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SpectralSequences :: Spectral sequences and hypercohomology calculations

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

See also