There are various base change theorems in algebraic geometry (flat in zariski topology, proper, smooth in etale topology etc). Often, especially in the etale site, it is easier to prove base change for specific morphisms (relative dimension one or projective instead of proper, for instance) and there is a formal mechanism for reducing from the generic case to the specific case. This post sketches a proof of that.
This post is very rough and is only to remind me of my chain of thought. Read at your own peril.
The first example of Iwasawa theory most people see is the historically first one of class groups for the cyclotomic $latex\mathbb Z_p$ tower of . The reason the idea works at all is that somehow, passing to the limit reduces the “amount of bad stuff” that can happen and the key to this is a sort of control theorem where, if we denote the inverse limit of class groups by which has an action of , then Iwasawa shows that the class groups at finite levels can be recovered by essentially doing where . This step very crucially uses class field theory in the identification of the class group with the maximal unramified extension.
Now one might object that perhaps we don’t need to work so hard. After all, where the limit is over the algebras where and it seems natural that . Unfortunately (or fortunately), this is not true at all and is where Iwasawa theory gets it’s great power from. The purpose of this post is to record how bad things can go without any control theorems.
In this post, we will consider the (nasty?) non linear differential equation:
and show that the power series:
is a solution using combinatorics! In particular, note that is the exponential generating function for the number of labelled trees on vertices by Cayley’s formula. Let us call this number , which is equal to by Cayley.
If we plug in the generating function into the differential equation, we find that in order to show that $h(t)$ is a solution to the differential equation, it is sufficient to show the following recurrence:
We will show this recurrence directly using combinatorics:
We will count the number of labelled trees on vertices with a distinguished edge with a distinguished direction. This number is clearly since there are edges to choose from and directions.
But if we imagine deleting the edge, this is the same as taking two labelled trees of sizes with a distinguished vertex on each and a choice of elements from . The choice of elements is because once we have two trees with labels from and we need to map the union of these two sets into and once we pick a element subset of , we can map the first set into it, preserving order and similarly for the complement.
Then, on a tree of size , there are distinct vertices to choose from.
This correspondence is clearly bijective and so the number we are counting is also equal to:
I do not know how to prove the recurrence directly! But there are many nice proofs of Cayley’s theorem available. One I particularly like is here.
When Serre and Grothendieck were coming up with an extension of the usual cohomology theory to varieties in algebraic geometry, an important example was given by Serre which showed that you couldn’t have a Weil cohomology theory with coefficients in in characteristic . I will explain the example and extend it to show something about the ordinary case.
I will cover a set of results that are connected and illustrate some interesting techniques about working with finiteness conditions (finitely generated as a ring, module etc) in algebraic geometry over a field.
Initially, I will prove Chevalley’s theorem that the image of a constructible set is constructible. This will involve proving Grothendieck’s generic freeness and Noetherian induction. Next, I will give a couple of applications of Chevalley’s theorem – I will prove that flat maps are open and provide a different proof of the Nullstellensatz.
Epistemic status: I have not checked this carefully for errors, it is entirely possible there are mistakes in this.
We will be mostly be concerned with Elliptic curves over number fields . However, let us start with a general lemma about Elliptic curves with endomorphism ring . In this post, I collect some properties about representations coming from Elliptic curves.
Hilbert’s Nullstellensatz plays a central role in algebraic geometry. It can be seen as the fundamental link between the modern theory of schemes and the classical theory of algebraic varieties over fields. Since this is one of the first results a novice in algebraic geometry learns and is often proved very algebraically, one often does not gain a good understanding of the proof till much later.
I will present three proofs of the Nullstellensatz found in the literature from a geometric perspective. This will highlight the role of the “spreading out and specializing” common to the proofs that might not be obvious from an algebraic presentation. The last proof is a very short, self contained demonstration of the techniques. Along the way, we will also see a geometric proof of Noether Normalization.
The proof of Hilbert’s lemma is usually broken up into the following two steps: 1) Prove the weak Nullstellensatz and 2) Derive the strong Nullstellensatz using the Rabinowitsch or other means. I will be focusing solely on the first step in this post. Nothing in this is original except for the presentation.