Functors of Artin Rings

Abstract

In the previous chapters we introduced the notion of infinitesimal deformations of a complex manifold together with an idea of its role in deformation theory. The notion of infinitesimal deformations extends to a wide class of algebro-geometric structures; for instance if R is an associative $\mathbb {C}$-algebra, one can define a deformation of R over an Artin local $\mathbb {C}$-algebra A as an isomorphism class of structures of associative A-algebra on the A-module $R\otimes _{\mathbb {C}}A$, such that the natural projection $R\otimes _{\mathbb {C}}A\rightarrow R$ is a morphism of algebras.