => 12
{1, 3} => 20
2
{2, -3} => 3h
{2, 0} => 3
{2, 1} => 9
{2, 2} => 18
{2, 3} => 30
2
{3, -3} => 4h
{3, 0} => 4
{3, 1} => 12
{3, 2} => 24
{3, 3} => 40
o8 : HashTable
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In case of hypercohomology, we write k instead of h-1, and use the cohomology ring ZZ[h,k].