Cech cohomology

Čech cohomology is a particular type of cohomology in mathematics. It is named for the mathematician Eduard Čech.

Construction

Let X be a topological space with open cover U={U_α}_α, where α is in I, a countable ordered set. We will simplify notation by writing intersections as U_α∩ U_β = U_αβ, and so on for higher intersections. For every intersection U_α0···αn there are n + 1 inclusions defined as follows:

∂_i : U_α0···αn→ U_{α0···α(i – 1)α(i + 1)···αn}

that is, ∂_i skips the i th open set. We now define the cochain groups for this cohomology.

Let F be a presheaf. The 0-cochains on U with values in F are functions assigning an element of F(U_α) to every open set U_α. Using the properties of products, we may write

C ⁰(U,F) = ∏_{α in I} F(U_α).

Then we define the 1-cochains to be elements of

C ¹(U,F) = ∏_αβ F(U_αβ)

and so on, so that

C ⁱ(U,F) = ∏_α0···αi F(U_α0···αi ).

Next, the differential. Let δ : C ⁱ(U,F)→C ^{i + 1}(U,F) be the alternating difference of the F(∂_i ):

δ = F(∂₀) – F(∂₁) + ··· + (-1) ^{i + 1}F(∂_i ).

One shows that δ² = 0, as required. By taking the cohomology of the chain complex formed we have the Čech cohomology of U with values in the presheaf F. We write

H_δC ^*(U,F) = H ^*(U,F).

Now, if V is a cover which refines U then there is a well-defined map in cohomology H ^*(U,F)→H ^*(V,F) and as such {H ^*(U,F)}_U is a direct system of groups. We call the direct limit of this system the Čech cohomology of X with values in F.

References

• R. Bott & L. Tu, Differential Forms in Algebraic Topology, Springer Graduate Texts in Mathematics 82, 1982.