I’ve been trying to learn something about deformations of abelian varieties lately. One of the big results is:

Theorem 1 The “local moduli space” of an abelian variety ${X/k}$ is smooth of dimension ${g^2}$, if ${\dim X = g}$.

Why might you care about this result? Let’s say you’re in the case ${g = 1}$, so then presumably you’re interested in the stack ${M_{1, 1}}$ of elliptic curves. This is a Deligne-Mumford stack: that is, there are enough étale maps ${\mathrm{Spec} R \rightarrow M_{1, 1}}$ from affine schemes, and as a result it makes sense to talk about the strict henselianization at a point (or the completion at a “point” from an algebraically closed field). Then, the point of the above theorem is that you can work out exactly what that is: it’s a one-dimensional thing. This isn’t too surprising, because the isomorphism class of an elliptic curve depends on one parameter (the ${j}$-invariant). So knowing the deformation theory of elliptic curves lets you say what ${M_{1, 1}}$ looks like, very locally.

Let’s make this a bit more precise. An actual formulation of the theorem would specify what “local moduli” really means. For us, it means deformations. A deformation of an abelian variety ${X/k}$ over an artin local ring ${R}$ with residue field ${k}$ is the data of an abelian scheme (that is, a proper flat group scheme with abelian variety fibers) ${X' \rightarrow \mathrm{Spec} R}$ together with an isomorphism of abelian varieties ${X' \times_{\mathrm{Spec} R} \mathrm{Spec} k \simeq X}$.

Theorem 2 Let ${k}$ be an algebraically closed field, and let ${X/k}$ be an abelian variety. Then the functor of deformations of ${X}$ is prorepresentable by ${W(k)[[t_1, \dots, t_{g^2}]]}$ for ${W(k)}$ the ring of Witt vectors over ${k}$.

In other words, to give a deformation of ${X}$ over an artin local ${R}$ with residue field ${k}$ is the same as giving a homomorphism of local rings

$\displaystyle W(k)[[t_1, \dots, t_{g^2}]] \rightarrow R .$

The relevance of ${W(k)}$ here essentially comes from the fact that every complete (e.g. artin) local ring with residue field ${k}$ is uniquely a continuous ${W(k)}$-algebra. If we restricted ourselves to ${k}$-algebras, we could replace ${W(k)}$ by ${k}$. (more…)

This is a continuation of the project outlined in this post yesterday of describing Grothendieck’s proof that the fundamental group of a smooth curve in characteristic $p$ has $2g$ topological generators (where $g$ is the genus). The first step, as I explained there, is to show that one may “lift” such smooth curves to characteristic zero, in order that a comparison may be made between the characteristic $p$ curve and something much more concrete in characteristic zero, that we can approach via topological methods. This post will be devoted to showing that such a lifting is always possible.

1. Introduction

It is a general question of when one can “lift” varieties in characteristic ${p}$ to characteristic zero. Doing so often allows one to bring in transcendental techniques (to the lift), as it will in this case of ${\pi_1}$. Let us thus be formal:

Definition 4 Let ${X_0}$ be a proper, smooth scheme of finite type over a field ${k}$ of characteristic ${p}$. We say that a lifting of ${X_0}$ is the data of a DVR ${A}$ of characteristic zero with residue field ${k}$, and a proper, smooth morphism ${X \rightarrow \mathrm{Spec} A}$ whose special fiber is isomorphic to ${X_0}$.

There are obstructions that can prevent one from making such a lifting. One example is given by étale cohomology. A combination of the so-called proper and smooth base change theorems implies that, in such a situation, the cohomology of the special fiber and the cohomology of the general fiber, with coefficients in any finite group without ${p}$-torsion, are isomorphic. As a result, if there is something funny in the étale cohomology of ${X_0}$, it might not be liftable. See this MO question.

In the case of curves, fortunately, it turns out there are no such problems, but still actually lifting one will take some work. We aim to prove:

Theorem 5 Let ${X_0 \rightarrow \mathrm{Spec} k}$ be a smooth, proper curve of finite type over the field ${k}$ of characteristic ${p}$. Then if ${A}$ is any complete DVR of characteristic zero with residue field ${k}$, there is a smooth lifting ${X \rightarrow \mathrm{Spec} A}$ of ${X_0}$.

One should, of course, actually check that such a complete DVR does exist. But this is a general piece of algebra, found for instance in Serre’s Local Fields.

The reason there won’t be any obstructions in the case of curves is that they are of dimension one, but we’ll see that the cohomological obstructions to lifting all live in ${H^2}$.

The strategy, in fact, will be to lift ${X_0 \rightarrow \mathrm{Spec} k}$ to a sequence of smooth schemes ${X_n \rightarrow \mathrm{Spec} A/\mathfrak{m}^n}$ (where ${\mathfrak{m} \subset A}$ is the maximal ideal) that each lift each other, using the local nilpotent lifting property of smooth morphisms.

This family ${\left\{X_n \rightarrow \mathrm{Spec} A/\mathfrak{m}^n\right\}}$ is an example of a so-called formal scheme, which for our purposes is just such a compatible sequence of liftings. Obviously any scheme ${X \rightarrow \mathrm{Spec} A}$ gives rise to a formal scheme (take the base-changes to ${\mathrm{Spec} A/\mathfrak{m}^n}$), but it is actually nontrivial (i.e., not always true) to show that a formal scheme is indeed of this form. But we will be able to do this as well in the case of curves. (more…)

In this post, we shall accomplish the goal stated earlier: we shall show that a formally smooth morphism of noetherian rings (which is essentially of finite type) is flat. We shall even get an equivalence: flatness together with smoothness on the fibers will be both necessary and sufficient to ensure that a given such morphism is formally smooth.

In order to do this, we shall use a refinement of the criterion in the first post for when a quotient of a formally smooth algebra is formally smooth. We shall need a bit of local algebra to do this, but the reward will be a very convenient Jacobian criterion, which will then enable us to prove (using the results from last time on lifting flatness from the fibers) the final characterization of smoothness.

3. The Jacobian criterion

So now we want a characterization of when a morphism is smooth. Let us motivate this with an analogy from standard differential topology. Consider real-valued functions ${f_1, \dots, f_p \in C^{\infty}(\mathbb{R}^n)}$. Now, if ${f_1, f_2, \dots, f_p}$ are such that their gradients ${\nabla f_i}$ form a matrix of rank ${p}$, then we can define a manifold near zero which is the common zero set of all the ${f_i}$. We are going to give a relative version of this in the algebraic setting.

Recall that a map of rings ${A \rightarrow B}$ is essentially of finite presentation if ${B}$ is the localization of a finitely presented ${A}$-algebra.

Proposition 5 Let ${(A, \mathfrak{m}) \rightarrow (B, \mathfrak{n})}$ be a local homomorphism of local rings such that ${B}$ is essentially of finite presentation. Suppose ${B = (A[X_1, \dots, X_n])_{\mathfrak{q}}/I}$ for some finitely generated ideal ${I \subset A[X_1, \dots, X_n]_{\mathfrak{q}}}$, where ${\mathfrak{q}}$ is a prime ideal in the polynomial ring.Then ${I/I^2}$ is generated as a ${B}$-module by polynomials ${f_1, \dots, f_k \in A[X_1, \dots, X_n]}$ whose Jacobian matrix has maximal rank in ${B/\mathfrak{n}}$ if and only if ${B}$ is formally smooth over ${A}$. In this case, ${I/I^2}$ is even freely generated by the ${f_i}$. (more…)

Ultimately, we are headed towards a characterization of formal smoothness for reasonable morphisms (e.g. the types one encounters in classical algebraic geometry): we want to show that they are precisely the flat morphisms whose fibers are smooth varieties. This will be a much more usable criterion in practice (formal smoothness is given by a somewhat abstract lifting property, but checking that a concrete variety is smooth is much easier).  This is the intuition between smoothness: one should think of a flat map is a “continuously varying” family of fibers, and one wishes the fibers to be regular. This corresponds to the fact from differential topology that a submersion has submanifolds as its fibers.

It is actually far from obvious that a formally smooth (and finitely presented) morphism is even flat. Ultimately, the idea of the proof is going to be write the ring as a quotient of a localization of a polynomial ring. The advantage is that this auxiliary ring will be clearly flat, and it will also have fibers that are regular local rings.  In a regular local ring, we have a large supply of regular sequences, and the point is that we will be able to lift the regularity of these sequences from the fiber to the full ring.

Thus we shall use the following piece of local algebra.

Theorem Let ${(A, \mathfrak{m}) \rightarrow (B, \mathfrak{n})}$ be a local homomorphism of local noetherian rings. Let ${M}$ be a finitely generated ${B}$-module, which is flat over ${A}$.

Let ${f \in B}$. Then the following are equivalent:

1. ${M/fM}$ is flat over ${A}$ and ${f: M \rightarrow M}$ is injective.
2. ${f: M \otimes k \rightarrow M \otimes k}$ is injective where ${k = A/\mathfrak{m}}$.

This is a useful criterion of checking when an element is ${M}$-regular by checking on the fiber. That is, what really matters is that we can deduce the first statement from the second. (more…)

I’ll now say a few words on formal smoothness. This happens to be closely related to the theory of the cotangent complex (namely, the cotangent complex provides a clean criterion for when a morphism is formally smooth). Ultimately, I would like to aim first for the result that a formally smooth morphism of finite presentation is flat, and thus to characterize such morphisms via the geometric idea of “smoothness” (even though the algebraic version of formally smooth is pure commutative algebra).

1. What is formal smoothness?

The idea of a smooth morphism in algebraic geometry is one that is surjective on the tangent space, at least if one is working with smooth varieties over an algebraically closed field. So this means that one should be able to lift tangent vectors, which are given by maps from the ring into ${k[\epsilon]/\epsilon^2}$.

This makes the following definition seem more plausible:

Definition 1 Let ${B}$ be an ${A}$-algebra. Then ${B}$ is formally smooth if given any ${A}$-algebra ${D}$ and ideal ${I \subset D }$ of square zero, the map

$\displaystyle \hom_A(B, D) \rightarrow \hom_A(B, D/I)$

is a surjection.

So this means that in any diagram

there exists a dotted arrow making the diagram commute. (more…)