Holmstrom Field with one element

Many people have expressed hopes for a definition of the “field with one element”, which would be useful in various aspects of arithmetic geometry. Some approaches: Soule and Durov

See stuff on the web page of Consani, and her talk from the Toronto workshop.

Blogs! See neverendingbooks and F_un mathematics

Cameron blog post

Field with one element folder, under N TH

arXiv:1103.1745 The geometry of blueprints. Part I: Algebraic background and scheme theory from arXiv Front: math.AG by Oliver Lorscheid A blueprint generalizes 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 semi-ring 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 and $K$-theory and a unified view on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.

F1 library folder under N TH

arXiv:1103.1235 Reconstructing the spectrum of F_1 from the stable homotopy category from arXiv Front: math.AT by Stella Anevski The finite stable homotopy category S_0 has been suggested as a candidate for a category of perfect complexes over the monoid scheme Spec F_1. We apply a reconstruction theorem from algebraic geometry to S_0, and show that one recovers the one point topological space. We also classify filtering subsets of the set of principal thick subcategories of S_0, and of its p-local versions. This is motivated by a result saying that the analogous classification for the category of perfect complexes over an affine scheme provides topological information.

arXiv:1009.0121 Construction of schemes over $F_1$, and over idempotent semirings: towards tropical geometry from arXiv Front: math.AG by Satoshi Takagi In this paper, we give some categorical description of the general spectrum functor, defining it as an adjoint of a global section functor. The general spectrum functor includes that of $F_1$ and of semirings.

Toen and Vaquie: Under Spec Z. Some notes: Idea: Think of commutative monoids in a symm monoidal cat C as models for affine schemes relative to C. If there is a reasonable symmetric monoidal functor from C to Z-modules, get a base change functor, and a notion of scheme under Spec(Z). Homotopical version of this requires C to have a model structure. Now have flat and Zariski topology. Can make sense of schemes: a functor with a Zariski covering. Stuff about toric varieties and GL. Brave new AG over the sphere spectrum, and the spectrum with one element.

arXiv:1102.4046 Congruence schemes from arXiv Front: math.AG by Anton Deitmar A new category of algebro-geometric objects is defined. This construction is a vast generalization of existing F1-theories, as it contains the the theory of monoid schemes on the one hand and classical algebraic theory, e.g. Grothendieck schemes, on the the other. It also gives a handy description of Berkovich subdomains and thus contains Berkovich’s approach to abstract skeletons. Further it complements the theory of monoid schemes in view of number theoretic applications as congruence schemes encode number theoretical information as opposed to combinatorial data which are seen by monoid schemes.

[arXiv:0907.3824] Algebraic groups over the field with one element from arXiv Front: math.AG by Oliver Lorscheid Remarks in a paper by Jacques Tits from 1956 led to a philosophy how a theory of split reductive groups over $\F_1$, the so-called field with one element, should look like. Namely, every split reductive group over $\Z$ should descend to $\F_1$, and its group of $\F_1$-rational points should be its Weyl group. We connect the notion of a torified variety to the notion of $\F_1$-schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive groups as $\Fun$-schemes. We endow the class of $\F_1$-schemes with two classes of morphisms, one leading to a satisfying notion of $\F_1$-rational points, the other leading to the notion of an algebraic group over $\F_1$ such that every split reductive group is defined as an algebraic group over $\F_1$. Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of $\GL(n)$ and Grassmann varieties are realized in this theory.

arXiv:0908.3124 A theory of base motives from arXiv Front: math.AG by Jack Morava A category of correspondences based on Waldhausen A-theory has interesting analogies, in the context of differential topology, to categories of mixed Tate motives studied in arithmetic geometry

In particular, the Hopf object S \wedge_A S (regarding A() as a kind of local ring over the sphere spectrum) has some similarities to a motivic group for this category; its associated rational Lie algebra is free, on odd-degree generators…

[arXiv:0909.0069] Mapping F_1-land:An overview of geometries over the field with one element from arXiv Front: math.AG by Javier López Peña, Oliver Lorscheid This paper gives an overview of the various approaches towards F_1-geometry. In a first part, we review all known theories in literature so far, which are: Deitmar’s F_1-schemes, Toën and Vaquié’s F_1-schemes, Haran’s F-schemes, Durov’s generalized schemes, Soulé’s varieties over F_1 as well as his and Connes-Consani’s variations of this theory, Connes and Consani’s F_1-schemes, the author’s torified varieties and Borger’s Lambda-schemes. In a second part, we will tie up these different theories by describing functors between the different F_1-geometries, which partly rely on the work of others, partly describe work in progress and partly gain new insights in the field. This leads to a commutative diagram of F_1-geometries and functors between them that connects all the reviewed theories. We conclude the paper by reviewing the second author’s constructions that lead to realization of Tits’ idea about Chevalley groups over F_1.

arXiv:1009.3235 On the Algebraic K-theory of Monoids from arXiv Front: math.KT by Chenghao Chu, Jack Morava Let $A$ be a not necessarily commutative monoid with zero such that projective $A$-acts are free. This paper shows that the algebraic K-groups of $A$ can be defined using the +-construction and the Q-construction. It is shown that these two constructions give the same K-groups. As an immediate application, the homotopy invariance of algebraic K-theory of certain affine $\mathbb{F}_1$-schemes is obtained. From the computation of $K_2(A),$ where $A$ is the monoid associated to a finitely generated abelian group, the universal central extension of certain groups are constructed.

nLab page on Field with one element