Schema
Overview
Ringed space (X, O_X) locally isomorphic to Spec(R). Grothendieck's replacement for varieties; allows nilpotents, generic points. Foundation for modern algebraic geometry.
Era: 1950s–1960s
Proposed at: Bures-sur-Yvette, France (IHES)
Formal structure
Domain: Ring R; prime ideals p ⊂ R; affine scheme Spec(R) = {p : p prime}
O_X(D(f)) = R_f; scheme X = ⋃ Spec(R_i) with gluing
Constraints: O_X sheaf of rings; stalks local rings; morphisms local homomorphisms
Objective: Study varieties via commutative algebra; cohomology