Complex Analytic and Differential Geometry 2025, 11 - Cup Product and Cartesian Products

Let R be a sheaf of commutative rings, A, B sheaves of R-modules on a space X. (A ⊗_R B)_x = A_x⊗_Rx B_x with weakest topology such that the range of any section is open in A ⊗_R B for all open sets U in X. Given f in A, g in B, the cup product is f∪g(x₀...x_p+q) = f(x₀..., x_p)(x_p+q)⊗g(x_p..., x_p+q) It is a cocycle if f, g are cocycles, so there exists a well-defined R(X)-bilinear morphism. H^p(X, A) × H^q(X, B) → H^p+q(X, A ⊗_R B). The Cup product is associative and anticommutative. It may be defined alternatively via the Cech cochains. Given the p and q-chains c, c', their cochain c∪c' ∈ C^p+q(U, A⊗_RB): (c∪c')_α0....αp+q = c_{α0... αp}⊗ c'_{αp... αp+q} on U_{α Straightforwardly, by Supp[f∪g] ⊂ Supp[f] ∩ Supp[g], the intersection of families of supports are again families of supports.}

A continuous map F: X → Y between topological spaces and a sheaf of abelian groups π: A → Y define the sheaf-space F^-1A = A ×_Y X = {(s, x): π(s) = F(x)} with projection π' = pr₂: F^-1A → X. The stalks are defined (F^-1A)_x = A_F(x) and the sections σ can be considered continuous mappings such that π○σ = F. Any section s in A(U) has a pull-back. F*s = s○F ∈ F^-1A(F^-1A)_x^[q]. For elements of A, F*v ∈ (F^-1A)_x^[q], F*v = v(F) ∈ (F^-1A)_x = A_F(x). Applications of F^-1 applied to sheaves on X retains exact sequences. Morphisms F* in this sequence commutes with the associated coholomogy exact sequences, and it preserves the cup product for cohomology classes.

A sheaf A on X and a strongly paracompact subspace S in X, if Ω ranges over open neighborhoods of S, H^q(S, A) = lim_{S ⊂ Ω} H^q(Ω, A). Define the external tensor product: A □_R B = pr₁^-1A ⊗_R pr₂^-1B The sheaf A □_R B is the sheaf on X × Y with stalks (A □_R B)_{(x, y)} = A_x ⊗_R B_y. For cohomology classes α ∈ H^q(X, A), β ∈ H^p(Y, B), the cartesian product α × β ∈ H^p+q(X × Y, A□_R B): α×β = (pr₁*α) ∪ (pr₂*β).