Complex Analytic and Differential Geometry 2025, 13 - Alexander-Spanier Cohomology

A topological space X, a commutative ring R and its R-module M have a constant sheaf X × M (write M). The Alexander-Spanier cohomology group with values in M is the sheaf cohomology group H^q(X, M). H⁰(X, M) is the set of locally constant functions from X to M, meaning it's equivalent with M^E, where E is the set of connected components of X. H⁰([0, 1], M) = M, H^q([0, 1], M) = 0 ∀ q ≠ 0. For any space X, the projection π: X × I → X and injections i_t: X → X × I, x → (x, t) induce inverse isomorphisms between H^q(X, M) and H^q(X × I, M) with I = [0, 1]. i*_t doesn't depend on t. For two homotopic maps f, g: X → Y: f* = g*: H^q(Y, M) → H^q(X, M). If X is a compact differentiable manifold, its cohomology groups are finitely generated.

A closure of T in S^○ as subspace of X has the restriction isomorphism H^q(X, S; M) → H^q(X \ T, S \ T; M). If S is open or strongly paracompact in X, the relative cohomology groups can be written in terms of cohomology groups with supports in X \ S. Locally compact X, Y have properly homotopic maps f, g between them if their homotopy lies in H: X × I → Y, hence f* = g*: H^qc_{(Y, M) → H^q}c(X, M).

Define the q-th torsion module of A, B: Tor_q(A, B) = H_q(K_˙) = H_q(C_˙ ⊗ B) = H_q(A ⊗ D_˙) With this, the algebraic Kuenneth formula for complexes of R-modules (K^&dit;, d'), (L^&dit;, d'') with (K ⊗ L)^˙ the simple complex associated to the double complex (K ⊗ L)^{p, q} = K^p ⊗ L^q. If K^˙, L^˙ are torsion-free, there is a split exact sequence utilizing μ({k^p} × {l^q}) = {k^p⊗ l^q} for cocycles in appropriate cohomology groups 0 → ⊗_p+q=; H^p(K^˙) ⊗ H^q(L^˙) →^μ H^l((L⊗L)^˙) → ⊗_p+q=l+1 H^p(K^˙)☆H^q(L^˙) → 0 If K^˙ is a complex of R-modules and M an R-module with either K^˙ or M torsion free, then 0 → H^p(K^˙) ⊗ M → H^p(K^˙⊗M) → H^p+1(K^˙)☆M→0.

Assume that either a sheaf A of R-modules, or an R-module M is torsion free and the underlying topological space X is compact or M is finitely generated, then 0 → H^p(X, A) ⊗ M → H^p(X, A ⊗ M) → H^p+1(X, A) ☆ M → 0 Assume that A is torsion free and Y is a compact topological space, and either X is compact or H^q(Y, B) are finitely generated R-modules, then 0 → ⊗_p+q=l H^p(X, A) ⊗ H^q(Y, B) →^μ H^l(X × Y, A□ B) → ⊗_{p+q = l+1}H^q(X, A) ☆ H^q(Y, B) → 0 with μ derived from the cartesian product. If A, B are torsion free constant sheaves, the Kuenneth formula holds when X or Y has the same homotopy type as a finite-cell complex.

Over a principal ring R, a module M is injective iff it's divisible. All modules can be embedded into such an injective module. The q-th extension module of A, B is written Ext^q_R(A, B) = H^q(K^˙) = H^q(Hom(A, D^˙)) = H^q(Hom(C_˙, B)) Particularly, Ext⁰(A, B) = Hom(A, B). If either A is projective, or B is injective, then Ext^q(A, B) = 0 ∀ q ≥ 1. For paracompact topological manifolds X with dim n, if L is an R-module and a locally constant sheaf of stalk L on X is a sheaf A so that all points have neighborhoods Ω with A_↑Ω is R-isomorphic to the constant sheaf L. A could be seen as a discrete fiber bundle over X with R-modules as fibers and R-linear transition automorphisms. A locally constant sheaf τ_X = H⁰(F^˙_{ℤ, ℤ}) of stalk ℤ is the orientation sheaf of X. It's given by homomorphism π₁(X) → {1, -1} and coincides with the trivial sheaf iff X is orientable. If M is also an R-module with Ext(L, M) = 0 then 0 → Ext(H_c^n-q+1(X, A), M) → H^q(X, τ_X⊗Hom(A, M)) → Hom(H_c^n-q(X, A), M) → 0 A connected topological manifold with dim n and R-module L, H_cn(X, τ_X ⊗ L) ≡ L and H_cⁿ(X, L) ≡ L/2L if X is not orientable