If M is a bigraded module over a bigraded polynomial ring representing a sheaf F on P^{n_1} x P^{n_2}, the script returns a block of the cohomology table, represented as a table of "cohomology polynomials" in $\mathbb Z[h,k]$ of the form $$\sum_{i=0}^{|n|} \, dim H^i(\mathcal F(c_1,c_2)) * h^i$$ in each position \{c_1,c_2\} for $a_1 \le c_1 \le b_1$ and $a_2 \le c_2 \le b_2$. In case M corresponds to an object in the derived category D^b(P^{n_1}x P^{n_2}), then hypercohomology polynomials are returned, with the convention that k stands for k=h^{ -1}.
The polynomial for \{b_1,b_2\} sits in the north-east corner, the one corresponding to (a_1,a_2) in the south-west corner.
In the case of a product of more (or fewer) projective spaces, or if a hash table output is desired, use cohomologyHashTable or eulerPolynomialTable instead.
The script computes a sufficient part of the Tate resolution for F, and then calls itself in the version for a Tate resolution. More generally, If T is part of a Tate resolution of F the function returns a matrix of cohomology polynomials corresponding to T.
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={0,0}; |
i5 : cohomologyMatrix(M,low,high) o5 = | 2h h 0 1 | | 0 0 0 0 | | 0 0 0 0 | | 2h3 h3 0 h2 | 4 4 o5 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) |
As a second example, consider the structure sheaf $\mathcal O_E$ of a nonsingular cubic contained in (point)xP^2. The corresponding graded module is
i6 : M = S^1/ideal(x_(0,0), x_(1,0)^3+x_(1,1)^3+x_(1,2)^3) o6 = cokernel | x_(0,0) x_(1,0)^3+x_(1,1)^3+x_(1,2)^3 | 1 o6 : S-module, quotient of S |
i7 : low = {-3,-3};high={0,0}; |
i9 : cohomologyMatrix(M,low,high) o9 = | h+1 h+1 h+1 h+1 | | 3h 3h 3h 3h | | 6h 6h 6h 6h | | 9h 9h 9h 9h | 4 4 o9 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) |
and the "1+h" in the Northeast (= upper right) corner signifies that that $h^0(\mathcal O_E) = h^1(\mathcal O_E) = 1.$
The object cohomologyMatrix is a method function.