Requiring only familiarity with the terminology of categories, this book will interest algebraic geometers and students studying schemes for the first time. Feit translates the geometric intuition of local structure into a purely categorical format, filling a gap at the foundations of algebraic geometry. The main result is that, given an initial category ${\mathcal C}$ of “local” objects and morphisms, there is a canonical enlargement of ${\mathcal C}$ to a category ${\mathcal C}^{gl}$ which contains all “global” objects whose local structure derives from ${\mathcal C}$ and which is functorially equivalent to the traditional notion of “global objects”. Using this approach, Feit unifies definitions for numerous technical objects of algebraic geometry, including schemes, Tate's rigid analytic spaces, and algebraic spaces.

Readership

Graduate students studying schemes for the first time; algebraic geometers with interest in foundational issues.