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SpectralSequences :: Spectral sequences and hypercohomology calculations

Spectral sequences and hypercohomology calculations

If F is a coherent sheaf on a smooth toric variety X then multigraded commutative algebra can be used to compute the cohomology groups Hi(X, F).

Indeed if B is the irrelevant ideal of X then the cohomology group Hi(X, F) can be relized as the degree zero piece of the multigraded module Exti(B[l], F) for sufficiently large l; here B[l] denotes the lth Forbenius power of B and F is any multigraded module whose corresponding sheaf on X is 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 = ℙ1 ×ℙ1 and F = OC(1,0) where C is a general divisor of type (3,3) on X. In this setting, H0(C,F) and H1(C, F) are both 2-dimensional vector spaces.

We first make the multi-graded coordinate ring of 1 ×ℙ1, 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 1 ×ℙ1. 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 OC(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