deformationfunktor

The deformationfunktor, often called the deformation functor, is a framework in deformation theory that associates to each Artinian local algebra A over a field k a set (or groupoid) of deformations of a fixed mathematical object defined over k. The functor viewpoint encodes all infinitesimal deformations and their compatibilities in a single algebraic structure, enabling the study of existence, uniqueness, and parameter spaces of deformations.

For a fixed scheme X over k, the deformationfunktor Def_X is defined on local Artinian k-algebras A with residue field k by sending A to the set of isomorphism classes of deformations of X over Spec A whose special fiber is X. Similarly, for a fixed coherent sheaf F on X, Def_F(A) consists of isomorphism classes of flat A-sheaves on X_A that specialize to F over k.

Tangent spaces and obstructions are described by Ext and cohomology groups. For deformations of F, the tangent space is Ext^1(F,F) and the obstructions lie in Ext^2(F,F). For deformations of X, the tangent space is H^1(T_X) and obstructions lie in H^2(T_X) (with T_X the sheaf of vector fields).

Schlessinger's criteria provide conditions (H1)–(H4) under which a deformationfunktor is prorepresentable, i.e., controlled by a formal object. If these hold, the functor possesses a hull or a miniversal deformation, giving a local moduli description near X or F.

Applications include the construction of moduli spaces, analysis of obstructions to deformability, and the study of the local structure of moduli problems in algebraic geometry and complex geometry.