This tag refers to the (non-existent) "field of one element".
2
votes
1answer
125 views
Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients?
This is a mostly philosophical question. Is it fair to think of usual binomial coefficients and their identities as an $F_1$ case of $q$-binomial coefficients and identities? Here $F_1$ is the field ...
14
votes
0answers
299 views
Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?
I recently gave a talk, where I talked about the tensor category
of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor.
Vaughan Jones, who was in the audience, later told me ...
5
votes
0answers
333 views
The logarithm over $\mathbb F_1$
In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'.
Question 1: can somebody explain or give ...
3
votes
0answers
126 views
Categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$
This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains ...
10
votes
1answer
427 views
Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category?
Following Anton Deitmar, let $\mathcal B$ be an "$\mathbb F_1$-linear category" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is balanced, pointed, contains finite products, kernels, and ...
20
votes
3answers
629 views
The non-simplicity of $SO(4)$ and $A_4$
It is well known that the alternating group $A_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the ...
13
votes
0answers
617 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 does make a lot of ...
13
votes
1answer
556 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 question it is ...
3
votes
1answer
251 views
affine and projective schemes over $\mathbf{F}_1$?
What should affine and projective schemes over $\mathbf{F}_1$ be?
30
votes
1answer
2k 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:
$\bullet$ In the approach by ...
1
vote
2answers
395 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 used in the work ...
4
votes
1answer
460 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 set of connected ...
6
votes
0answers
535 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 subtrivial. For ...
2
votes
1answer
257 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 $M \coprod N$ generates $M ...
2
votes
0answers
547 views
Spec$\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.
25
votes
2answers
2k 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 differ in their ...
5
votes
2answers
429 views
Are the closures of the tori in the decomposition of a torified variety toric varieties?
In "Torified varieties and their geometries over $\mathbb{F}_1$", J. L. Pena and O. Lorscheid define a torified variety as a variety $X$ over $\mathbb{Z}$ along with a family of locally closed ...
18
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 $p=1$. Should one expect ...
64
votes
2answers
4k 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 algebraic geometry over ...
24
votes
1answer
1k views
Tomita-Takesaki versus Frobenius: where is the similarity?
I've often heard Alain Connes say that the modular flow of Tomita-Takesaki theory should be thought of as a characteristic zero analog of the Frobenius endomorphism. ... can anyone justify this ...
12
votes
1answer
502 views
Are centrally extended p-adic groups defined over F_1?
Let G be a semisimple algebraic group.
Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor G : Rings → Groups by the second algebraic ...
20
votes
2answers
1k views
Tannaka formalism and the étale fundamental group
For quite a while, I have been wondering if there is a general principle/theory that has
both Tannaka fundamental groups and étale fundamental groups as a special case.
To elaborate: The theory of ...
18
votes
4answers
2k views
Applications of algebraic geometry over a field with one element
I would like to understand at least one of the several existing approaches to algebraic geometry over $\mathbb{F}_1$ (the field with one element). Is there an example of an "interesting" theorem that ...
12
votes
2answers
1k views
Why does the Gamma function satisfy a functional equation?
In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
12
votes
2answers
991 views
What does Faltings' theorem look like over function fields?
Minhyong Kim's reply to a question John Baez once asked about the analogy between $\text{Spec } \mathbb{Z}$ and 3-manifolds contains the following snippet:
Finally, regarding the field with one ...
13
votes
5answers
1k views
Elliptic Curves over F_1?
Is there an notion of elliptic curve over the field with one element? As I learned from a previous question, there are several different versions of what the field with one element and what schemes ...
28
votes
3answers
2k views
Topological Langlands?
In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...
6
votes
1answer
498 views
What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be?
What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least ...
21
votes
2answers
2k 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 element.
I saw that the ...
8
votes
2answers
1k views
What is $\overline{\text{Spec}\mathbb{Z}}$?
In Connes work on the Riemann Hypothesis he talks about constructing $\overline{\text{Spec}\mathbb{Z}}$ as a curve over the field with one element. I just want to know what Spec means. Is the same as ...
62
votes
7answers
6k 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 schemes is ...
14
votes
3answers
2k 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?
16
votes
12answers
2k views
Homological Algebra for Commutative Monoids?
Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
