algebra, algebraic geometry

Galois Deformations 1: Schlessinger

Today we’ll begin a short series on deformations of Galois representations. I know very little about this topic in general, but since I brought up Taniyama-Shimura and this is one of the main tools used in proving it I thought it would be interesting to take a look at what these are. It will also be a nice example of why abstraction is often better. Today we’ll review standard deformation theory from a geometric viewpoint, but we’ll frame it in very abstract terms (the way Schlessinger did in his fantastic paper). This abstraction will allow us to apply certain results that come from geometric intuition to any functor that satisfies certain criteria.

I’ve posted a little about deformation theory around here. I won’t recall it because our framework will be slightly different. Instead of thinking about individual deformations, let’s do something more Grothendieckian and see what happens when we consider all deformations at the same time. In other words, fix some smooth varitey {X/k}, then given some “deformation ring” {A} (to be precise later), we consider the set of all deformations of our variety {\tilde{X}/A}. Here is where our formalism comes in. Our set changes with {A} nicely enough that {Def_X: _\Lambda Art_k \rightarrow Set} actually forms a functor.

Let’s talk about the category of deformation rings now. Let {\Lambda} be a Noetherian ring. The category labelled {_\Lambda Art_k} means the category of local Artinian {\Lambda}-algebras together with a choice of augmentation map to {k}. This last part is important, because the morphisms between two of these rings must be both {\Lambda}-algebra maps and maps that commute with the augmentation to {k}. The reason for this is that when I pick a deformation {\tilde{X}/A} I’ve done more than just specified an abstract deformation. I also am saying that the pullback square gives me a choice of isomorphism of the special fiber:

{\begin{matrix} X & \hookrightarrow & \tilde{X} \\ \downarrow & & \downarrow \\ Spec k & \rightarrow & Spec A \end{matrix}}

where that bottom arrow is the one on spectra induced by the augmentation {A\rightarrow k}. Based on this geometric picture, we abstract as little as possible and call any functor {F: _\Lambda Art_k\rightarrow Set} a deformation functor if it satisfies certain properties that formally correspond to having a single deformation over {k} (exercise for the uninitiated: prove that if X has a non-trivial automorphism, then the functor will not satisfy this condition if the seemingly strange augmentation condition is dropped. I’ve never seen this exercise written down, but it seems important to me), being able to glue when you ought to be able to, and this gluing being unique over first order infinitesimal neighborhoods. The exact conditions can be found here (I don’t feel bad linking to that nLab page since I wrote it).

A key theorem due to Schlessinger that will come up next time is that under mild conditions to check on this functor we actually get that it is prorepresentable. This just means if we “complete” the category formally by making a new functor {\widehat{F}: _\Lambda Noeth_k\rightarrow Set} by {\widehat{F}(R)=\lim F(R/m^n )} the functor is actually representable here. We won’t dwell on this because when thinking about Galois representations we will have stronger things going on by continuity of our maps which will make this point less important. All this will be made more precise next time, but it is worth pointing out how amazing the forsight of Schlessinger was to formulate this in terms of functors so that it would apply all over math. His criterion says if we check certain conditions based on geometric intuition the functor will be representable. Or more intuitively, we will have a “space” that universally parametrizes everything. You can read more about this at that nLab page.

Next time we’ll bring this back from this abstraction and talk about what this means for being able to “deform” Galois representations.

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