SCHLESSINGER’S CRITERIA

RAVI VAKIL

I’m tired of writing this on the board repeatedly, so here it is in the form of a handout.

Fix our functor F : C → Sets.

Let A′ → A and A′′ → A be morphisms in C, and consider the map

F (A′ ×_{A} A′′) → F (A′) ×_{F (A)} F (A′′). (1)

(1) F has a hull iff F has properties H1–H3:

H1. (You can glue.) (1) is a surjection whenever A′′ → A is a small extension.

Equivalently whenever A′′ → A is any surjection. H2. (Uniqueness of gluing on k[ϵ]/ϵ^{2}.) (1) is a bijection when A = k, A′′ = k[ϵ]/ϵ^{2}.

Equivalently, A′′ = k[V]. Then by previous lemma, t_{F} is a k-vector space. H3. (finite-dimensional tangent space) dim_{k}(t_{F}) < ∞.

(2) F is pro-representable if and only if F has the additional property

H4. (bijection for gluing a small extension to itself)

F (A′ ×_{A} A′) → F (A′) ×_{F (A)} F (A′). (2)

is a bijection for any small extension A′ → A.

Important comment. Assume F satisfies H1–H3. Now given a fairly small extension p : A′ → A. Given any a ∈ F (A), i.e. family over A, the set of lifts to F (A′) has a transitive action by the group t_{F} ⊗ I. H4 is precisely the condition that this set is a principal homogeneous space under t_{F} ⊗ I.

Date: Thursday, November 2, 2000.