Deformationsfunktoren

Deformationsfunktoren, also known as deformation functors, are mathematical objects used in deformation theory, a branch of algebraic geometry and commutative algebra. They provide a way to study the deformations of algebraic structures, such as rings, modules, and schemes, by encoding the infinitesimal deformations of these structures in a functorial manner.

The concept of deformationsfunktoren was introduced by Alexander Grothendieck in the 1960s as part of his foundational work on algebraic geometry. The idea is to associate to a given algebraic structure a functor that captures the possible ways in which the structure can be deformed. This functor, called the deformation functor, takes as input a local Artinian ring and outputs the set of deformations of the original structure over that ring.

Deformationsfunktoren are particularly useful in the study of moduli problems, where one wants to understand the space of all possible deformations of a given algebraic structure up to isomorphism. The deformation functor provides a way to encode this information in a functorial manner, making it possible to apply techniques from category theory and homological algebra to study moduli problems.

One of the key properties of deformationsfunktoren is their pro-representability. This means that the deformation functor is represented by a formal scheme, which is a scheme that is locally isomorphic to the spectrum of a power series ring. The pro-representability of the deformation functor allows one to study the deformations of an algebraic structure by studying the geometry of the corresponding formal scheme.

Deformationsfunktoren have found applications in various areas of mathematics, including algebraic geometry, commutative algebra, and representation theory. They have been used to study the deformations of algebraic varieties, modules over commutative rings, and representations of Lie algebras. The theory of deformationsfunktoren continues to be an active area of research, with new developments and applications emerging in recent years.