Lemma of the day

Let F be a predeformation category which has a versal formal object. Then

  1. F has a minimal versal formal object,
  2. minimal versal objects are unique up to isomorphism, and
  3. any versal object is the pushforward of a minimal versal object along a power series ring extension.

See Lemma Tag 06T5.

What is fun about this lemma is that it produces a minimal versal object (as defined in Definition Tag 06T4) from a versal one without assuming Schlessinger’s axioms. If Schlessinger’s axioms are satisfied and one is in the classical case (see Definition Tag 06GC), then a minimal versal formal object is a versal formal object defined over a ring with minimal tangent space. This is discussed in Section Tag 06IL.