Section 2.9 - Formal Schemes

One feature which clearly distinguishes the theory of schemes from the older theory of varieties is the possibility of having nilpotent elements in the structure sheaf of a scheme. In particular, if Y is a closed subvariety of a variety X, defined by a sheaf of ideals I , then for any n ≥ 1 we can consider the closed subscheme Yn defined by the nth power I n of the sheaf of ideals I . For n ≥ 2, this is a scheme with nilpotent elements. It carries information about Y together with the infinitesimal properties of the embedding of Y in X.