From now on the main object (the irony of this terminology will become clear) of study is ${\mathbb{F}_1}$. I will first elaborate on what Kapranov-Smirnov calls the “folklore imagery”, trying to understand the motivation behind a statement like

A vector space of ${\mathbb{F}_1}$ is just a set.

Afterwards I will draw some conclusions from these observations, motivating some of the approaches outlined in Mapping ${\mathbb{F}_1}$-land.

Demystifying the ${\mathbb{F}_1}$-lore

In Projective geometry over ${\mathbb{F}_1}$ and the Gaussian binomial coefficient a nice build-up is given. I will summarise and give my own viewpoint.

We already know that ${\mathbb{F}_1}$ doesn’t exist as a field (because we need ${0\neq 1}$ by the axioms). But what if we loosen up the axioms and say the trivial ring should be taken as ${\mathbb{F}_1}$? In that case modules over this ring are in relationship with vector spaces over ${\mathbb{F}_1}$. But there are only trivial modules over the trivial ring, hence no nontrivial vector spaces. This approach obviously doesn’t work. A conclusion I like to make after vastly generalising this approach:

Setting ${\mathbb{F}_1}$ to be something doesn’t work.

The idea of looking at vector spaces looks promising though: given a field it is the most obvious structure built upon it, so we want to make sense of it over ${\mathbb{F}_1}$. We know that the cardinality of an ${n}$-dimensional vector space ${V}$ over a finite field ${\mathbb{F}_q}$ is ${q^n}$. Applying this to ${\mathbb{F}_1}$ we get a single point, for all ${n}$. So far so good, but we also know that a basis for ${V}$ consists of ${n}$ elements. The problem with this approach is that it’s too direct: we cannot construct objects over ${\mathbb{F}_1}$, we need to get there by an analogy that avoids contradictions like this. Applying this for instance to ${\mathbb{F}_1[t]}$ we see that this doesn’t yield any satisfying definition either. The conclusion after another round of generalisation:

Look at induced objects, not constructions.

The same applies to noncommutative geometry by the way. But let’s focus on ${\mathbb{F}_1}$ for the moment.

So we need a simple object over ${\mathbb{F}_1}$ where we can avoid an explicit construction, getting facts about that object only by analogy without running into contradictions. Let’s try this for ${\mathbb{P}^n/\mathbb{F}_1}$. The construction of this over an actual field ${k}$ consists of constructing ${\mathbb{A}^{n+1}/k}$ (an ${n+1}$-dimensional vector space over ${k}$) and setting ${\mathbb{P}^n/k}$ to be the set of lines through the origin.

If we take ${k=\mathbb{F}_q}$ the number of points in ${\mathbb{A}^{n+1}/\mathbb{F}_q}$ is ${q^{n+1}}$. The number of lines through the origin is ${(q^{n+1}-1)/(q-1)}$: just take any point in ${\mathbb{A}^{n+1}/\mathbb{F}_q\setminus\left\{ 0 \right\}}$, this defines a line through the origin, but there are ${q}$ points (including the origin) on this line, therefore we divide by ${q-1}$. If we write down the polynomial function counting the number of points in ${\mathbb{P}^n/\mathbb{F}_q}$ (i.e., ${q^n+q^{n-1}+\ldots+1}$) and evaluate in ${q=1}$ we see something that leads to a nontrivial object! To make this analogy really sound you have to write down what you actually want from a finite projective space and axiomatise it. But let’s rejoice for now, and conclude

The projective geometry ${\mathbb{P}^1/\mathbb{F}_1}$ contains ${n}$ points.

What we have actually done is changing the construction of ${\mathbb{P}^n/\mathbb{F}_1}$ from an algebraic-geometric viewpoint to a combinatorial-geometric viewpoint, something that is quintessential in finite geometry (and I guess I’m the angs+’er with the most background in this kind of stuff ). If people are interested in a write-up about this, I’d be happy to provide one but I suspect my fellow seminarians are not that into finite geometry and combinatorics. The only important observation we need to make is that a line in ${\mathbb{P}^n/\mathbb{F}_1}$ contains exactly two points in this sense, as a line in ${\mathbb{P}^n/\mathbb{F}_q}$ contains ${q+1}$ points by the axioms of a combinatorial-geometric projective space.

Now taking ${\mathbb{F}_1}$-vector spaces to be sets actually makes sense: ${\mathbb{A}^{n+1}/\mathbb{F}_1}$ is an ${(n+1)}$-set, taking any point as the distinguished base point (or origin) and considering “lines” through the origin, or more appropriately ${2}$-subsets, of which we have exactly ${n}$ as we started with ${n+1}$ elements and fixed an origin. But this (inherently geometric) idea of “fixing an origin” has its downsides in what follows, where we will adjoin an origin which is more in the sense of algebra as adding a zero vector (which doesn’t exist over ${\mathbb{F}_1}$) doesn’t change the dimension or cardinality of a base for a vector space.

Implications

Now we can switch our attention to Kapranov’s and Smirnov’s unfinished paper. If Wittgenstein were an algebraic geometer interested in ${\mathbb{F}_1}$-geometry I guess he would have written a Tractatus Absoluto-Geometricus, which could have looked (in a very crude sense) like this

1. 1 Geometry over ${\mathbb{F}_1}$ can only be understood through induced objects.
2. 2 Vector spaces over ${\mathbb{F}_1}$ are plain sets.
   1. 2.1 Dimension equals cardinality.
   2. 2.2 ${\mathrm{GL}_n(\mathbb{F}_1)=\mathrm{S}_n}$.
   3. 2.3 ${\mathrm{SL}_n(\mathbb{F}_1)=\mathrm{A}_n}$.
   4. 2.4 ${\det\colon\mathrm{GL}_n(\mathbb{F}_1)\rightarrow\mathbb{F}_1^\times}$ is the sign homomorphism.
   5. 2.5 The Grassmannian ${\mathrm{Gr}(k,n)(\mathbb{F}_1)}$ is the set of ${k}$-subsets.
   6. 2.6 There is no harm in formally adjoining a zero vector in a ${\mathbb{F}_1}$-vector space turning it into a pointed set, just be careful with interpretations.
3. 3 The polynomial ring ${\mathbb{F}_1[t]}$ can only be understood through its automorphisms.
   1. 3.1 Polynomial automorphisms are a generalisation of field automorphisms by evaluation at “zero”.
   2. 3.2 We have ${\mathrm{GL}_n(\mathbb{F}_1[t])\rightarrow\mathrm{GL}_n(\mathbb{F}_1)}$.
   3. 3.3 ${\mathrm{GL}_n(\mathbb{F}_1[t])=\mathrm{B}_n}$ the braid group on ${n}$ strings, by analogy of the canonical ${\mathrm{B}_n\rightarrow\mathrm{S}_n}$.
   4. 3.4 For more information I refer you to the grandmaster himself and his blog post ${\mathbb{F}_1}$ and braid groups.
4. 4 Finite fields have finite extensions and algebraic closures and so does ${\mathbb{F}_1}$.
   1. 4.1 ${\mathbb{F}_{1^n}}$ as a vector space over ${\mathbb{F}_1}$ is the (pointed) set ${\mu_n}$ consisting of the ${n}$-th roots of unity and an adjoined zero (which will serve as the point of the pointed set).
   2. 4.2 By choosing a primitive root in ${\mu_n}$ we get a non-canonical isomorphism ${\mathrm{C}_n\cong\mu_n}$.
   3. 4.3 ${\mathbb{A}^1/\mathbb{F}_1}$ as a scheme is ${\mathrm{Spec}\,\mathbb{F}_1[t]}$.
   4. 4.4 ${\mathrm{Spec}\,\mathbb{F}_1[t]}$ describes the algebraic closure of ${\mathbb{F}_1}$.
   5. 4.5 ${\overline{\mathbb{F}_1}}$ therefore corresponds to ${\mathrm{\mu}_\infty\cup\left\{ 0 \right\}}$.
5. 5 We can generalise linear algebra to finite extensions.
   1. 5.1 The action of ${\mathbb{F}_{1^n}}$ on ${V}$ is the action of ${\mu_n}$ on ${V\setminus\left\{ 0 \right\}}$.
   2. 5.2 ${\mathbb{F}_{1^n}}$-vector spaces are nothing but ${\mu_n}$-sets.
   3. 5.2 A ${d}$-dimensional ${\mathbb{F}_1}$-space contains ${dn}$ points.
   4. 5.4 The ${\mu_n}$-action is strictly multiplicative, there is no additive structure.
   5. 5.5 The lack of additive structure agrees with the notion of vector spaces as (pointed) sets.
   6. 5.6 A ${\mathbb{F}_{1^n}}$-basis is a set containing a representative of every orbit under the ${\mu_n}$-action.
6. 6 We can interpret other finite objects over ${\mathbb{F}_1}$.
   1. 6.1 ${\mathbb{F}_q}$ is a ${\mathbb{F}_{1^n}}$-vector space if ${q\equiv 1\bmod n}$ because ${(\mathbb{F}_q^\times,\cdot)\cong(\mathrm{C}_{nd},\cdot)}$ for ${d=(q-1)/n}$, and therefore a ${\mathbb{F}_{1^n}}$-algebra.
   2. 6.2 This vector space structure induces the (necessarily unique) ${\mathbb{F}_{1^n}}$-vector space structure on ${\mathbb{F}_q^e}$ of dimension ${ed}$.
   3. 6.3 All development of techniques should coincide with this observation.
   4. 6.4 For a construction of exact sequences over ${\mathbb{F}_1}$ I refer you to Absolute linear algebra.

For the real Wittgenstein aficionado, my apologies for not hitting the right tone and ideas. This post mostly served as a way for me to get all my ${\mathbb{F}_1}$-folklore straight, so that I can explain to a complete outsider why stuff in the ${\mathbb{F}_1}$ is taken as what it is. If you feel any gaps present, please tell me so.

If you followed me this far you should be familiar enough with the ideas and twists of mind necessary to understanding some parts of Kapranov-Smirnov, or you have made up your mind and will take all ${\mathbb{F}_1}$-stuff to be dadaist nonsense. But before tackling Kapranov-Smirnov I suggest you read the series of posts developing the same theme as this one did but in greater depth:

1. The ${\mathbb{F}_1}$ folklore
2. Absolute linear algebra
3. ${\mathbb{F}_1}$ and braid groups

For the next (and last) post of this series, the main idea will be Stuff over ${\mathbb{F}_1}$ contains multiplicative structure, not additive structure.

About Pieter Belmans

Besides my mathematical interests which range from finite and algebraic geometry to number theory and their computational aspects I entertain myself with LaTeX, music theory, playing guitar and reading (what I believe to be) (post)modern literature.

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