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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.
Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic functions.
Algebraic geometry based on formal schemes is called formal algebraic geometry.
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