This tag refers to the (non-existent) "field of one element".

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### Laurent and power series over the field with one element?

Question. Is there a suitable notion of the Laurent series ring $\mathbb{F}_1((t))$ and power series ring $\mathbb{F}_1[[t]]$ in some framework for the field with one element $\mathbb{F}_1$?
For ...

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**1**answer

156 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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votes

**1**answer

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### 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 ...

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votes

**3**answers

717 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 ...

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639 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

**1**answer

565 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 ...

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votes

**1**answer

259 views

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

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

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**1**answer

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### 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 ...

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vote

**2**answers

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 ...

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votes

**1**answer

484 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 ...

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**0**answers

543 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 ...

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votes

**1**answer

262 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 ...

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**0**answers

567 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.

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**2**answers

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### 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 ...

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votes

**2**answers

441 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 ...

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votes

**1**answer

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 ...

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votes

**2**answers

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### 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 ...

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**1**answer

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### 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 ...

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**1**answer

505 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 ...

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### 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 ...

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votes

**4**answers

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### 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 ...

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votes

**2**answers

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### 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 ...

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votes

**2**answers

1k 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 ...

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votes

**5**answers

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 ...

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votes

**3**answers

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 ...

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votes

**1**answer

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### 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 ...

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votes

**2**answers

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### 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 ...

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votes

**2**answers

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 ...

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votes

**7**answers

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### 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 ...

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votes

**3**answers

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### 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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### 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 ...