Questions tagged [field-with-one-element]
This tag refers to the (non-existent) "field of one element".
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Does the $\mathbb{F}_1$ point of view lead to any testable predictions?
In number theory we can informally consider number rings as curves over something like a field with one element. For example it is mentioned here by Kedlaya.
The question is does this perspective lead ...
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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 ...
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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 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 ...
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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" ...
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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 ...
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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 $G$ ...
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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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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 ...
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In what sense do the real and complex places correspond to setting q equal to 1 or -1?
It often happens that if we have a scheme $X/\mathbb Z$ (or an open subset thereof) and we denote by $p(q) = X(\mathbb F_q)$, then $p(1)$ and $p(-1)$ compute the euler characteristic of $X(\mathbb C)$ ...
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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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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 ...
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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 ...
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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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Lax monoidal structure on the right Kan extension of a partially monoidal Γ-set
First some preliminaries. Let me write $Fin_\ast$ for the skeleton of the category of finite pointed sets and pointed maps between them on the objects $n_+=\{0,1,...,n\}$, where $0$ is the base point (...
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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 ...
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Intermediate arithmetic results in F_1 geometry
Much is made of the search for a proof of the Riemann hypothesis via the field with one element. Are there any lesser classic arithmetic results that have been proved with F_1 geometry, such as the ...
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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 ...
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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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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 ...
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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\...
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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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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: $4$ ...