Algebraized

Algebraized is an adjective used in mathematics to describe an object or process that can be described, analyzed, or obtained through algebraic means, or that admits an algebraic model. In algebraic geometry and related fields, the term is often used to indicate that a given analytic or formal object comes from an algebraic one via a standard construction.

One common use concerns complex analytic spaces. A complex analytic space is said to be algebraizable if it is isomorphic to the analytification of an algebraic variety over the complex numbers. This idea is central to the spirit of GAGA-type results, which compare algebraic and analytic categories. In general, not every analytic space is algebraizable; algebraization is a restrictive property that depends on the object and the context.

Another important vein is formal geometry. A formal scheme over a complete local ring is algebraizable if it is isomorphic to the formal completion of an algebraic scheme along a closed subscheme. This notion provides a bridge between formal geometry and algebraic geometry and is used in deformation theory and related topics.

Morphism algebraization refers to the situation where a morphism of formal schemes (or of analytic spaces) lifts to a morphism of algebraic schemes. Proving algebraization in this sense often involves foundational results such as Grothendieck’s formal geometry, Artin’s approximation, and related theorems.

See also: algebraization, algebraic geometry, analytic geometry, GAGA, Chow’s theorem, formal schemes, completions.