The geometry of blueprints: Part I: Algebraic background and scheme theory

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp. congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and F1-schemes (after Kato, Deitmar and Connes–Consani). Beside this unification, the category of blueprints contains new interesting objects as “improved” cyclotomic field extensions Fn1 of F1 and “archimedean valuation rings”. It also yields a notion of semiring schemes.This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Titsʼ idea of Chevalley groups over F1, congruence schemes, sheaf cohomology, K-theory and a unified view on analytic geometry over F1, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.