The geometry of blueprints. Part I: Algebraic background and scheme theory
By Oliver Lorscheid. 2011
78 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/c9099529f0dad79508adca2b1c2542bb69a9e49b
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In Algebra and Number Theory
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Citations in the corpus
- Equations of tropical varieties (2013, background): “xactly to the relations satisfied by a valuation , φis well-defined. Uniqueness is immediate. 3. F1-SCHEMES AND SEMIRING SCHEMES 3.1. Construction of F1-schemes and semiring schemes. The papers [TV09], [Lor12], and [Dur07] each construct categories of schemes over semirings and some notion of F1. For ”
- Algebraic Geometry Over Hyperrings (2015, background): “This motivated Connes and Consani to search possible other candidates rather than monoids and the study of algebraic geometry over more general algebraic objects by many others (cf. [3], [13], [28], [35], [37]).”
- Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “authors of [GG16] work in the category of semiring scheme and consider the base change X∆×F 1 T, where Tdenotes the semi-ring of tropical numbers. 6 In [Lor15], based on his theory of blueprints (see [Lor12]), Lorscheid proposes a much larger framework for tropicalization that generalizes both Gians”