Atlas of Fundamental Concepts

Presheaves encode restriction; sheaves impose local identity and gluing. Stalks isolate germs at points. Ring-valued sheaves create ringed and locally ringed spaces; localization produces local coordinate rings; $\operatorname{Spec}(R)$ packages prime ideals, topology, and a structure sheaf into an affine scheme. Schemes are spaces locally modeled on affine spectra, and additional hypotheses produce Noetherian, reduced, integral, regular, separated, projective, and variety-like geometries.