Complex Analytic and Differential Geometry 2025, 10 - Cech Coholomogy

Let U = (U_α)_α∈I an open covering of X. The group C^q(U, A) of Cech q-cochains is the set of families c = (c_α) ∈ Π_(α)∈I^q+1 A(U). Its group structure is the obvious one deduced from the addition law on sections of A. The Cech-differential δ^q: C^q(U, A) → C^q+1(U, A) is defined (δ^qc)_α = Σ_0≤j≤q+1 (-1)^j c_{α, ↑Uα} so that C^q(U, A) = 0, δ^q = 0 for q < 0. Also, δ^q+1○δ^q = 0. The Cech cohomology group of A relative to U is H'^q(U, A) = H^q(C(U, A)), which is the direct limit of Cech cohomology groups over all open coverings of U. Its elements are the equivalence classes in the disjoint union of the cohomology groups over the coverings. Elements of two such groups are identified if their images coincide for some refinement of their coverings. For a sheaf A on X that is either flabby, or if X is paracompact and A is a sheaf of modules over a soft sheaf or rings R on X, then H'^q(U, A) = 0 for all q ≥ 1, and every open covering U of X.

Assuming that H^s(U_α, A) = 0 for all indices, and s ≥ 1, then there is an isomorphism H'^q(U, A) ⇄ H^q(X, A). The covering U is acyclic wrt. A. For paracompact X and an exact sequence of sheaves 0 → A → B → C → 0 induces an exact sequence H'^q(X, A) → H'^q(X, B) → H'^q(X, C) → H'^q+1(X, A) →... An open covering U of a paracompact X and c ∈ C^q(U, C) creates a finer covering V and a refinement map ρ: J → I such that ρ^qc∈ C^q_B(V, C). On paracompact spaces, the canonical morphism is an isomorphism.

A family of supports on a topological space X is a collection Φ of closed subsets of X so that if F, F' ∈ Φ, then F∪F'∈Φ. If F ∈ Φ, and F' ⊂ F closed, then F' is also in Φ. For any sheaf A and family of supports Φ on X, A_Φ (X) denotes the set of all sections f of A(X) with Supp[f] ∈ Φ. The cohomology groups of A with supports in Φ are H^q_Φ(X, A) = H^q(A_Φ(X)). The cohomology groups with compact supports are denoted H^q_c and those with supports in a subset S H^q_S. Assume X is separable and locally compact with locally finite covering U by relatively compact open subsets.. The subgroup of cochains C^q_c(U, A) with only finitely many non-zero coefficients, defines the Cech cohomology groups with compact support: H'^q_c(U, A) = H^q(C_c(U, A)) H'^q_c(X, A) = lim H^q(C_c(U, A))