Here is the polymath collection of links on Mochizuki.
This is Ellenberg’s blog with comments on Mochizuki.
Today we’re going to finish going over preliminaries and notation from
the Mochizuki masterpiece, Geometry of Frobenioids I: The general theory.
If we are successful at this, we may then be in a position to read the definition of Frobenioid.
Our groups will be Hausdorff topological groups. Subgroups have centralizers.
To a given profinite group is associated a category of finite sets with a action. This category is a Galois category. If every open subgroup of has trivial centralizer, we say that $\Pi$ is slim. This is all we need to know about group theory.
Next we learn about categories. We consider their objects and arrows. We think about one object and one morphism categories. Has anyone ever read Paul Pot’s book, “The one bullet manager?”
We discuss some restrictions of categories to single objects by looking at morphisms all of whose images (or all of whose domains) are a single object. Morphisms of the original category form functors between these restricted categories.
We define what it means for an arrow to be fiberwise surjective. (Basically when it can be put in a commutative diagram with any other arrow having the same target.) A category is of FSM type if every fiberwise surjective morphism is an isomorphism. We define that an endomorphism is a sub-automorphism if there is an automorphism and a map so that applying the automorphism then the map is the same as applying the map then the endomorphism.
Various other notions on categories are discussed. It’s hard to see where this is going. Time for the definition of Frobenioid.
We now begin describing the most dramatic part yet of Mochizuki’s epic paper The Geometry of Frobenioids I: General Theory. We will go through section 0: preliminaries and notation.
Recall that the whole paper may be found here:
Mochizuki starts saying some things that make sense but are not very deep.
Given a partially ordered set E, the category ORD(E) has underlying objects elements of E and morphisms which are inequalities. (You’d almost think we should think about inequality as being related.)
Within number systems we can think about some monoids. The positive integers are a monoid under multiplication and the primes generate it.
You could do this with integers, rationals or reals and look at the positive versions of any of the three. If a monoid is isomorphic to the nonnegative (reals,rationals or integers) as an additive monoid we refer to it as a monoprime monoid. (It is not clear that this helps.)
One can complete monoprime monoids from rational type to real type.
A number field is a finite extension of the rationals.
Monoids are a category with one object the monoid so that elements of the monoid are endomorphisms.
Next we consider the category of all monoids. We can go from monoids to their invertible elements with the +- superscript. (This is important!!! The +- superscript appears throughout Mochizuki’s description of theatres later on.)
You can mod out by invertible elements to get the characteristic and groupify in the normal way to get the group.
A monoid is: Torsion free if no torsion elements. Sharp if no invertible elements. Integral if groupification injective. Saturated if whenever a positiveinteger multiple of an element of the group lives in the monoid so does the element.
One can perfect monoids through an inductive limit under multiplication by all positive integers. This basically makes multiplication bijective (adds fractions.)
Now our attention turns to sharp, saturated, and integral modules and something important is going on. There is a partial order in such a monoid if there is with . An element of such a monoid is prime if it has no factors. (Where’s 1?) Thus we can rebuild the notion of primality within this kind of monoids.
Next time, we’ll go over Mochizuki’s preliminaries on Topological groups and Categories.
The goal of this post, a fool’s errand really, is to try to give a summary of the introduction to Mochizuki’s paper, The Geometry of Frobenioids I: The General Theory, which may be found here:
I would like to add that there is also a wikipedia entry describing these strange objects known as Frobenioids:
I would like to point out that one thing which is odd about this wikipedia entry is that it does not go so far as to define Frobenioids precisely. Instead, it borrows language from the technical summary of the paper. The technical summary is just the first section of the introduction. Roughly, that summary says that there is something called an elementary Frobenioid which we should think of like a semi-direct product of Monoids and a Frobenioid is going to be a category equipped with a functor to elementary Frobenioids with enough properties to guarantee that this functor can be reconstructed category-theoretically. The philosophy is supposed to be that the “base category” from which the monoids are taken is a model for field extensions of a finite field where as the functor assigns divisors.
All of this is contained in section I1, the “technical summary” of the paper. And nothing outside of section I1 is contained in the wikipedia article. If section I1 seems not to make sense, that is because not all technical details in the definitions have been given there. The remainder of this post will consist of a brief summary of the remaining sections of the introduction.
After the technical summary, the paper contains some further intriguing introductory sections. The next of these is I2 which is intriguingly titled “Abstract combinatorialization of Arithmetic geometry.” This section is a kind of philosophical essay about how Galois categories allow one to abstract all info about Field extensions into combinatorial info about the Galois groups. Mochizuki regards this Galois theory as very good for working over nonarchimedean primes, instead one needs divisors for working over the archimedean ones. The idea is that Frobenioids are a kind of compromise that lets you do both. No actual mathematics happens in this section.
Section I3 seems to be difficult to follow if one does not already understand Mochizuki’s rather impenetrable p-adic Teichmuller theory. Mochizuki laments that for some reason the morphism p —> p^n does not extend to all the integers. He says it would make sense if one worked with Monoids instead !!! He relates the idea that the theory of Frobenioids will encode a substantial part of scheme theory, but not, not, not, all of scheme theory.
Section I4 briefly describes a dichotomy between Frobenius and Etale structures. The idea is, apparently, that the Frobenioids will suppress the etale structure of schemes leaving only the other part.
Stay tuned for more posts on later sections of this riveting paper.
This is mainly a test post to check whether the LaTeX support works here and to discuss briefly what may in broadest terms be thought of as the philosophy of Mochizuki’s proof.
In polynomials, there is a very easy proof of the ABC conjecture (more correctly, the Mason-Stothers theorem) due to Noah Snyder. We give that proof here. If
with relatively prime polynomials over the complex numbers then we also have . From this we see that
Another way of saying that is that the wronskian of and is the same as . We let be the product of all factors which are common to either and or and or and . Then is the quotient of by the radical. If for instance has degree and has degree and has degree then the Wronskian has degree at most so the radical has degree at least . All other cases follow in the same way.
What we learn from this is that being able to take the derivative helps to prove the abc conjecture. In the integer case, we should be taking the derivative in the direction of choices of primes. It is probably for this reason that Mochizuki wants to monkey around with the definition of schemes and introduce so many objects. He hopes to find a way to make sense of this differentiation process.