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 => sumi hi(F(a))*hi ∈ZZ[h,k], where k represents h-1, 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 polyomials 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. In general the script cohomologyHashTable gives the same information as this script, but in a less compact form.
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 |
i2 : M = S^1 1 o2 = S o2 : S-module, free |
i3 : low = {-3,-3};high = {3,3}; |
i5 : H' = cohomologyHashTable(M, low,high); |
i6 : H = eulerPolynomialTable H' o6 = HashTable{{-1, -1} => 0 } {-1, -2} => 0 {-1, -3} => 0 {-1, 0} => 0 {-1, 1} => 0 {-1, 2} => 0 {-1, 3} => 0 {-2, -1} => 0 {-2, -2} => 0 3 {-2, -3} => h {-2, 0} => h {-2, 1} => 3h {-2, 2} => 6h {-2, 3} => 10h {-3, -1} => 0 {-3, -2} => 0 3 {-3, -3} => 2h {-3, 0} => 2h {-3, 1} => 6h {-3, 2} => 12h {-3, 3} => 20h {0, -1} => 0 {0, -2} => 0 2 {0, -3} => h {0, 0} => 1 {0, 1} => 3 {0, 2} => 6 {0, 3} => 10 {1, -1} => 0 {1, -2} => 0 2 {1, -3} => 2h {1, 0} => 2 {1, 1} => 6 {1, 2} => 12 {1, 3} => 20 {2, -1} => 0 {2, -2} => 0 2 {2, -3} => 3h {2, 0} => 3 {2, 1} => 9 {2, 2} => 18 {2, 3} => 30 {3, -1} => 0 {3, -2} => 0 2 {3, -3} => 4h {3, 0} => 4 {3, 1} => 12 {3, 2} => 24 {3, 3} => 40 o6 : HashTable |
i7 : H = eulerPolynomialTable (M, low, high) o7 = HashTable{{-1, -1} => 0 } {-1, -2} => 0 {-1, -3} => 0 {-1, 0} => 0 {-1, 1} => 0 {-1, 2} => 0 {-1, 3} => 0 {-2, -1} => 0 {-2, -2} => 0 3 {-2, -3} => h {-2, 0} => h {-2, 1} => 3h {-2, 2} => 6h {-2, 3} => 10h {-3, -1} => 0 {-3, -2} => 0 3 {-3, -3} => 2h {-3, 0} => 2h {-3, 1} => 6h {-3, 2} => 12h {-3, 3} => 20h {0, -1} => 0 {0, -2} => 0 2 {0, -3} => h {0, 0} => 1 {0, 1} => 3 {0, 2} => 6 {0, 3} => 10 {1, -1} => 0 {1, -2} => 0 2 {1, -3} => 2h {1, 0} => 2 {1, 1} => 6 {1, 2} => 12 {1, 3} => 20 {2, -1} => 0 {2, -2} => 0 2 {2, -3} => 3h {2, 0} => 3 {2, 1} => 9 {2, 2} => 18 {2, 3} => 30 {3, -1} => 0 {3, -2} => 0 2 {3, -3} => 4h {3, 0} => 4 {3, 1} => 12 {3, 2} => 24 {3, 3} => 40 o7 : HashTable |
We can print just the entries representing nonzero cohomology groups:
i8 : trimH = hashTable(select(pairs H, p-> p_1!=0)) 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 |
In the case of two factors (t=2), the same information can be read conveniently from a matrix
i9 : cohomologyMatrix(M, low, high) o9 = | 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 o9 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) |
where the entry in the a= {a0,a1}place is sumi hi(F(a)*hi ∈ZZ[h].
In case of hypercohomology, we write k instead of h-1, and use the cohomology ring ZZ[h,k].