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

Abstract

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 ${\mathbb{F}}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as "improved" cyclotomic field extensions ${\mathbb{F}}_{1^n}$ of ${\mathbb{F}}_1$ 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: Tits' idea of Chevalley groups over ${\mathbb{F}}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unified view on analytic geometry over ${\mathbb{F}}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.

Slightly revised and extended version as in print. 51 pages

References in corpus (7)

• New Approach to Arakelov Geometry
• Lambda-rings and the field with one element
• Mapping F_1-land:An overview of geometries over the field with one element
• Congruence schemes
• Algebraic groups over the field with one element
• Non-Additive Prolegomena (to any future Arithmetic that will be able to present itself as a Geometry)
• Sheaves and $K$-theory for ${\mathbb{F}}_1$-schemes

Cited by in corpus (5)

• Functorial tropicalization of logarithmic schemes: The case of constant coefficients
• Analytic geometry over F_1 and the Fargues-Fontaine curve
• On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields
• Morita theory of systems
• Toward Homological Characterization of Semirings by e-Injective Semimodules