Instead, F1 refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects.
Mathematical objects such as vector spaces and polynomial rings can be carried over into these new theories by mimicking their abstract properties.
One of the defining features of theories of F1 is that these new foundations allow more objects than classical abstract algebra does, one of which behaves like a field of characteristic one.
In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes.
This is analogous to the condition in classical projective geometry that a line must contain at least three points.
In the late 1980s, Alexander Smirnov gave a series of talks in which he conjectured that the Riemann hypothesis could be proven by considering the integers as a curve over a field with one element.
By 1991, Smirnov had taken some steps towards algebraic geometry over F1,[3] introducing extensions of F1 and using them to handle the projective line P1 over F1.
[4] In 1993, Yuri Manin gave a series of lectures on zeta functions where he proposed developing a theory of algebraic geometry over F1.
[5] He suggested that zeta functions of varieties over F1 would have very simple descriptions, and he proposed a relation between the K‑theory of F1 and the homotopy groups of spheres.
The first published definition of a variety over F1 came from Christophe Soulé in 1999,[6] who constructed it using algebras over the complex numbers and functors from categories of certain rings.
[7] Deitmar suggested that F1 should be found by forgetting the additive structure of a ring and focusing on the multiplication.
[8] Toën and Vaquié built on Hakim's theory of relative schemes and defined F1 using symmetric monoidal categories.
[14] It has also been suggested to have connections to the unique games conjecture in computational complexity theory.
[15] Oliver Lorscheid, along with others, has recently achieved Tits' original aim of describing Chevalley groups over F1 by introducing objects called blueprints, which are a simultaneous generalisation of both semirings and monoids.
is a blue scheme G with a group operation that is a morphism in the Tits category, whose base extension is
The expansion of the q‑binomial coefficient into a sum of powers of q corresponds to the Schubert cell decomposition of the Grassmannian.
Morally, it mimicks the theory of schemes developed in the 1950s and 1960s by replacing commutative rings with monoids.
This construction achieves many of the desired properties of F1‑geometry: Spec F1 consists of a single point, so behaves similarly to the spectrum of a field in conventional geometry, and the category of affine monoid schemes is dual to the category of multiplicative monoids, mirroring the duality of affine schemes and commutative rings.
Furthermore, this theory satisfies the combinatorial properties expected of F1 mentioned in previous sections; for instance, projective space over F1 of dimension n as a monoid scheme is identical to an apartment of projective space over Fq of dimension n when described as a building.
This is non-naturally isomorphic to the cyclic group of order n, the isomorphism depending on choice of a primitive root of unity:[28] Thus a vector space of dimension d over F1n is a finite set of order dn on which the roots of unity act freely, together with a base point.
From this point of view the finite field Fq is an algebra over F1n, of dimension d = (q − 1)/n for any n that is a factor of q − 1 (for example n = q − 1 or n = 1).