nLab
deformation functor

Contents

Idea and Motivation

One motivation for studying deformations over Artin rings is that one may wish to classify/parameterize or create a moduli space of some particular type of object. Often it is very easy to write down a functor of points for the space. For instance, curves of genus gg, vector bundles on some fixed variety, maps between varieties, closed subschemes, quotients of a coherent sheaf, and the list goes on.

But then one must ask, is this really a space? In other words, is it representable? This rarely happens, but even if you don’t have a legitimate space, you can see what the “points” are formally locally. If everything in your original problem was noetherian, then completion of the local ring at a point gives rise to the Artin rings we consider. The deformations of the object over infinitesimal thickenings gives us a way to tell what the points near our given point look like on the original moduli space.

This reduces us to now asking for a given deformation problem to be representable, but this is almost always too much to hope for as well. Schlessinger’s criterion refers to two weaker notions of representability, namely that of hulls and that of prorepresentability. The criterion gives a beautifully simple necessary and sufficient way to determine when we get each.

Once we determine these formal local points, we must then go backwards and see if these correspond to actually points in the original space. This is known as effectivization.

Definitions

Λ\Lambda will denote a complete noetherian ring.

Unless otherwise specified, everything will be in the category of local Artin Λ\Lambda-algebras with residue field kk and will be denoted ΛArt k_\Lambda Art_k.

Define ΛNoeth k_\Lambda Noeth_k to be the category of complete noetherian local Λ\Lambda-algebras with residue field kk.

A thickening is a surjection AfA0A' \stackrel{f}{\to} A\to 0 such that I=ker(f)I=ker(f) satisfies m AI=0m_{A'}I=0.

A small thickening is a thickening in which II is principal. Note that in our category any surjection can be factored as a sequence of small thickenings.

A predeformation functor is a functor F: ΛArt kSetF: _\Lambda Art_k \to \text{Set} such that F(k)={*}F(k)=\{*\} is a singleton. This condition is the notion that we are really over a “point”, and so any deformation must be the trivial one.

The functor overhatF: ΛNoeth kSet\overhat{F}: _\Lambda Noeth_k\to \text{Set} is defined to be overhatF(A)=lim nF(R/𝔪 n)\displaystyle \overhat{F}(A)= \lim_n F(R/\mathfrak{m}^n).

A predeformation functor is prorepresentable if overhatF\overhat{F} is representable.

A hull for FF is a pair (R,η)(R, \eta) where R ΛNoeth kR\in _\Lambda Noeth_k and ηoverhatF(R)\eta\in \overhat{F}(R) such that h RFh_R\to F is formally smooth and we have a bijection on tangent spaces T h RT FT_{h_R}\to T_F.

k[ϵ]k[\epsilon] is defined to be the ring k[ϵ]/(ϵ 2)k[\epsilon]/(\epsilon^2) with the trivial Λ\Lambda-algebra structure.

The Conditions

Note that whenever we are given a predeformation functor FF, and two maps of rings: AAA'\to A and AAA''\to A we get an induced map F(A× AA)F(A)× F(A)F(A)F(A'\times_A A'')\to F(A')\times_{F(A)} F(A'') by functoriality of FF and the universal property of a pullback. Call this map (*).

Schlessinger gives four conditions in his paper called (H1) - (H4).

(H1) If AAA''\to A is a small thickening then (*) is surjective.

(H2) If A=kA=k and A=k[ϵ]A''=k[\epsilon], then (*) is bijective.

(H3) T FT_F is finite-dimensional

(H4) If A=AA''=A' and AAA'\to A is a small thickening, then (*) is bijective

Geometrically the conditions can informally be thought of as follows:

One can think of (H1) as being able to “glue”.

One can think of (H2) as gluing being unique over infinitesimal neighborhoods.

(H3) is having a finite dimensional tangent space

One can think of (H4) as gluing being unique on a small thickening over itself.

Deformation Functors

We call a predeformation functor a deformation functor if it satisfies (H1) and (H2). Here we make precise what was meant in the motivation section. Suppose you have a moduli functor F˜:SchSet\tilde{F}: \text{Sch}\to \text{Set} and you would like to know what the moduli space looks like in a neighborhood of some point say η 0F¯(Speck)\eta_0\in \overline{F}(\text{Spec} k). Then if we consider the functor of deformations of η 0\eta_0, we get F: ΛArt kSetF: _\Lambda Art_k \to \text{Set} by the following prescription F(A)={ηF¯(SpecA):η| Speck=η 0}F(A)=\{ \eta \in \overline{F}(\text{Spec} A): \eta|_{\text{Spec} k}=\eta_0\}. This FF can be seen to be a deformation functor.

Of course, this doesn’t always work if we’re parametrizing objects with automorphisms, but this general approach does work for most of the examples in the motivation section and explains why this class of functors is what we consider.

Schlessinger’s Criterion

The main theorem of Schlessinger’s paper says that (H1), (H2), and (H3) are satisfied if and only if FF has a hull. The second part of the theorem is that FF is prorepresentable if and only if in addition (H4) is satisfied.

Examples

Let Def XDef_X be the predeformation functor parametrizing flat deformations of XX. Then (H1) and (H2) are always satisfied and hence it is a deformation functor. If XX is proper, then (H3) is satisfied and it has a hull. Def XDef_X is prorepresentable if and only if every automorphism extends over a small thickening.

Consider the node X=Speck[[x,y]]/(xy)X=\text{Spec} k[ [ x,y ] ] /(xy). Then the pair (k[[t]],Speck[[x,y,t]]/(xyt))(k[ [ t ] ], \text{Spec} k [ [ x,y,t ] ] /(xy - t)) is a hull for Def XDef_X. Note that this functor is not prorepresentable.

Several more can be added.

Tangent and Obstruction Theories

I haven’t come up with a nice succinct way to do this yet. I think there is a nice streamlined gerbe/stacky way to think about it in the Brian Osserman article cited below.

Effectivization

Given a deformation functor, FF, that came from a moduli problem and some object ηoverhatF(R)\eta\in \overhat{F}(R), we’d like to tell when it corresponds to a point on the original moduli space over SpecR\text{Spec}R, or if (R,η)(R,\eta) is a hull for FF does there exist a universal object over SpecR\text{Spec}R?

An example where we do have effectivization is Grothendieck’s existence theorem for coherent sheaves. Let f:XSpecAf:X\to \text{Spec}A be proper and AA complete, local, noetherian ring. Let A n=A/m nA_n=A/m^n and X n=X AA nX_n=X\otimes_A A_n the thickenings over A nA_n. If n\mathcal{F}_n is a compatible collection of coherent sheaves on X nX_n, then there exists a coherent sheaf \mathcal{F} on XX whose restriction to each X nX_n is n\mathcal{F}_n.

References

See also deformation theory and references therein.

  • B. Osserman, Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, IX (2010), no. 3, 581-633.

  • M. Schlessinger, Functors of Artin rings, Trans. AMS 130, 208-222 (1968)

this was a groundbreaking article at the time, still much cited.

Revised on January 19, 2011 03:37:53 by hilbertthm90 (24.19.0.156)