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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 26th 2009

created field with one element with two useful references

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 21st 2014

added to field with one element some of the original references (Tits, Manin, Soulé) which were missing, with some brief words in the text.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 22nd 2014
• (edited Jul 22nd 2014)

Where in the literature, if anywhere, might I find the statement or claim or proposal that $Spec(\mathbb{Z}) \cup place_{\infty}$ is $\mathbb{P}_{\mathbb{F}_1}$ ?

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJul 22nd 2014

I thought the whole basis of the original observation of Tits was that formulae for number of points in projective spaces over $F_q$, made sense with $q = 1$. Projective $n$-space is a set of size $n$, vector spaces of dimension $n$ over $F_1$ are pointed sets of size $n+1$, etc. as in week 259.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 22nd 2014
• (edited Jul 22nd 2014)

This however is not the statement that the function field analogy suggests, is it, it is not the statement that $\mathbb{P}_{\mathbb{F}_1} = Spec(\mathbb{Z}) \cup place_{\infty}$

At least some people go around claiming that $\mathbb{F}_1$-geometry is what makes the function field analogy work (e.g. Peter Arndt in reply to your question on MO here, or Duff in an MO question here). I am getting the impression that however the bulk of the literature on $\mathbb{F}_1$ does not really relate to that. (?)

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJul 22nd 2014

Hmm, mind you here

The space $\mathbb{P}_1$ has three points, $c_X$ , $c_Y$ , $\eta$, two closed and one open. (p. 10)

I guess that agrees with John’s

The projective line over $F_1$ has just two points (or more precisely, two closed points).

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 22nd 2014
• (edited Jul 22nd 2014)

Thanks for providing these links. But so all of this is not (!) what the function field analogy suggests.

Or am I missing something?

Let’s step back, before discussing the projective line, let’s consider the affine line:

In which approaches to $\mathbb{F}_1$ is $\mathbb{Z} = \mathbb{F}_1[x]$ ?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 22nd 2014
• (edited Jul 22nd 2014)

Hm, maybe nobody does. I thought I saw this somewhere, but now I don’t find it anymore.

But what then do people say concerning interpreting the function field analogy in terms of geometry over $\mathbb{F}_1$? Clearly zeta functions are discussed all over the place in this context, but what beyond that? Does anyone discuss any other entries of the analogy table in terms of $\mathbb{F}_1$?

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJul 22nd 2014

Kapranov and Smirnov opt for

…we introduce the affine line over $F_1$, to consist of 0 and the roots of unity of all orders. So as a set it is identified with the algebraic closure of $F_1$. Also, the affine line should be regarded as the spectrum of the non-existing ring $F_1[t]$.

Soulé’s affine line over $F_1$ is bottom of p. 4 here.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeJul 22nd 2014

Perhaps you saw the first footnote to these mathcamp notes.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJul 22nd 2014
• (edited Jul 22nd 2014)

Thanks again, David!

So neither Kapranov-Smirnov nor Soulé say anything resembling Z=F1[x], or am I missing something?

But that footnote does!! That’s excellent. Now I am not feeling so alone anymore :-)

But I am perplexed: the whole starting point of the function field analogy and apparently also the community’s subconcious expectation is that F1[x] should be like Z. But at the same time none of the dozens of proposals for F1 gives something like this? Is this possible?

• CommentRowNumber12.
• CommentAuthorZhen Lin
• CommentTimeJul 22nd 2014

It would be very nice if the affine line over $\mathbb{F}_1$ “is” $\operatorname{Spec} \mathbb{Z}$… but that seems unlikely to me, because $\operatorname{Spec} \mathbb{Z}$ is somehow not very “homogeneous”. I also would not say that the function field analogy starts with $\mathbb{F}_1$; if anything, it is $\mathbb{F}_1$ that starts with the function field analogy!

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJul 22nd 2014

The function field analogy starts with saying that $Z$ is like $F_q[x]$ for $q\to 1$.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJul 22nd 2014
• (edited Jul 22nd 2014)

Hm, actually Michael Rosen’s book on the analogy “Number theory in function fields” starts with saying on its pages 1 and 2 that $\mathbb{Z}$ is like $F_q[x]$ for q=3. Er? Maybe it’s time for me to get another coffee.

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeJul 22nd 2014

Does what Connes writes on p. 3 of this help us a little with the conundrum of #14?

$|\mathbb{P}^d(\mathbb{F}_1)| = \frac{3^{d-1} - 1}{2} = |\mathbb{P}^d(\mathbb{F}_3)|.$

Some restriction of points to degree zero gets us to the $d + 1$ expected.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeSep 30th 2014
• (edited Sep 30th 2014)

I have added pointer to Durov’s text and in particular pointer to equation (0.4.24.2) there, where the definition of modules over $\mathbb{F}_1$ being modules over the maybe monad finally appears. (Also added the analogous pointer to maybe monad.)

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeJan 17th 2018

I added a reference

• CommentRowNumber18.
• CommentAuthorTim_Porter
• CommentTimeJun 16th 2018

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeSep 10th 2018
• (edited Sep 10th 2018)

Which authors admit that algebraic K-theory over the field with one element is… stable cohomotopy?

I see Guillot 06 “Adams operations in cohomotopy” says this

Apparently also Deitmar 06, but now I don’t find this.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeSep 11th 2018
• (edited Sep 11th 2018)

You saw this MO question? Peter Arndt’s answer mentions Deitmar’s role.

Riepe suggests Priddy “Transfer, symmetric groups, and stable homotopy theory” for an early appearance.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeSep 11th 2018

am starting a section “Algebra over $\mathbb{F}_1$” (here), to record some basics.

What’s a canonical source to cite for the idea or fact that modules over $\mathbb{F}_1$ are pointed finite sets (to go here)?

Sometimes this is apparently attributed to

• Christophe Soulé, Les varietes sur le corps a un element Mosc. Math. J., 4(1):217-244, 312, 2004 (pdf)

but I didn’t spot that exact statement in there. Wikipedia credits Noad Snyder’s secret seminar here (at the bottom). (?)

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeSep 11th 2018

David, our messages overlapped! Thanks for the pointer to Deitmar’s article!

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeSep 11th 2018

The latter from 1973 starts out with

K-theory of the category of finite sets and permutations together with the composition law of disjoint union corresponds to stable cohomotopy theory.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeSep 11th 2018
• (edited Sep 11th 2018)

Hi David,

so Priddy in his article, when he says “K-theory”, means K-theory of a permutative category, and the line you quote then refers to this result by Segal: $K(FinSet) \simeq \mathbb{S}$.

What I am trying to write out is how this is the first step in the “proof” that $\mathbb{S} \simeq K \mathbb{F}_1 \coloneqq K(\mathbb{F}_1 Mod)$, where on the right I mean the K-theory of the permutative category of $\mathbb{F}_1$-modules, in accord with this example.

Therefore my question for a good source of the claim or definiton that $\mathbb{F}_1 Mod = FinSet$ (or maybe $= FinSet^{\ast/}$, but that doesn’t make a difference for the K-theory of a permutative category, since it only sees the isomorphisms).

Now looking more closely at Noah Snyder’s old post (here) I get the impression that his argument is meant to be a review of Durov 07.

Now in Durov 07 I see mentioning of this statement as a parenthetical inside remark 2.5.6. Does Durov’s article make this more explicit in the sections before?

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeSep 11th 2018

John Baez told me about modules over $F_1$ being pointed sets at least as far back as 2006. I’ll see if I can trace back.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeSep 11th 2018

Thanks. I suppose the statement is folklore, but it would be nice to find some “canonical” reference that could be cited in publications.

• CommentRowNumber27.
• CommentAuthorDavid_Corfield
• CommentTimeSep 11th 2018

TWF 187 perhaps but for some reason I see a blank at John’s page.

• CommentRowNumber28.
• CommentAuthorTim_Porter
• CommentTimeSep 11th 2018

According to John on Azimuth, yesterday:

You may have noticed that all the pictures in my lectures are gone. That’s because they’re held on the U.C. Riverside math department server math.ucr.edu, and this server is down. It’s been down for several days since a power outage on campus. I’m trying to get people to restart it, but so far with no luck!

This explains the TWF observation of David_R.

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeSep 11th 2018

Gee, that sounds bad. Hopefully they have a backup…

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeSep 11th 2018
• (edited Sep 11th 2018)

added pointer also to

• Jack Morava, Rekha Santhanam, Power operations and absolute geometry, 2012 (pdf)

which sort of builds on

and his interpretation of stable cohomotopy as algebraic K-theory over $\mathbb{F}_1$ (and which seems to be the only article citing it?!)

• CommentRowNumber31.
• CommentAuthorDavid_Corfield
• CommentTimeSep 11th 2018

Using cached pages, I see even I knew it in 2005. In 2002, (TWF184) John was speaking of the projective spaces over $F_1$ as finite sets. But as you say in #26, it’s folklore.

On my travels, I see that John Berman in his thesis (Chap. 4) – Categorified algebra and equivariant homotopy theory – is after

a picture that unites noncommutative motivic homotopy theory (arising from the Eilenberg-Watts theorem) with global equivariant homotopy theory

which pairs ’Algebraic K-theory’ with ’Equivariant sphere spectrum’.

We anticipate that this analogy could be made rigorous by an understanding of noncommutative motives over the field with one element.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeSep 11th 2018

Thanks for the pointer to Berman (p. 92 I suppose?) I have added pointers here and here

• CommentRowNumber33.
• CommentAuthorDavid_Corfield
• CommentTimeSep 11th 2018

Yes p.92.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeOct 4th 2019

• Chenghao Chu, Oliver Lorscheid, Rekha Santhanam, Sheaves and K-theory for $\mathbb{F}_1$-schemes (https://arxiv.org/abs/1010.2896)
• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeOct 4th 2019
• (edited Oct 4th 2019)

So after the sentence

The perspective that the K-theory $K \mathbb{F}_1$ over $\mathbb{F}_1$ should be stable Cohomotopy has been highlighted in (Deitmar 06, p. 2, Guillot 06).

I have added this sentence:

Generalized to equivariant stable homotopy theory, the statement that equivariant K-theory $K_G \mathbb{F}_1$ over $\mathbb{F}_1$ should be equivariant stable Cohomotopy is discussed in Chu-Lorscheid-Santhanam 10, 5.3.

Am adding this now also to the respective points at stable Cohomotopy and equivariant stable Cohomotopy.

• CommentRowNumber36.
• CommentAuthorDavid_Corfield
• CommentTimeDec 9th 2020

Changed Berman article link to the arXiv version.