Minimal Cech coverings

Please don’t read this unless you want to be distracted in these difficult times and you enjoy thinking about elementary problems in etale cohomology. I am going to pose a challenge computing etale cohomology by Cech coverings.

The setting. We have a quasi-projective scheme X over an algebraically closed field k, a prime number ℓ different from the characteristic of k and a constructible sheaf F of Z/ℓ Z vector spaces. Next we have a cohomology class ξ in the etale cohomology group H^i(X, F). By a result of Artin, the cohomology of F over X is the same as the Cech cohomology with respect to etale coverings, but (of course) you have to take the colimit of all etale coverings. Thus we know there exists an etale covering cU = {U_i —> X} and a degree i Cech cohomology class ξ’ for cU and F which maps to ξ in H^i(X, F).

The problem I want to think about is this: is there a notion of minimal etale covers cU such that you can represent ξ by a Cech cohomology class for that covering.

Let’s do an example. Take X = P^1 the projective line and take F the constant sheaf Z/ ℓ Z. Then there is a nonzero class ξ in H^2(X, F). Then we can try and minimize (one at a time) the following numerical invariants of Cech coverings \cU = {U_i —> X}_{i = 1, …, n} such that ξ can be represented as a Cech cohomology class on them:

  1. the sum of the number of connected components of U_i,
  2. the sum of the geometric genera of the components of the U_i,
  3. the sum of all the betti numbers of the U_i, or
  4. the sum of the degrees of the morphisms U_i —> X.

I will edit this and put other suggestions here in the future

Rmk: Observe that if cU = {U_i —> X}_{i =1, …, n} is an etale covering, then so is {U —> X} where U is the disjoint union of the U_i and these covering have identical Cech complexes. Hence whatever invariant we use, it shouldn’t distinguish between these coverings, for example we shouldn’t use “n” as an invariant.

Rmk: for etale cohomology we have to use “regular” Cech cohomology, not alternating Cech cohomology.

Rmk: It might be better to work with etale hypercoverings and define numerical invariants for those.

Back to the example, here is a covering that works. Let t be a coordinate on P^1. Let U_0 = G_m with coordinate x mapping to t = x^ℓ in P^1. Let U_1 = G_m with coordinate y mapping to t = y^ℓ + 1. Let U_2 = G_m with coordinate z mapping to t = z^ℓ / (z^ℓ – 1). The (double) fibre product of U_0 and U_1 over P^1 is connected and similarly for U_0 and U_2 and for U_1 and U_2. However, the (triple) fibre product of U_0 and U_1 and U_2 over P^1 is disconnected because a computation shows that (x^{-1}yz)^ℓ = 1 and hence x^{-1}yz has to be an ℓth root of 1. This suggests that we can write down an interesting Cech cohomology class in degree 2 for this etale Cech covering and indeed this is true. To actually prove that ξ can be represented by a Cech cohomology class for this covering, you can proceed as follows: first compute the obstruction to representing ξ by a Cech cohomology class for the open covering P^1 = (P^1 – {0}) cup (P^1 – {∞}). This is an element of H^1(P^1 – {0, ∞}, F). Then show this obstruction dies when passing to the covering {U_0 –> P^1, U_1 —> P^1, U_2 —> P^1} which refines the two part open covering because the H^1 element dies on U_i \times_{P^1} U_j for all i, j. Anyway, my solution gives the invariants as in list above:

  1. For this invariant I get 3 because U_0, U_1, U_2 are connected
  2. For this invariant I get 0 because U_0, U_1, U_2 are rational
  3. For this invariant I get 6 because U_0, U_1, U_2 are G_m
  4. For this invariant I get 3ℓ because the maps U_i —> P^1 have degree ℓ

Challenge: Can you improve for any of these invariants on the numbers I get? I’ve tried to improve on the first invariant without success so far.