All Questions
Tagged with field-with-one-element ag.algebraic-geometry
36 questions
8
votes
1
answer
1k
views
Why is $\operatorname{Spec}(\mathbb Z)$ supposed to lie over $\operatorname{Spec}(\mathbb F_1)$ rather than the other way around?
$\DeclareMathOperator\Spec{Spec}$I understand that one major motivation for the field with one element is supposed to be that there should be a map $\Spec(\mathbb Z) \to \Spec(\mathbb F_1)$, which has ...
- 63.9k
2
votes
0
answers
180
views
Is there a homotopical analogue of short exact sequence?
For $R$-modules for a commutative ring $R$, submodules and quotients are put on roughly the same footing; the kernel of a quotient is an injection into the source, and the cokernel of this injection ...
- 1,252
7
votes
0
answers
295
views
Connections between Borger's absolute geometry and Connes' and Consani's $\Gamma$-spaces
As the idea of an absolute geometry over the field with one element $\mathbb{F}_1$ becomes more clear, two approaches seem to have crystallized, being based on different assumptions and going into ...
6
votes
0
answers
469
views
Étale cohomology of the field with one element
In the function field - number field analogy, some expect progress on RH to come from reproducing various aspects of the Grothendieck program in a way where $\mathbb{Z}$ could be treated as a function ...
user30211
5
votes
0
answers
536
views
What do we know about $\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}$ and the spectral DM Stack $\mathrm{Spét}(\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z})$?
These days I've been trying to wrap my head around the current proposed approaches to algebraic geometry over the elusive "field with one element", one of whose main objects of interest is ...
- 11.8k
2
votes
0
answers
199
views
Does the category of rings embed fully faithfully into the category of $\mathbb{F}_{1}$-algebras?
The idea of a theory of algebraic geometry over the "field with one element" $\mathbb{F}_{1}$ is to give a fully faithfully embedding of categories
$$\mathsf{Sch}_{\mathbb{Z}}\hookrightarrow\...
- 11.8k
41
votes
1
answer
3k
views
Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry
Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
- 511
35
votes
2
answers
2k
views
Durov approach to Arakelov geometry and $\mathbb{F}_1$
Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry ...
- 14.7k
5
votes
1
answer
247
views
Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?
There is the following beautiful formula (see Qiaochu Yuan excellent blog):
$$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...
- 24.8k
7
votes
1
answer
390
views
Combinatorial/probabilistic statements having $F_{\text{un}}$/$F_q$ geometric interpetation
$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years.
One of the points is that it provides bridge between geometrical and ...
- 24.8k
7
votes
1
answer
434
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 ...
- 24.8k
14
votes
1
answer
801
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:
$$ \binom{ a }{ b}_{\!\!q} $$
...
- 24.8k
10
votes
0
answers
343
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 $G$ ...
- 7,789
25
votes
1
answer
1k
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 ...
- 3,309
15
votes
0
answers
2k
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 ...
- 3,309
9
votes
3
answers
941
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 ...
- 24.8k
26
votes
1
answer
816
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 ...
- 32.4k
33
votes
2
answers
1k
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 $q\to1$ ...
- 24.8k
1
vote
0
answers
709
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: $4$ ...
25
votes
1
answer
2k
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 ...
- 2,303
6
votes
0
answers
294
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 ...
user1437
8
votes
0
answers
508
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 ...
- 838
14
votes
0
answers
913
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 ...
- 20.2k
3
votes
1
answer
323
views
affine and projective schemes over $\mathbf{F}_1$?
What should affine and projective schemes over $\mathbf{F}_1$ be?
user19475
54
votes
2
answers
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 ...
- 63.1k
3
votes
1
answer
850
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 ...
- 63.1k
32
votes
2
answers
3k
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 ...
- 6,209
6
votes
2
answers
540
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 ...
- 3,290
18
votes
1
answer
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 $$\...
- 370
101
votes
2
answers
11k
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 ...
- 5,232
36
votes
3
answers
3k
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 ...
- 4,450
25
votes
4
answers
3k
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 ...
- 1,304
16
votes
5
answers
2k
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 ...
- 27.5k
6
votes
1
answer
875
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 ...
- 4,593
29
votes
2
answers
4k
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 ...
- 4,593
108
votes
7
answers
21k
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 one-...
- 5,062