# Spectral sequences and hypercohomology calculations

If $\mathcal{F}$ is a coherent sheaf on a smooth toric variety $X$ then multigraded commutative algebra can be used to compute the cohomology groups $H^i(X, \mathcal{F})$.

Indeed if $B$ is the irrelevant ideal of $X$ then the cohomology group $H^i(X, \mathcal{F})$ can be realized as the degree zero piece of the multigraded module $Ext^i(B^{[l]}, F)$ for sufficiently large $l$; here $B^{[l]}$ denotes the $l$th Forbenius power of $B$ and $F$ is any multigraded module whose corresponding sheaf on $X$ is $\mathcal{F}$.

Given the fan of $X$ and $F$ a sufficiently large power of $l$ can be determined effectively. We refer to sections 2 and 3 of the paper "Cohomology on Toric Varieties and Local Cohomology with Monomial Supports" for more details.

In this example, we consider the case that $X = \mathbb{P}^1 \times \mathbb{P}^1$ and $F = \mathcal{O}_C(1,0)$ where $C$ is a general divisor of type $(3,3)$ on $X$. In this setting, $H^0(C,F)$ and $H^1(C, F)$ are both $2$-dimensional vector spaces.

We first make the multi-graded coordinate ring of $\mathbb{P}^1 \times \mathbb{P}^1$, the irrelevant ideal, and a sufficentily high Frobenus power of the irrelevant ideal needed for our calculations. Also the complex $G$ below is a resolution of the irrelevant ideal.

 i1 : R = ZZ/101[a_0..b_1, Degrees=>{2:{1,0},2:{0,1}}]; -- PP^1 x PP^1 i2 : B = intersect(ideal(a_0,a_1),ideal(b_0,b_1)) ; -- irrelevant ideal o2 : Ideal of R i3 : B = B_*/(x -> x^5)//ideal ; -- Sufficentily high Frobenius power o3 : Ideal of R i4 : G = res image gens B ;

We next make the ideal, denoted by $I$ below, of a general divisor of type $(3,3)$ on $\mathbb{P}^1 \times \mathbb{P}^1$. Also the chain complex $F$ below is a resolution of this ideal.

 i5 : I = ideal random(R^1, R^{{-3,-3}}) ; -- ideal of C o5 : Ideal of R i6 : F = res comodule I 1 1 o6 = R <-- R <-- 0 0 1 2 o6 : ChainComplex

To use hypercohomology to compute the cohomology groups of the line bundle $\mathcal{O}_C(1,0)$ on $C$ we twist the complex $F$ above by a line of rulting and then make a filtered complex whose associated spectral sequence abuts to the desired cohomology groups.

 i7 : K = Hom(G , filteredComplex (F ** R^{{1,0}})) ; -- Twist F by a line of ruling and make filtered complex whose ss abuts to HH OO_C(1,0) i8 : E = prune spectralSequence K ; --the spectral sequence degenerates on the second page i9 : E^1 +-----------------------------------------------+---------------------------------------------+ | 1 | 1 | o9 = |R |R | | | | |{0, 0} |{1, 0} | +-----------------------------------------------+---------------------------------------------+ |cokernel {-11, 0} | a_1^5 a_0^5 0 0 | |cokernel {-8, 3} | a_1^5 a_0^5 0 0 | | | {-1, -10} | 0 0 b_1^5 b_0^5 | | {2, -7} | 0 0 b_1^5 b_0^5 | | | | | |{0, -1} |{1, -1} | +-----------------------------------------------+---------------------------------------------+ |cokernel {-11, -10} | b_1^5 b_0^5 a_1^5 a_0^5 ||cokernel {-8, -7} | b_1^5 b_0^5 a_1^5 a_0^5 || | | | |{0, -2} |{1, -2} | +-----------------------------------------------+---------------------------------------------+ o9 : SpectralSequencePage i10 : E^2 ; -- output is a mess

The cohomology groups we want are obtained as follows.

 i11 : basis({0,0}, E^2_{0,0}) -- == HH^0 OO_C(1,0) o11 = {-1, 0} | a_0 a_1 | o11 : Matrix i12 : basis({0,0}, E^2_{1,-2}) -- == HH^1 OO_C(1,0) o12 = {-8, -1} | 0 0 | {-8, -1} | 0 0 | {-8, -1} | 0 0 | {-7, -2} | 0 0 | {-7, -2} | 0 0 | {-7, -2} | 0 0 | {-6, -3} | 0 0 | {-6, -3} | 0 0 | {-6, -3} | 0 0 | {-6, -3} | 0 0 | {-5, -4} | 0 0 | {-5, -4} | 0 0 | {-5, -4} | 0 0 | {-4, -5} | 0 0 | {-4, -5} | 0 0 | {-6, -3} | 0 0 | {-6, -3} | 0 0 | {-6, -3} | 0 0 | {-5, -4} | 0 0 | {-5, -4} | 0 0 | {-5, -4} | 0 0 | {-5, -4} | 0 0 | {-4, -5} | 0 0 | {-4, -5} | 0 0 | {-4, -5} | 0 0 | {-4, -5} | 0 0 | {-4, -5} | 0 0 | {-3, -6} | 0 0 | {-3, -6} | 0 0 | {-3, -6} | 0 0 | {-2, -7} | 0 0 | {-2, -7} | 0 0 | {-2, -7} | a_1^2b_0^4b_1^3 a_1^2b_0^3b_1^4 | o12 : Matrix