# cohomologyMatrix -- cohomology groups of a sheaf on P^{n_1}xP^{n_2}, or of (part) of a Tate resolution

## Synopsis

• Usage:
H=cohomologyMatrix(M,low,high)
H=cohomologyMatrix(T,low,high)
• Inputs:
• T, , free complex over the exterior algebra
• M, , graded module representing a sheaf on a product of projective spaces
• low, a list,
• high, a list, two lists low=\{a_1,a_2\}, high=\{b_1,b_2\} representing bidegrees
• Outputs:
• H, , a (1+b1-a1)x(1+b2-a2) matrix of ring elements in $\mathbb Z[h,k]$

## Description

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 $h^0(\mathcal O_E) = h^1(\mathcal O_E) = 1.$