Questions tagged [f-1]

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

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27
votes
2answers
815 views

Durov approach to Arakelov geometry and $\mathbb{F}_1$

Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry ...
9
votes
0answers
244 views

The term “absolute geometry”

My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
4
votes
1answer
170 views

Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?

There is the following beautiful formula (see Qiaochu Yuan excellent blog): $$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...
12
votes
2answers
362 views

$q$ as a prime power and a root of unity

The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer $$[n]_q := \frac{q^n-1}{q-1}.$$ In analogy, the number of ...
15
votes
1answer
510 views

Schur-Weyl duality and q-symmetric functions

Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...
20
votes
1answer
789 views

What is $\mathbb{Q}_1$, the “field of $1$-adic numbers”?

(Disclaimer: I'm totally ignorant about $\mathbb{F}_1$ theory) There are now (several) working definitions of the "field with one element" $\mathbb{F}_1$ (not literally a field, of course), and ...
7
votes
0answers
208 views

Combinatorial/probabilistic statements having $F_{un}$/$F_q$ geometric interpetation

There were lots of "Fun with $F_{un}$" (field with one element) recent years. One of the points is that it provides bridge between geometrical and combinatorial questions, i.e. in the limit case $q=1$...
7
votes
1answer
352 views

Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

Let's start from a little bit far. Basic probability theory - chain rule reads: $$ P(AB) = P(A)P(B|A)$$ Example: consider n+m balls, where n - white balls, m - black balls, consider A - first ...
14
votes
1answer
515 views

Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?

The q-Vandermonde identity reads: $$ \binom{m + n}{k}_{\!\!q} =\sum_{j} \binom{m}{k - j}_{\!\!q} \binom{n}{j}_{\!\!q} q^{j(m-k+j)} $$ The q-binomial coefficients: $$ \binom{ a }{ b}_{\!\!q} $$ ...
9
votes
0answers
283 views

What are the analogs of a Levi/Parabolic/Borel/Bruhat over the field with 1 element?

This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway. If I understand correctly, for any reductive algebraic group $G$ ...
22
votes
1answer
811 views

How is Borger's approach to $\mathbb{F_{1}}$ related to previous approaches (e.g. Deitmar's)?

The "traditional" approach to the so-called "field with one element" $\mathbb{F}_{1}$ is by using monoids, or, to put it in another way, by forgetting the additive structure of rings. In Deitmar's ...
14
votes
0answers
1k views

Inter-Universal Teichmuller Theory and the Field with One Element

The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems ...
5
votes
2answers
205 views

Field with one element look at counting index-$n$ subgroups in terms of Homs to $S_n$, generalization to $F_{1^k}$?

Main idea shortly: As we discussed recently MO272045, there is beautiful fomula which counts index-n subgroups in terms of homomorphisms to $S_n$. Let me give "field with one element" interpretation ...
10
votes
3answers
698 views

Buildings, projective geometry - what led Tits to think of “the field with one element”?

The mysterious object "field with one element" seems to appear first in J. Tits papers on buildings. It is mentioned in almost any text on $\mathbb{F}_1$. However, I have never seen any exposition of ...
26
votes
1answer
709 views

What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?

The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
32
votes
2answers
889 views

Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element". Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
11
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0answers
498 views

Bijection between conjugacy classes and irreducible representation of Weyl group = Langlands correspondence over “field with one element”

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group there is well-known "natural bijection" ...
11
votes
1answer
482 views

Gauss, Jacobi, Kloosterman sums and representation theory in the $\mathbb F_1$-world

This question is inspired by Why are Bessel function and Kloosterman sum similar? - it developed in me desire to understand Kloosterman sums better. There seems to be common knowledge that Gauss, ...
3
votes
0answers
192 views

Can MacLane's notion of universality inform $\mathbb{F}_1$?

MacLane (1939) calls a field $F$ universal if every other field $F'$ of the same cardinality and characteristic as $F$ is a subfield of $F$. He then exhibits an example, viz. a field of generalized ...
13
votes
1answer
695 views

What “should” be the absolute galois group of a field with one element

As far as I know there is many "suggestions" of what should be a "field with one element" $\mathbf{F}_{1}$. My question is the following: How we should think or what should be the "absolute Galois ...
5
votes
2answers
183 views

Descent of flatness from algebras to monoids

Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-...
10
votes
1answer
538 views

Is an ordinary scheme in Borger's Absolute Geometry the same as a “scheme over F1” with a map to Spec(Z)?

$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\F}{\mathbb{F}_1}$ $\newcommand{\spec}{\operatorname{Spec}}$ If I understand correctly, in Borger's paper about the field with one element, the category of "...
1
vote
0answers
268 views

What is the significance of the $-1$-simplex?

The number of $k$-simplex elements in an $n$-simplex is counted by the binomial coefficient $\binom{n+1}{k+1}$. For example, the $3$-simplex is the tetrahedron, which has the following elements: $4$ ...
6
votes
0answers
255 views

What is known about the $q$-analogue of the simplex?

I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ...
24
votes
1answer
1k views

A geometric theory of Blueprints? (Algebras over the field with one element)

In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F_1$, I was just reading through Lorscheid's paper The geometry of blueprints. I certainly like the idea a ...
6
votes
0answers
258 views

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 ...
2
votes
1answer
226 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 ...
22
votes
1answer
767 views

Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?

As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...
14
votes
0answers
408 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 ...
7
votes
0answers
451 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
250 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 ...
11
votes
1answer
687 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 ...
24
votes
3answers
2k 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 ...
14
votes
0answers
815 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 ...
15
votes
1answer
684 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
312 views

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

What should affine and projective schemes over $\mathbf{F}_1$ be?
47
votes
2answers
4k 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
421 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 "...
3
votes
1answer
647 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 ...
7
votes
0answers
601 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
288 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
665 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.
30
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
482 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
3k 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 $$\...
86
votes
2answers
8k 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 ...
27
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 claim?...
12
votes
1answer
534 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 K-...
31
votes
3answers
2k 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 ...
23
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 ...