# December 9, 2007 {#week259}

This week I'll talk about the "field with one element" - even though
it doesn't exist. It's a mathematical phantom.

But first: the Egg Nebula.

In ["Week 257"](#week257) and ["Week 258"](#week258) I talked
about interstellar dust. As I mentioned, lots of it comes from
"asymptotic giant branch" stars - stars like our Sun, but later in
their life, when they're big, red, pulsing, and puffing out elements
like hydrogen, helium, carbon, nitrogen, and oxygen.

The pulsations grow wilder and wilder until the star blows off its
entire outer atmosphere, forming a big cloud of gas and dust
misleadingly called a "planetary nebula". It leaves behind its dense
inner core as a hot white dwarf. Intense radiation from this core
eventually heats the gas and dust until they glow.

Back in ["Week 223"](#week223) I showed my favorite example of a
planetary nebula: the Cat's Eye.

::: {align="center"}
[![](cats-eye_nebula.jpg)](http://heritage.stsci.edu/2004/27/index.html)
:::

And I quoted the astronomer Bruce Balick on what will happen here when
our Sun becomes a planetary nebula 6.9 billion years from now:

> Here on Earth, we'll feel the wind of the ejected gases sweeping
> past, slowly at first (a mere 5 miles per second!), and then picking
> up speed as the spasms continue - eventually to reach 1000 miles per
> second!! The remnant Sun will rise as a dot of intense light, no
> larger than Venus, more brilliant than 100 present Suns, and an
> intensely hot blue-white color hotter than any welder's torch. Light
> from the fiendish blue "pinprick" will braise the Earth and tear
> apart its surface molecules and atoms. A new but very thin
> "atmosphere" of free electrons will form as the Earth's surface
> turns to dust.

Eerie!

Here's a "protoplanetary nebula" - that is, a planetary nebula
that's just getting started:

::: {align="center"}
![](egg_nebula.jpg){width="600"}
:::

1) Rainbow image of a dusty star, NASA,
`http://hubblesite.org/newscenter/archive/releases/nebula/planetary/2003/09/`

It's called the "Egg Nebula". You can see layers of dust coming out
puff after puff, shooting outwards at about 20 kilometers per second,
stretching out for about a third of a light year. The colors - red,
green and blue - aren't anything you'd actually see. They're just an
easy way to depict three different polarizations of light. I don't know
why the light is polarized that way.

You can see a dark disk of thicker dust running around the star. It
could be an "accretion disk" spiralling into the star. The "beams"
shining out left and right are still poorly understood. Maybe they're
jets of matter ejected from the north and south poles of the disk? This
idea seem more plausible when you look at this photo taken by NICMOS,
which is Hubble's "Near Infrared Camera and Multi-Object
Spectrometer":

::: {align="center"}
![](egg_nebula_infrared.jpg){width="500"}
:::

2) Raghvendra Sahai, Egg Nebula in polarized light, Hubble Heritage
Project, `http://heritage.stsci.edu/2003/09/supplemental.html`

This near-infrared image also shows a bright spot called "Peak A"
about 500 AU from the central star. An "AU", or astronomical unit, is
the distance from the Sun to the Earth.

Nobody knows what this bright spot is. Some argue that it's just a
clump of dust reflecting light from the main star. Others advocate a
more exciting theory: it's a white dwarf orbiting the main star, which
exploded in a "thermonuclear burst" after accreting a bunch of dust.

3) Joel H. Kastner and Noam Soker, The Egg Nebula (AFGL 2688):
deepening enigma, to appear in Asymmetrical Planetary Nebulae III, eds.
M. Meixner, J. Kastner, N. Soker, and B. Balick, ASP Conference Series.
Available as
[`arXiv:astro-ph/0309677`](http://arXiv.org/abs/astro-ph/0309677).

I hope to say more about planetary nebulae in future Weeks, mainly
because they're so beautiful.

But now: the field with one element!

A [field](http://en.wikipedia.org/wiki/Field_%28mathematics%29) is a
mathematical structure where you can add, multiply, subtract and divide
in ways that satisfy these familiar rules:

     x + y = y + x        (x + y) + z = x + (y + z)   x + 0 = x

     xy = yx              (xy)z = x(yz)               1x = x

     x(y + z) = xy + xz 

     every element x has an element -x with x + (-x) = 0
     
     every element x that's not 0 has an element 1/x with x (1/x) = 1

You'll note that the last clause is the odd man out. Addition,
subtraction and multiplication can all be described as everywhere
defined operations. Division cannot, since we can't divide by 0. This
is the funny thing about fields, which is what causes the problem we'll
run into.

Everyone who has studied math knows three examples of fields: the
rational numbers Q, the real numbers R and the complex numbers C. There
are a lot more, too - for example, function fields, number fields, and
finite fields.

Let me say a tiny bit about these three kinds of fields.

The simplest sort of "[function
field](http://en.wikipedia.org/wiki/Function_field_of_an_algebraic_variety)"
consists of rational functions of one variable - that is, ratios of
polynomials, like this:

(2z^3 + z + 1)/(z^2 - 7)

Here the coefficients of your polynomials should lie in some field F you
already know about. The resulting field is called F(z). If F is the
complex numbers, we can think of F(z) = C(z) as consisting of functions
on the Riemann sphere. In ["Week 201"](#week201), I explained how
the symmetries of this field form a group important in special
relativity: the Lorentz group!

It's also very interesting to study the field of functions on a surface
fancier than the sphere, but still defined by algebraic equations, like
the surface of a doughnut or $n$-holed doughnut. Number theorists and
algebraic geometers spend a lot of time thinking about these fields,
which are called "function fields of complex curves".

For example, different ways of describing the surface of a doughnut by
algebraic equations give different "[elliptic
curves](http://en.wikipedia.org/wiki/Elliptic_curve)". This is
terminology is bound to puzzle beginners! They're called "curves"
even though it's $2$-dimensional, because it takes one *complex* number
to say where you are on a little patch of a surface, just as it takes
one *real* number to say where you are on an ordinary curve like a
circle. That's the origin of the term "complex curve". And, they're
called "elliptic" because they first showed up when people were
studying elliptic integrals, which are generalizations of trig functions
from circles to ellipses.

I explained more about elliptic curves in ["Week 13"](#week13) and
["Week 125"](#week125). Lurking behind this, there's a lot of
fascinating stuff about function fields of elliptic curves.

The simplest sort of "[number
field](http://en.wikipedia.org/wiki/Algebraic_number_field)" comes from
taking the rational numbers and throwing in the solutions of a
polynomial equation. For example, in ["Week 20"](#week20) I talked
about the "golden field", which consists of all numbers of the form

a + b √ 5

where a and b are rational.

One of the most beautiful ideas in math is the analogy between number
fields and function fields - the idea that numbers are like functions on
some sort of "space". I began explaining this in
["Week 205"](#week205), ["Week 216"](#week216) and
["Week 218"](#week218), but there's much more to say about what's
known... and also many things that remain mysterious.

In particular, it's pretty well understood how number fields resemble
function fields of complex curves, and how this relates number theory to
*2-dimensional* topology. But, there are also many analogies between
number theory and *3-dimensional* topology, which I began listing in
["Week 257"](#week257). It seems these analogies are doomed to
remain mysterious until we get a handle on the field with one element.
But more on that later.

The simplest sort of "[finite
field](http://en.wikipedia.org/wiki/Finite_field)" comes from choosing
a prime number p and taking the integers modulo p. The result is
sometimes called Z/p, especially when you're just concerned with
addition. But when you think of it as a field, it's better to call it
F_p.

The reason is that there's a finite field of size q whenever q is a
*power* of a prime, and this field is unique - so it's called F~q~. You
build F~q~ sort of like how you build the complex numbers starting from
the real numbers, or number fields starting from the rational numbers.
Namely, to construct F~p^n~, you take F_p and throw in the roots of a
well-chosen polynomial of degree n: one that doesn't have any roots in
F_p, but "wants" to have n different roots.

Okay: that was a tiny bit about function fields, number fields and
finite fields. But now I need to point out some slight lies I told!

I said there was a finite field with q elements whenever q was a prime
power. You might think this should include q = 1, since 1 is the
*zeroth* power of *any* prime.

So, is there a field with one element?

If so, it must have 1 = 0. That doesn't violate the definition of a
field that I gave you... does it? The definition said any element
that's not 0 has a reciprocal. In this particular example, 0 also has a
reciprocal, since we can set 1/0 = 1 and not get into any
contradictions. But that's not a problem: in usual math practice,
saying "we can divide by anything that's not zero" doesn't deny the
possibility that we can divide by 0.

Unfortunately, allowing a field with 1 = 0 causes nothing but grief. For
example, we can define vector spaces using any field (people say
"over" any field), and there's a nice theorem saying two vector
spaces are isomorphic if and only if they have the same dimension. And
normally, there's one vector space of each dimension. But the last part
isn't true for a field with 1 = 0. In a vector space over such a field,
every vector v has

v = 1 v = 0 v = 0

So, every vector space is 0-dimensional!

To prevent such problems, people add one extra clause to the definition
of a field:

     1 is not equal to 0

This clause looks even more tacked-on and silly than the clause saying
everything *nonzero* has a reciprocal... but it works fairly well.

However, the field with one element still wants to exist! Not the silly
field with 1 = 0, but something else, something more mysterious...
something that Gavin Wraith calls a "mathematical phantom":

4) Gavin Wraith, Mathematical phantoms,
`http://www.wra1th.plus.com/gcw/rants/math/MathPhant.html`

What's a mathematical phantom? According to Wraith, it's an object
that doesn't exist within a given mathematical framework, but
nonetheless "obtrudes its effects so convincingly that one is forced to
concede a broader notion of existence".

Like a genie that talks its way out of a bottle, a sufficiently powerful
mathematical phantom can talk us into letting it exist by promising to
work wonders for us. Great examples include the number zero, irrational
numbers, negative numbers, imaginary numbers, and quaternions. At one
point all these were considered highly dubious entities. Now they're
widely accepted. They "exist". Someday the field with one element will
exist too!

Why?

I gave a lot of reasons in ["Week 183"](#week183),
["Week 184"](#week184), ["Week 185"](#week185),
["Week 186"](#week186) and ["Week 187"](#week187), but let me
rapidly summarize.

It's all about "q-deformation". In physics, people talk about
q-deformation when they're taking groups and turning them into
"quantum groups". But it has a closely related aspect that's in some
ways more fundamental. When we count things involving $n$-dimensional
vector spaces over the finite field F~q~, we often get answers that are
polynomials in q. If we then set q = 1, the resulting formulas count
analogous things involving $n$-element sets!

So, finite sets want to be finite-dimensional vector spaces over the
(nonexistent) field with one element... or something like that. We can
be more precise after looking at some examples.

Here's the simplest example. Say we count lines through the origin in
an $n$-dimensional vector space over F~q~. We get the "q-integer"

     qn - 1
    ------- = 1 + q + q2 + ... + qn-1
     q - 1

which I'll write as \[n\] for short.

\mathsf{Set}ting q = 1, we get n. This is the number of points in an $n$-element
set. Sure, that sounds silly. But, I'm trying to make a point here! At
q = 1, stuff about $n$-dimensional vector spaces over F~q~ reduces to
stuff about $n$-element sets, and the q-integer \[n\] reduces to the
ordinary integer n.

This may not be the best way to understand the pattern, though. Lines
through the origin in an $n$-dimensional vector space are the same as
points in an $(n-1)$-dimensional projective space. So, the real analogy
may be between "points in a projective space" and "points in a set".

Here's a more impressive example. Pick any uncombed Young diagram D
with n boxes. Here's one with 8 boxes:

    X                     1 box in the first row
    X X                   3 boxes in the first two rows
    X X X                 6 boxes in the first three rows
    X X                   8 boxes in the first four rows

Then, count the "D-flags on an $n$-dimensional vector space over F~q~".
In our example, such a D-flag is:

        a $1$-dimensional subspace
     of a $3$-dimensional subspace
     of a $6$-dimensional subspace
     of a $8$-dimensional vector space over Fq

If you actually count these D-flags you'll get some formula, which is a
polynomial in q. And when you set q = 1, you'll get the number of
"D-flags on an $n$-element set". In our example, such a D-flag is:

       a 1-element subset 
    of a 3-element subset
    of a 6-element subset
    of a 8-element set

For details, and a proof that this really works, try:

5) John Baez, Lecture 4 in the Geometric Representation Theory Seminar,
October 9, 2007. Available at
`http://math.ucr.edu/home/baez/qg-fall2007/qg-fall2007.html#f07_4`

These examples can be generalized. In ["Week 187"](#week187) I
showed how to get one example for each subset of the dots in any Dynkin
diagram! This idea goes back to Jacques Tits, who was the first to
suggest that there should be a field with one element. Dynkin diagrams
give algebraic groups over F~q~... but he noticed that these groups
reduce to "Coxeter groups" as q \to 1. And, if you mark some dots on a
Dynkin diagram you get a "flag variety" on which your algebraic group
acts... but as q \to 1, this reduces to a finite set on which your
Coxeter group acts.

If you don't understand the previous paragraph, don't worry - it's
over now. It's great stuff, but my main point is that there seems to be
an analogy like this:

     q = 1                      q = a power of a prime number    

     $n$-element set              $(n-1)$-dimensional projective space over Fq
     integer n                  q-integer [n]
     permutation groups Sn      projective special linear group \mathrm{PSL}(n,Fq)
     factorial n!               q-factorial [n]!

This opens up lots of questions. For example, if projective spaces over
F_1 are just finite sets, what should *vector spaces* over F_1 be?

People have thought about this, and the answer seems to be "pointed
sets" - sets with a distinguished point, which you can think of as the
"origin". A pointed set with n+1 elements seems to act like an
n-dimensional vector space over F~q~.

For more clues, and an attempt to do algebraic geometry using the field
with one element, try this:

6) Christophe Soulé, On the field with one element, Talk given at the
Arbeitstagung, Bonn, June 1999, IHES preprint available at
[http://www.ihes.fr/\~soule/f1-soule.pdf](%20http://www.ihes.fr/~soule/f1-soule.pdf)

Soulé tries to define "algebraic varieties" over F_1, namely curves
and their higher-dimensional generalization. And, he talks a lot about
zeta functions for such varieties. He goes into more detail here:

7) Christophe Soulé, Les varietes sur le corps a un element, Moscow
Math. Jour. 4 (2004), 217-244, 312.

The theme of zeta functions - see ["Week 216"](#week216) - is
deeply involved in this business. For more, try these papers:

8) N. Kurokawa, Zeta functions over F_1, Proc. Japan Acad. Ser. A
Math. Sci. 81 (2006), 180-184.

9) Anton Deitmar, Remarks on zeta functions and K-theory over F_1,
available as
[`arXiv:math/0605429`](http://arXiv.org/abs/arXiv:math/0605429).

But instead of talking about zeta functions, I'd like to talk about two
approaches to giving a formal definition of the field with one element.
Both of them involve taking the concept of field and modifying it so it
doesn't necessary involve the operation of addition. The first one, due
to Deitmar, simply throws out addition! The second, due to Nikolai
Durov, allows for a wide choice of operations - and thus a wide supply
of "exotic fields".

For Deitmar's approach, try these:

10) Anton Deitmar, Schemes over F_1, available as
[`arXiv:math/0404185`](http://arXiv.org/abs/arXiv:math/0404185).

F_1-schemes and toric varieties, available as
[`arXiv:math/0608179`](http://arXiv.org/abs/arXiv:math/0608179).

The usual approach to fields treats fields as specially nice commutative
rings. A "[commutative
ring](http://en.wikipedia.org/wiki/Commutative_ring)" is a gadget where
you can add and multiply, and these rules hold:

     x + y = y + x        (x + y) + z = x + (y + z)   x + 0 = x

     xy = yx              (xy)z = x(yz)               1x = x

     x(y + z) = xy + xz 

     every element x has an element -x with x + (-x) = 0

Deitmar throws out addition and treats fields as specially nice
commutative monoids. A commutative
"[monoid](http://en.wikipedia.org/wiki/Commutative_ring)" is a gadget
where you can multiply, and these rules hold:

     xy = yx              (xy)z = x(yz)               1x = x

For Deitmar, the field with one element, F_1, is just the commutative
monoid with one element, namely 1. A "vector space over F_1" is just
a set on which this monoid acts via multiplication... but that amounts
to just a plain old set. The "dimension" of such a "vector space" is
just its cardinality.

All this so far is quite trivial, but Deitmar makes a nice attempt at
redoing algebraic geometry to include this field with one element. One
reason to do this is to understand the mysterious $3$-dimensional aspect
of number theory.

To explain this, I need to say a bit about
"[schemes](http://en.wikipedia.org/wiki/Scheme_%28mathematics%29)". In
ordinary algebraic geometry, we turn commutative rings into spaces to
think about them geometrically. I explained this back in
["Week 199"](#week199) and ["Week 205"](#week205), but let me
review quickly, and go further:

We can think of elements of a commutative ring R as functions on certain
space called the
"[spectrum](http://en.wikipedia.org/wiki/Spectrum_of_a_ring)" of R,
Spec(R). This space has a topology, so we can also talk about functions
that are defined, not on all of Spec(R), but just *part* of Spec(R) -
namely some open set. Indeed, for each open set U in Spec(R), there's a
commutative ring \mathrm{O}(U) consisting of those functions defined on U. These
commutative rings are related in nice ways:

1.  If the open set V is smaller than U, we can restrict functions from
    U to V, getting a ring homomorphism \mathrm{O}(U) \to \mathrm{U}(V)
2.  If U is covered by a bunch of open sets U_i, and we have a function
    f_i in each \mathrm{O}(U_i), such that f_i and f_j agree when restricted
    to the set U_i ∩ U_j, then there's a unique function f in \mathrm{O}(U)
    that restricts to each of these functions f_i.

Something satisfying condition 1 is called a
"[presheaf](http://en.wikipedia.org/wiki/Presheaf#Definition_of_a_presheaf)"
of commutative rings; something also satisfying condition 2 is called a
"[sheaf](http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29)" of
commutative rings.

So, Spec(R) is not just a topological space, it's equipped with a sheaf
of commutative rings. People call this a "[ringed
space](http://en.wikipedia.org/wiki/Ringed_space)".

Whenever we have a ringed space, we can ask if it comes from a
commutative ring R in the way I just sketched. If so, we call it an
"affine scheme". Affine schemes are just a fancy geometrical way of
talking about commutative rings!

More interestingly, whenever we have a ringed space, we can ask if it's
*locally* isomorphic to one coming from a commutative ring. In other
words: does every point have a neighborhood that, as a ringed space,
looks like Spec(R) for some commutative ring R? Or in other words: is
our ringed space *locally* isomorphic to an affine scheme? If so, we
call it a "scheme".

A classic example of a scheme that's not an affine scheme is the
Riemann sphere. There aren't any rational functions defined on the
whole Riemann sphere, except for constants - the rest all blow up
somewhere. So, it's hopeless trying to think of the Riemann sphere as
an affine scheme.

But, for any open set U in the Riemann sphere there's a commutative
ring \mathrm{O}(U) consisting of rational functions that are defined on U. So,
the Riemann sphere becomes a ringed space. And, it's *locally*
isomorphic to the complex plane, which is the affine scheme
corresponding to the commutative ring of complex polynomials in one
variable. So, the Riemann sphere is a scheme!

For more on schemes, try this nice introduction, which actually has lots
of pictures:

11) David Eisenbud and Joe Harris, The Geometry of Schemes, Springer,
Berlin, 2000.

Now, we can talk about schemes "over a field F", meaning that each
commutative ring \mathrm{O}(U) is also a vector space over F, in a well-behaved
way, giving us a "sheaf of commutative rings over F". For example, the
Riemann sphere is a scheme over C.

There's a secret $3$-dimensional aspect to the affine scheme Spec(Z),
where Z is the commutative ring of integers. As explained in the Addenda
to ["Week 257"](#week257), we might understand this if we could see
Spec(Z) as a scheme over the field with one element! For more, see this:

12) M. Kapranov and A. Smirnov, Cohomology determinants and reciprocity
laws: number field case, available at
`http://wwwhomes.uni-bielefeld.de/triepe/F1.html`

So, we really need a theory of schemes over the field with one element.
The problem is, F_1 isn't really a field. In Deitmar's approach,
it's just a commutative monoid.

So, let me sketch how Deitmar gets around this. In a nutshell, he takes
advantage of the fact that a lot of basic algebraic geometry only
requires multiplication, not addition!

He starts by defining a "commutative ring over F_1" to be simply a
commutative monoid. The simplest example is F_1 itself.

Now, watch how he gets away with never using addition:

He defines an "ideal" in a commutative monoid R to be a subset I for
which the product of something in I with anything in R again lies in I.
He says an ideal P is "prime" if whenever a product of two elements in
R is in P, at least one of them is in P.

He defines the "spectrum" Spec(R) of a commutative monoid R to be the
set of its prime ideals. He gives this the "Zariski topology". That's
the topology where the closed sets are the whole space, or any set of
prime ideals that contain a given ideal.

He then shows how to get a sheaf of commutative monoids on Spec(R). He
defines a "scheme" to be a space equipped with a sheaf of commutative
monoids that's *locally* isomorphic to one of this sort.

If you know algebraic geometry, these definitions should seem very
familiar. And if you don't, you can just replace the word "monoid" by
"ring" everywhere in the previous three paragraphs, and you'll get
the standard definitions in algebraic geometry!

Deitmar shows how to build a scheme over F_1 called the "projective
line". The projective line over C is just the Riemann sphere. The
projective line over F_1 has just two points (or more precisely, two
closed points). This is good, because the projective line over the field
with q elements has

\[2\] = 1 + q

points, and we're doing the q = 1 case.

Deitmar's construction seems like a lot of work to get ahold of the
2-element set, if that's all it secretly is. But, I need to think about
this more. After all, he doesn't just get a space; he gets a sheaf of
commutative monoids on this space! And what's that like? I should work
it out.

Deitmar also shows how to relate schemes over F_1 to the usual sort of
schemes.

From a commutative ring, we can always get a commutative monoid just by
forgetting the addition. This process has a kind of reverse, too.
Namely, from a commutative monoid, we can get a commutative ring simply
by taking formal integral linear combinations of elements. Using this,
Deitmar shows how we can turn ordinary schemes into schemes over
F_1... and conversely. He says an ordinary scheme is "defined over
F_1" if it arises in this way from a scheme over F_1.

Okay, that's a taste of Deitmar's approach. For Durov's approach, try
this mammoth 568-page paper:

13) Nikolai Durov, New approach to Arakelov geometry, available as
[`arXiv:0704.2030`](http://arxiv.org/abs/0704.2030).

or read our discussions of it at the $n$-Category Café, starting here:

14) David Corfield, The field with one element,
`http://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html`

Durov defines a "generalized ring" to be what Lawvere much earlier
called an "algebraic theory". What is it? Nothing scary! It's just a
gadget with a bunch of abstract $n$-ary operations closed under
composition, permutation, duplication and deletion of arguments, and
equipped with an identity operation.

So, for example, if our gadget has a binary operation

(x,y) \mapsto f(x,y)

we can compose this with itself to get the ternary operation

(x,y,z) \mapsto f(f(x,y),z)

and the 4-ary operation

(w,x,y,z) \mapsto f(w,f(x,f(y,z)))

and so on. We can then permute arguments in our 4-ary operation to get
one like this:

(w,x,y,z) \mapsto f(z,f(x,f(w,y)))

or duplicate some arguments to get a binary operation like this

(x,y) \mapsto f(x,f(x,f(y,y)))

From this we can then form a 3-ary operation by deleting an argument,
for example like this:

(x,y,z) \mapsto f(x,f(x,f(y,y)))

If you know about
"[operads](http://en.wikipedia.org/wiki/Operad_theory)", this kind of
gadget is just a specially nice operad where we can duplicate and delete
operations.

Now, a generalized ring is said to be "commutative" if all the
operations commute in a certain sense. (I'll let you guess what it
means for an $n$-ary operation to commute with an m-ary operation.) We get
an example of a commutative generalized ring from a commutative ring R
if we let the $n$-ary operations be "n-ary R-linear combinations", like
this:

(x_1, ..., x_n) \mapsto r_1 x_1 + ... + r_n x_n

We also get a very similar example from any commutative
"[rig](http://en.wikipedia.org/wiki/Semiring)", which is a gizmo
satisfying rules like those of a commutative ring, but without
negatives:

     x + y = y + x        (x + y) + z = x + (y + z)   x + 0 = x

     xy = yx              (xy)z = x(yz)               1x = x

     x(y + z) = xy + xz 

And, we get an example from any commutative monoid, where we only have
1-ary operations, coming from multiplication by elements of our monoid:

(x_1) \mapsto r x_1

So, Durov's framework generalizes Deitmar's! But, it includes a lot
more examples: exotic hothouse flowers like the "tropical rig", the
"real integers", and more. He develops a theory of schemes for all
these generalized rings, and builds it "up to construction of algebraic
K-theory, intersection theory and Chern classes" - fancy things that
algebraic geometers like.

What I don't yet see is how either Deitmar's or Durov's approach
helps us understand the secret $3$-dimensional nature of Spec(Z). I may
just need to read their papers more carefully and think about them more.

Finally, here's yet another approach to the field with one element:

15) Bertrand Toen and M. Vaquie, Under Spec(Z), available as
[`arXiv:math/0509684`](http://arxiv.org/abs/math/0509684).

16) Shai Haran, Non-additive geometry, Composito Mathematica 143
(2007), 613-638.

Toen describes interesting relations between algebra over F_1 and
stable homotopy theory. Haran even suggests that the Riemann Hypothesis
could be proved if we understood enough about the geometry of schemes
over F_1! This is fascinating... I don't understand it, but I want
to.

In short, a mathematical phantom is gradually taking solid form before
our very eyes! In the process, a grand generalization of algebraic
geometry is emerging, which enriches it to include some previously
scorned entities: rigs, monoids and the like. And, this enrichment holds
the promise of shedding light on some otherwise impenetrable mysteries:
for example, the deep inner meaning of q-deformation, and the
3-dimensional nature of Spec(Z).

------------------------------------------------------------------------

**Addenda**: I thank Thomas Riepe and David Corfield for drawing my
attention to the paper by Shai Haran. Thomas Riepe also recommends the
following online introduction to schemes:

17) Marc Levine, Summer course in motivic homotopy theory, available at
`http://www.math.neu.edu/~levine/publ/SummerSchoolAG.pdf`

Kevin Buzzard has a word of advice about the "generic point":

>       We can think of elements of a commutative ring R as functions on
>       certain space called the "spectrum" of R, Spec(R).
>
> So this is the set of all *prime* ideals of R, right? Not just the
> maximal ones? So...
>
>       So, the Riemann sphere is a scheme!
>
> Well, you have to throw in a mystical extra "generic point" if you
> really want to make it a scheme :-) Corresponding to the zero ideal.
> My impression is that most non-algebraic geometers think that the
> generic point is either confusing or just plain daft. But believe me,
> it's a *really* good idea! For decades in the literature in algebraic
> geometry people were using the word "generic" to mean "something
> that was true most of the time" - in fact a "generic point" is
> probably another really good example of a phantom! For example a
> meromorphic function on the Riemann sphere that wasn't zero would be
> "generically non-zero" to people like Borel and Weil, and if you
> asked them for a definition they would say that it just meant
> something like "the zero locus in the space had a smaller dimension
> than the whole space" or "the zero locus was nowhere dense in any
> component of the space" or something, and of course people could even
> make rigorous definitions that worked in particular cases and so on,
> but then Grothendieck came along with his "generic point",
> corresponding to the zero ideal \[note to sub: check to see whether
> the idea was in the literature pre-Grothendieck!\] and suddenly a
> function that was "generically non-zero" was just a function which
> was non-zero on the generic point! Such a cool way of doing it :-)
>
> Kevin

If you get stuck on my puzzle "what does it mean for an $n$-ary operation
to commute with an m-ary operation?", let me just show you what it
means for a binary operation f to commute with a ternary operation g. It
means:

g(f(x_1,x_2), f(x_3,x_4), f(x_5,x_6)) = f(g(x_1,x_2,x_3),
g(x_4,x_5,x_6))

I hope this example gives away the general pattern.

If this is confusing, look at the case where we start with a ring R and
take as our $n$-ary operations the "n-ary R-linear combinations"

(x_1, ..., x_n) \mapsto r_1 x_1 + ... + r_n x_n

with r_i in R. Here an example of a binary operation is addition:

(x_1, x_2) \mapsto x_1 + x_2

while every unary operation is multiplication by some element of R:

x_1 \mapsto r x_1

To say "addition commutes with multiplication by an element of R"
means that

r(x_1 + x_2) = rx_1 + rx_2

This is just the distributive law so it holds for any ring R.

But, for the unary operations to commute with each other, we need R to
be commutative, since this says:

r(s x_1) = s (r x_1)

(In the calculations I just did, we can either think of the x_i as
elements of a specific R-module, or more abstractly as "dummy
variables" used to describe the ring R as a generalized ring in
Durov's sense - what Lawvere calls an algebraic theory.)

For more discussion, go to the [$n$-Category Café](http://golem.ph.utexas.edu/category/2007/12/this_weeks_finds_in_mathematic_19.html).

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*The analogy between number fields and function fields finds a basic
limitation with the lack of a ground field. One says that Spec(Z) (with
a point at infinity added, as is familiar in Arakelov geometry) is like
a (complete) curve, but over which field?* - Christophe Soulé