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 Hq(X, M). H0(X, M) is the set of locally constant functions from X to M, meaning it's equivalent with ME, where E is the set of connected components of X. H0([0, 1], M) = M, Hq([0, 1], M) = 0 ∀ q ≠ 0. For any space X, the projection π: X × I → X and injections it: X → X × I, x → (x, t) induce inverse isomorphisms between Hq(X, M) and Hq(X × I, M) with I = [0, 1]. i*t doesn't depend on t. For two homotopic maps f, g: X → Y: f* = g*: Hq(Y, M) → Hq(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 Hq(X, S; M) → Hq(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*: Hqc(Y, M) → Hqc(X, M).
Define the q-th torsion module of A, B:
Torq(A, B) = Hq(K˙) = Hq(C˙ ⊗ B) = Hq(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 = Kp ⊗ Lq. If K˙, L˙ are torsion-free, there is a split exact sequence utilizing μ({kp} × {lq}) = {kp⊗ lq} for cocycles in appropriate cohomology groups
0 → ⊗p+q=; Hp(K˙) ⊗ Hq(L˙) →μ Hl((L⊗L)˙) → ⊗p+q=l+1 Hp(K˙)☆Hq(L˙) → 0
If K˙ is a complex of R-modules and M an R-module with either K˙ or M torsion free, then
0 → Hp(K˙) ⊗ M → Hp(K˙⊗M) → Hp+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 → Hp(X, A) ⊗ M → Hp(X, A ⊗ M) → Hp+1(X, A) ☆ M → 0
Assume that A is torsion free and Y is a compact topological space, and either X is compact or Hq(Y, B) are finitely generated R-modules, then
0 → ⊗p+q=l Hp(X, A) ⊗ Hq(Y, B) →μ Hl(X × Y, A□ B) → ⊗p+q = l+1Hq(X, A) ☆ Hq(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.