Artin approximation and moduli

Posted on May 29, 2015 by Dima

tags: henselian rings, formal geometry, moduli

One is often interested whether a covariant functor \(F\) from the category of (affine) schemes to the category of sets can be represented by a scheme so that \(F(X) = h_S(X) = \mathrm{Hom}(X, S)\). It is natural to assume that all sensible functors are locally of finite presentation, that is, preserve inverse limits.

Passing to the opposite category of rings, it means that the contravariant functor \(F^{\mathrm op}\) from rings to sets turns direct limits into inverse limits. This is going to be useful for the following reason: any ring is a direct limit of its finitely generated (over some base) subrings, and so, any object \(\xi \in F^{\mathrm op}(R)\) will be defined over a finitely generated subring \(\xi \in F^{\mathrm op}(R_i)\), for \(F^{\mathrm op}(\varinjlim R_i) \cong \varprojlim F^{\mathrm op}(R_i)\). Example: let \(\mathcal{A}\) be the functor on the category of \(k\)-algebras such that \(\mathcal{A}(R)\) be the set of isomorphism classes of polirized Abelian varieties over \(X=\mathrm{Spec}\,R\). Then a particular \(\xi \in \mathcal{A}^{\mathrm op}(R)\) is defined by some equations with coefficients that belong to a finitely generated over \(k\) algebra \(R_i\).

Let \(R\) be a local ring with the maximal ideal \(\mathfrak m\). An infinitesimal deformation is an element \(\xi \in F(\mathrm{Spec}\,R/(\mathfrak{m}^n)\) for some \(n > 1\). A formal deformation is a sequence of compatible elements \(\xi_i \in F(\mathrm{Spec}\,R/(\mathfrak{m}^n))\). An effective formal deformation is an element \(\xi \in F(\mathrm{Spec}\,\hat R)\), where \(\hat R = \varprojlim R/(\mathfrak{m}^n)\) so in particular an effective formal deformation is a deformation.

A formal deformation \(\{\xi_n \in F(X_n)\}\) is called (uni)versal if for any infinitesimal deformation \(\eta \in F(Y)\) and any map \(Y \to X_n\) such that the induced map maps \(\xi_n\) to \(\eta\), and any \(Z\) that dominates \(Y\) and an element \(\theta \in F(Z)\) that maps to \(\eta\), there is a collection of maps \(Z \to X_n\) such that \(\xi_n\) maps to \(\theta\) (universal, if this collection of maps is unique). In a slogan: any map of an infinitesimal deformation to a (uni)versal one extends to any infinitesimal base (uniquely).

[from now on for notational conevenience let’s stick to contravariant functors on rings]

In general, not all formal deformations are effective. It follows from deformation theory that the universal deformation of an Abelian variety of dimension 2 has base of dimension 4, while the dimension of the base of a universal deformation of an Abelian variety endowed with an ample line bundle (polarization) is 3. It turns out that formal deformations that do not preserve polarization do not come from an element \(\xi \in F(\hat R)\).

But if a functor satisfies the property that every formal deformation is effective, one can apply the following strategy to represent it.

First ingredient is the following approximation theorem by Artin.

Theorem (Artin approximation). Let \(R\) be a henselian local ring with the maximal ideal \(\mathfrak m\), and let \(\hat R\) be its completion. Let \(f_1, \ldots, f_l \in R[x_1, \ldots, x_m]\) be polynomials with coefficients in \(R\). If \(\xi \in \hat R\) is a solution to the system of equations \[ \begin{array}{l} f_1(\xi)=0\\ \ldots\\ f_l(\xi)=0\\ \end{array} \] then for each \(n>0\) there exists \(\xi_n \in R\) which is a solution no the same system of equations and \(\xi \equiv \xi_n (\mathrm{mod}\,n)\).

An easy but important observation is that this statement can be extended to points of locally finitely presented functors. Let \(R\) be a henselian ring as above. Let \(\xi \in F(\hat R)\) for a contravariant locally finitely presented functor \(F\) from rings to sets. Then there exists a finitely generated over \(R\) ring \(R'\) such that \(\xi\) is an image of \(\xi' \in F(R')\). Let \(\xi_n\) be approximations to a system of equations \(f_1(x_1, \ldots, x_m)=0 \ldots f_l(x_1, \ldots, x_m)=0\) where \(f_1, \ldots, f_l\) are generators of the ideal \(I\) such that \(R' \cong R^m/I\). Images of \(\xi_n\) in \(F(\hat R)\) under the image induced by the map \(R' \to \hat R\) are approximations to \(\xi\).

Let us now see how one can algebrize a moduli space given by a functor \(F\) having an approximation statement and a universal formal deformation at hand. More precisely, we aim for the following statement.

Theorem. Assume that a universal effective formal deformation \(\xi\) over the base \(\mathrm{Spec}\,\hat R\) exists. Then there is a scheme \(X'\) of finite type, \(\xi' \in F(X')\), a point \(x\) and an isomorphism \(\mathcal{O}_{X',x} \cong \mathrm{Spec}\,\hat R\) such that localization of \(\xi\) and \(x\) is isomorphic to the versal deformation. The isomorphism is unique if \(\xi\) is universal.

The proof is actually almost tautological in the case when \(\hat R=k[[x_1, \ldots, x_m]]\) (so \(z\) is smooth). Take some morphism \(\mathrm{Spec}\,R \to X\), where \(X\) is some smooth variety of appropriate dimension, for example, \(\mathbb{A}^n\). Without loss of generality we may assume that \(R\) is a henselianization of \(\mathcal{O}_{Z,z}\), so approximation therem implies that for some 'etale cover \(X' \to X\) there is a \(\xi' \in F(X')\) such that its reduction mod \(\mathfrak{m}^2\) coincides with \(\xi\). Now using the defining property of universal deformation we can get a family of automorphisms of \(\hat R/\mathfrak{m}^n\) such that \(\xi' \mathrm{mod}\, \mathfrak{m}^n\) maps to \(\mathrm{mod}\, \mathfrak{m}^n\), their limit is an automorphism of \(\hat R\), which applied to the completion of the local ring \(\mathcal{O}_{X',x'}\) maps \(\xi'\) to \(\xi\).

In general, a universal deformation is not known to have a smooth base and how to deal with that is the content of Artin’s article ``Algebraization of formal moduli I’’.