Sheaf Cohomology

A sheaf F on X assigns to each open U ⊆ X a set/group/ring F(U), with restriction maps F(U)→F(V) for V⊆U, satisfying the gluing axiom: local sections that agree on overlaps glue to a unique global section.

Čech cohomology: Hˡ(U, F) computed from cocycles/coboundaries on nerve of cover U.

Ȟ⁰(X,F) = ker(δ⁰) = global sections
Ȟ¹(X,F) = ker(δ¹)/im(δ⁰) = obstructions to gluing

H¹ measures: can local data be assembled globally? For circle with Z-coefficients: H¹(S¹,ℤ) = ℤ.