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={Uα}α, where α is in I, a countable ordered set. We will simplify notation by writing intersections as Uα∩ Uβ = Uαβ, and so on for higher intersections. For every intersection Uα0···αn there are n + 1 inclusions defined as follows:
- ∂i : Uα0···αn→ Uα0···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(Uα) to every open set Uα. Using the properties of products, we may write
- C 0(U,F) = ∏α in I F(Uα).
Then we define the 1-cochains to be elements of
- C 1(U,F) = ∏αβ F(Uαβ)
and so on, so that
- C i(U,F) = ∏α0···αi F(Uα0···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 + 1F(∂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
- HδC *(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.