Cohomology-sheaf exact sequence
= Cohomology-sheaf exact sequence
A short exact sequence of <cochain complexes> of <sheaves> $0\to E\to F\to G\to0$ induces $\cdots\to\mathcal H^i(E)\to\mathcal H^i(F)\to\mathcal H^i(G)\to\mathcal H^{i+1}(E)\to\cdots$. Taking stalks is exact, so the construction and exactness follow from the ordinary long exact sequence of <cohomology> of complexes at every stalk. The stalkwise connecting maps come from the lift-and-differentiate construction and form a <sheaf> morphism.