Formal Scheme

In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes.

A locally noetherian scheme is a locally noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally noetherian formal schemes contains all locally noetherian schemes.