Complex Analytic and Differential Geometry 2025, 09 - Simplicial Flabby Resolution of Sheafs

A cohomology resolution of A is a differential complex of sheaves (L, d) with L^q = 0, d^q = 0 for q < 0 so that there is an exact sequence o → A →^j L⁰ →^d0 L¹ →.... if φ: A→B is a morphism of sheaves and (M, d) a resolution of B a morphism of resolutions φ, the above exact sequence can be one-to-one mapped to the exact sequence emerging from (M, d) via φ. A sheaf F is "flabby" if for all open subsets U of X, all sections of the restriction map F on U can be extended to X. The complex (A, d) is a resolution of the sheaf A called the resolution of A. For all rational q the q-th cohomology group of X with values in A is H^q(X, A) = H^q(A(X)) Any exact sequence of sheaves is associated with a long exact sequence of cohomology groups. A surjective sheaf morphism B → C with kernel A, then H¹(X, A) = 0, then B(X) → C(X) is surjective.

A closed subset S of X and U = X \ S, along with a sheaf on A defines a presheaf Ω → A(S∩Ω), if Ω ⊂ X is open. (A^S)_x = A_x, if x is in S and 0 if x is in U. Complementarily, (A_U)_x = 0 if x is in S, and A_x if x is in U. For any sheaf A on X and open sets U₁, U₂ subsets of X, set U = U₁∪U₂, V = U₁∩U₂, then H^q(U, A) → H^q(U₁, A) ⊕ H^q(U₂, A) → H^q(V, A) → H^{q + 1}(U, A) ... A sheaf A is acyclic on an open subset U, if H^q(U, A) = 0 for q ≥ 1. A sheaf where all sections f of A on an open subset U in X and all points x, have a neighborhood Ω of x and section h in A(Ω) so that h = f on U∩Ω, then A is flabby. Flabby sheafs are acyclic on all open sets U ⊂ X.

A subspace S is strongly paracompact in X, if S is Hausdorff and if for all coverings (U_α) of S by open sets in X there is another such covering (V_β) and a neighborhood W such that its intersection with the closure of V_β is contained in some U_α and all points of S have a neighborhood intersecting only finitely many sets V_β. Otherwise, S is strongly paracompact in X if it's paracompact and S is closed and X is paracompact, S has a fundamental family of paracompact neighborhoods, or S is paracompact had has homeomorphic neighborhoods to some product in which it's embedded as a slice. A sheaf A and a strongly paracompact subspace S on X, sees all sections f of A extendable to a section of A on an open neighborhood of A. If X is paracompact, then all sections of A(S) for S closed extend to a neighborhood Ω of S. A sheaf A on X is soft if all sections f of A on a closed set S can be extended to X.

A sheaf A is soft on X iff it's soft in a neighborhood of all points in X.Let 0 → A → B → C → 0 be exact. If A is soft, the map B(S) → C(S) is onto for any closed subset S of X. If A, B are soft, then C is soft. On a paracompact space, a soft sheaf is acyclic on all closed subsets. The support of a section f in A(X) is defined Supp[f] = {x∈X: f(x) ≠ 0}. It's always a closed set. Let (U_α)_α∈I an open covering of X. If A is soft and f in A(X), there is a partition of f subordinate to (U_α). If R is soft, every sheaf A of R-modules is soft. Let A be a sheaf of E-modules on a paracompact differentiable manifold X, then H^q(X, A) = 0 for all q ≥ 1.