Tagged Questions

12
votes
0answers
497 views

The symmetric group and the field with one element

I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This …
11
votes
1answer
509 views

Are there F_un Lie algebras ?

Background See WP-article on F_1 = F_{un} = Field with one element (and also this MO question). Paraphrasing someone: we do not know what is it, but it is not a field :). For this …
24
votes
2answers
1k views

Is the moduli space of curves defined over the field with one element?

There are various frameworks around which enlarge the category of rings to include more exotic objects such as the 'field with one element,' $\mathbb{F}_1$. While these frameworks …
25
votes
1answer
1k views

Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?

As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general: $\bullet …
52
votes
2answers
3k views

Riemann hypothesis via absolute geometry

Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute alge …
4
votes
1answer
392 views

Connected components of schemes over $\mathbb{F}_1$

I'm reading Deitmar's paper on Schemes over $\mathbb{F}_1$. Proposition 2.4. states that for a scheme $X$ over $\mathbb{F}_1$ there is a bijection between $X(\mathbb{F}_1)$ and the …
52
votes
7answers
4k views

What is the field with one element?

I've heard of this many times, but I don't know anything about it. What I do know is that it is supposed to solve the problem of the fact that the final object in the category of …
17
votes
1answer
2k views

Field with one element example?

$$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$ This formula holds for the unit ball in $\mathbb{Q_p}$. This formula also holds for $\mathbb{R}$ when $p=1$. Shou …
3
votes
1answer
210 views

affine and projective schemes over $\mathbf{F}_1$?

What should affine and projective schemes over $\mathbf{F}_1$ be?
19
votes
2answers
1k views

What is the algebraic closure of the field with one element?

If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one elemen …
2
votes
1answer
237 views

Coproducts of modules over an algebraic monad

Coproducts of modules over an algebraic monad $\Sigma$ are described in Section 4.16.14/15 in Durov's thesis. It is claimed there that for $\Sigma$-modules $M,N$, the set $M \copro …
6
votes
0answers
492 views

Invertible elements in generalized fields

Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are …
2
votes
2answers
373 views

Terminology for certain monoids which are to monoids like fields are to rings

Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus {0}$ is very similar to the condition for a commutative ring to be a field. This analogy is also us …
2
votes
0answers
485 views

Spec$\mathbb{Z}$ in absolute geometry

What are the obstacles that prevent from defining Spec$\mathbb{Z}$ in absolute geometry? By absolute geometry I mean the geometry over the field with one element F1.
12
votes
3answers
1k views

K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

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