field with one element



Various identities coming from algebraic geometry (and particularly algebraic groups) over finite fields 𝔽 q\mathbb{F}_q turn out to make perfect sense as expressions in qq when extrapolated to the case q=1q=1, and to reflect interesting (combinatorial, representation theoretical…) facts, even though, of course, there is no actual field with a single element.

Motivated by such observations, Jacques Tits envisioned a new kind of geometry adapted to the explanation of these identities. Christophe Soulé then expanded on Tits’ ideas by introducing the notion of field with one element and studying its fine arithmetic invariants. While there is no field with a single element in the standard sense of field, the idea is that there is some other object, denoted 𝔽 1\mathbb{F}_1, such that it makes sense to speak of “geometry over 𝔽 1\mathbb{F}_1”. Following the French pronunciation one also writes F unF_{un} (and is thus led to the inevitable pun).

In the relative point of view the SS-schemes are schemes with a morphism of schemes over a base scheme SS; but every SS-scheme is a scheme over Spec()Spec(\mathbb{Z}). In absolute algebraic geometry all “generalized schemes” should live over Spec(F 1)Spec(F_1) and Spec(F 1)Spec(F_1) should live below Spec()Spec(\mathbb{Z}); this is similar to the fact that the quotient stacks like [*/G][*/G] live below the single point ** (there is a direct image functor from sheaves on a point to sheaves over [*/G][*/G]. One of the principal and very bold hopes is that the study of F unF_{un} should lead to a natural proof of Riemann conjecture; namely some attractive strategies to such a proof already exist (see MathOverflow here).


After the very first observations by Tits, pioneers were Soulé and Kapranov and Smirnov. More recently there are extensive works by Alain Connes and Katia Consani, Nikolai Durov, James Borger and Oliver Lorscheid.

References and links

See also Lambda-ring, blue scheme and tropical geometry.


A survey of the various competing theories is

more details

An approach in terms of Lambda-rings is

Revised on September 5, 2013 18:52:26 by Zoran Škoda (