Deformation theory and Lie algebras

1 Formal moduli problems

The purpose of deformation theory is to study the local behaviour of moduli problems. In a moduli problem, one studies smoothly varying families of some kind of geometric object. We focus here on moduli problems over a field \(k\) of characteristic zero.

As a very simple example, one can consider the moduli problem of one dimensional vector spaces with n labelled points, not all of which are zero. A family of such objects over a variety \(X\) is the same as giving a line bundle on \(X\) with \(n\) sections that don’t simultaneously vanish. The moduli problem associated to this is the functor \(F:\CAlg ^{\heart } \to \Set \), where \(\CAlg ^{\heart }\) is the category of (discrete) commutative \(k\)-algebras, and the functor takes \(R\) to the set of line bundles with n non simultaneously vanishing sections. This moduli problem has an associated moduli space, namely \(\PP ^{n-1}\). This means that the functor \(F\) is represented by \(\PP ^{n-1}\), i.e \(F\) is naturally isomorphic to \(\Hom (\Spec (R) ,\PP ^{n-1})\).

In general, we think of a moduli problem as being represented by some kind of geometric object, such as a stack. For example, we can consider the moduli problem of families of elliptic curves. It is a bad idea to encode this moduli problem as a \(\Set \)-valued functor, because elliptic curves have automorphisms. Instead, we make our functor at least valued in \(\Gpd \), the category of groupoids, which in particular allows us to recover the isomorphism classes of elliptic curves as \(\pi _0\). The groupoid valued functor is much better behaved: for example it satisfied descent and is represented by a Deligne-Mumford stack. We might as well have our moduli problems be valued in \(\cS \), the category of spaces, since this is the category of \(\infty \)-groupoids, and in particular contains \(\Gpd \) and \(\Set \) fully faithfully in it.

To motivate how we will study the local behaviour of moduli problems, consider a \(k\)-variety \(X\), with functor of points \(X: \CAlg ^{\heart } \to \cS \), and let \(p \in X(k)\). We can form the completion of \(X\) at \(p\), i.e the functor \(X^{\wedge }_p :\CAlg ^{\heart } \to \cS \), whose value on \(R\) is the subset of maps \(\Spec (R) \to X\) such that the image on the Zariski spectra is contained in \(\{p\}\). This functor sees all of the local behaviour of \(X\) near \(p\). Any map in \(X^{\wedge }_p(R)\) is given by a map of rings \(\cO _{X,p} \to R\) that sends the maximal ideal \(m\) to nilpotent elements¹. Since \(m\) is finitely generated, this means that some large power \(m^n\) is sent to zero. Thus the map factors through \(\cO _{X,p}/m^n\) for \(n\gg 0\), which is an Artinian local ring that is finite dimensional with residue field \(k\). Thus we see in particular that the functor \(X^{\wedge }_p\) is actuallly determined by its restriction to the subcategory \(\CAlg ^{\heart }_{\art }\) of Artinian local finite dimensional rings with residue field \(k\).

To give an example of what kind of results are proven in deformation theory, we can consider the moduli problem of deformations of \(X\), some smooth proper variety over \(k\). This functor \(F:\CAlg ^{\heart }_{\art } \to \cS \) sends \(R\) to the groupoid of lifts of \(X\) to a smooth proper \(R\)-scheme². Some facts about this functor are:

1. \(\pi _0F(\Lambda [\ee _0]) = H^1(X;T_X)\).
2. \(\pi _1F(\Lambda [\ee _0]) = H^0(X;T_X)\).
3. Given an object in \(F(\Lambda [\ee _0])\), there is an obstruction in \(H^2(X;T_X)\) that vanishes iff it can be lifted to \(F(k[\ee _0]/\ee ^3)\).

In other words, isomorphism classes of first order deformations of \(X\) correspond to \(H^1(X;T_X)\), automorphisms of first order deformations correspond to \(H^0(X;T_X)\), and the obstructionn to lifting a first order deformationn of \(X\) to a second order deformation of \(X\) lies in \(H^2(X;T_X)\).

These facts, especially \((3)\), can be explained via spectral algebraic geometry. To do so we need to extend the functor \(F\) to the subcategory \(\CAlg _{art}\) of \(\CAlg \) consisting of connective, bounded above commutative \(k\)-algebras with \(\pi _i\) finite dimensional over \(k\), \(\pi _0\) local Artinian with residue field \(k\).

Our functor \(F: \CAlg _{\art } \to \cS \) now sends \(R\) to the space of smooth proper spectral schemes over \(R\) lifting \(X\). \(F\) now satisfies the following properties:

(F1) \(F(k) = *\)
(F2) \(F\) preserves pullback diagrams of the form

with \(f\) and \(g\) surjective on \(\pi _0\).

Via the pullback squares

(-tikz- diagram)

where \(\Lambda [\ee _n]\) is the trivial square zero extension in degree \(n\), we learn that \(F(\Lambda [\ee _n]) = \Omega F(\Lambda [\ee _{n+1}])\). Thus \(F(\Lambda [\ee _n])\) fit togehter to form a spectrum \(T_F\) satisfying \(\Omega ^{\infty -n}T_F = F(\Lambda [\ee _{n+1}])\). \(T_F\) is the tangent complex of \(F\).

Now, \((1)-(3)\) are all exaplained by the fact that \(\pi _i T_F = H^{1-i}(X;T_X)\). Indeed this immediately explaines \((1)\) and \((2)\) given the definition of \(T_F\). For \((3)\), there is a pullback square

(-tikz- diagram)

coming from the fact that \(k[\ee _0]/\ee _0^3\) is a square zero extension of \(\Lambda [\ee _0]\). Applying \(F\) gives the pullback square of spaces

(-tikz- diagram)

So if we have a class in \(\pi _0F(\Lambda [\ee _0])\), we learn from the fibre square above that the obstruction to lifting it to \(F(k[\ee _0]/\ee _0^3)\) lies in \(\pi _0F(\Lambda [\delta _1]) = H^2(X;T_X)\), explaining \((3)\). Note to give this explanation we needed to use \(\Lambda [\delta _1]\), which is not discrete.

The notion of a formal moduli problem axiomitizes the properties of this example.

¹ Indeed, this is equivalent to the image being contained in \(p\) since \(m\) consists of functions that vanish at \(p\), and the nilpotent elements are those that vanish on all of \(R\), so the condition says that if a function vanishes at \(p\), its pullback to \(\Spec (R)\) vanishes everywhere.

² The lift is equipped with an isomorphism of its reduction to \(k\) with \(X\).

Definition 1.1. A formal moduli problem (or FMP) is a functor \(F:\CAlg _{\art } \to \cS \) satisfying \((F1)\) and \((F2)\) above.