## June 13, 2018

### Fun for Everyone

#### Posted by John Baez

There’s a been a lot of progress on the ‘field with one element’ since I discussed it back in “week259”. I’ve been starting to learn more about it, and especially its possible connections to the Riemann Hypothesis. This is a great place to start:

Abstract. This text serves as an introduction to $\mathbb{F}_1$-geometry for the general mathematician. We explain the initial motivations for $\mathbb{F}_1$-geometry in detail, provide an overview of the different approaches to $\mathbb{F}_1$ and describe the main achievements of the field.

Lorscheid’s paper describes various approaches. Since I’m hoping the key to $\mathbb{F}_1$-mathematics is something very simple and natural that we haven’t fully explored yet, I’m especially attracted to these:

• Deitmar’s approach, which generalizes algebraic geometry by replacing commutative rings with commutative monoids. A lot of stuff in algebraic geometry, like the ideals and spectra of commutative rings, or the theory of schemes, doesn’t really require the additive aspect of a ring! So, for many purposes we can get away with commutative monoids, where we think of the monoid operation as multiplication. Sometimes it’s good to use commutative monoids equipped with a ‘zero’ element. The main problem is that algebraic geometry without addition seems to be approximately the same as toric geometry — a wonderful subject, but not general enough to handle everything we want from schemes over $\mathbb{F}_1$.

• Toën and Vaquié’s approach, which goes further and replaces commutative rings by commutative monoid objects in symmetric monoidal categories (which work best when they’re complete, cocomplete and cartesian closed). If our symmetric monoidal category is $(\mathbf{AbGp}, \otimes)$ we’re back to commutative rings, if it’s $(\mathbf{Set}, \times)$ we’ve got commutative monoids, but there are plenty of other nice choices: for example if it’s $(\mathbf{CommMon}, \otimes)$ we get commutative rigs, which are awfully nice.

One can also imagine ‘homotopy-coherent’ or ‘$\infty$-categorical’ analogues of these two approaches, which might provide a good home for certain ways the sphere spectrum shows up in this business as a substitute for the integers. For example, one could imagine that the ultimate replacement for a commutative ring is an $E_\infty$ algebra inside a symmetric monoidal $(\infty,1)$-category.

However, it’s not clear to me that homotopical thinking is the main thing we need to penetrate the mysteries of $\mathbb{F}_1$. There seem to be some other missing ideas….

Lorscheid’s own approach uses ‘blueprints’. A blueprint $(R,S)$ is a commutative rig $R$ equipped with a subset $S \subseteq R$ that’s closed under multiplication, contains $0$ and $1$, and generates $R$ as a rig.

I have trouble, just on general aesthetic grounds, believing that blueprints are final ultimate solution to the quest for a generalization of commutative rings that can handle the “field with one element”. They just don’t seem ‘god-given’ the way commutative monoids or commutative objects are. But they do various nice things.

Maybe someone has answered this already, since it’s a kind of obvious question:

Question. Is there a symmetric monoidal category $\mathbf{C}$ in which blueprints are the commutative monoid objects?

Maybe something like the category of ‘abelian groups equipped with a set of generators’?

Of course you should want to know what morphisms of blueprints are, because really we should want the category of commutative monoid objects in $\mathbf{C}$ to be equivalent to the category of blueprints. Luckily Lorscheid’s morphisms of blueprints are the obvious thing: a morphism $f : (R,S) \to (R',S')$ is a morphism of commutative rigs $f: R \to R'$ with $f(S) \subseteq S'$.

Anyway, there’s a lot more to say about $\mathbb{F}_1$, but Lorscheid’s paper is a great way to get into this subject.

Posted at June 13, 2018 6:17 AM UTC

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### Re: Fun for Everyone

With regard to the Riemann hypothesis, the point which seems to me to be crucial is what the analogue of the Frobenius morphism in characteristic $p$ should be. I don’t think any of the approaches to algebraic geometry over $\mathbb{F}_{1}$ which have been tried so far have even any kind of suggestion for this.

Shai Haran’s work (see for example this paper and this one) is that which seems to me to be deepest with regard to this kind of question.

Some people are cynical about the approach to the Riemann hypothesis via $\mathbb{F}_{1}$. However, it is a truly beautiful picture, and a proof in this way would open up a new kind of mathematics, probably providing a way to make all kinds of dichotomies (e.g. mixed Hodge modules vs perverse sheaves) precise. Thus it is worth striving for.

Posted by: Richard Williamson on June 13, 2018 8:56 AM | Permalink | Reply to this

### Re: Fun for Everyone

Richard wrote:

With regard to the Riemann hypothesis, the point which seems to me to be crucial is what the analogue of the Frobenius morphism in characteristic $p$ should be. I don’t think any of the approaches to algebraic geometry over $\mathbb{F}_{1}$ which have been tried so far have even any kind of suggestion for this.

This sounds like a fun bone to gnaw on. It’s a minor miracle that squaring is linear in characteristic 2, etc. It’s less surprising that raising to the first power is linear in characteristic 1. The surprise is that it’s not the identity.

Some people are cynical about the approach to the Riemann hypothesis via $\mathbb{F}_{1}$.

I’m cynical about lots of things (most of which boil down to “human nature”), but not about the capacity of mathematics to surprise us by being simpler than we expected.

Posted by: John Baez on June 13, 2018 4:50 PM | Permalink | Reply to this

### Re: Fun for Everyone

I am fond of Borger’s suggestion that one should study $\lambda$-rings as a model for $\mathbb{F}_1$-algebras. The argument is not that $\lambda$-rings are $\mathbb{F}_1$-algebras, but rather that they are $\mathbb{Z}$-algebras together with descent data along $\mathrm{Spec}(\mathbb{Z}) \to \mathrm{Spec}(\mathbb{F}_1)$.

Posted by: Theo Johnson-Freyd on June 13, 2018 3:12 PM | Permalink | Reply to this

### Re: Fun for Everyone

There’s a nLab page Borger’s absolute geometry, and a general one for field with one element.

Contributions always welcome.

Posted by: David Corfield on June 13, 2018 3:50 PM | Permalink | Reply to this

### Re: Fun for Everyone

The definition of blueprints piqued my interest because it reminded me of something similar that shows up in a (seemingly) very different context.

Namely, if you restrict to the case that $R$ is a ring, $S$ contains $-1$, and the nonzero elements of $S$ are invertible, you get what matroid theorists call a partial field. They come up naturally in matroid representability problems, where they essentially serve the purpose of being something “field-like” but with the property that they can have homomorphisms into fields of different characteristics. (Which, I guess, is basically what $\mathbb{F}_1$ is supposed to be about too.)

Partial fields can also be axiomatized, as their name suggests, as something like “fields except the addition can be partially defined”, which perhaps makes them look slightly more “god-given”. If partial fields are to blueprints as fields are to commutative rigs, perhaps blueprints are commutative monoids in “partial commutative monoids”, or something along those lines.

Incidentally, I initially assumed this was basically a coincidence, but thinking about it further it occurred to me that there actually ought to be something to this connection. Matroid representations over fields correspond to points in Grassmannians, so if Lorscheid’s work leads to a good definition of Grassmannians over partial fields there ought to be something to say. Apparently I am not the first to have this thought (see section 9.7).

Posted by: Nick Olson-Harris on June 13, 2018 8:46 PM | Permalink | Reply to this

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