Formal Scheme - Definition

Definition

Formal schemes are usually defined only in the Noetherian case. While there have been several definitions of non-Noetherian formal schemes, these encounter technical problems. Consequently we will only define locally noetherian formal schemes.

All rings will be assumed to be commutative and with unit. Let A be a (Noetherian) topological ring, that is, a ring A which is a topological space such that the operations of addition and multiplication are continuous. A is linearly topologized if zero has a base consisting of ideals. An ideal of definition for a linearly topologized ring is an open ideal such that for every open neighborhood V of 0, there exists a positive integer n such that . A linearly topologized ring is preadmissible if it admits an ideal of definition, and it is admissible if it is also complete. (In the terminology of Bourbaki, this is "complete and separated".)

Assume that A is admissible, and let be an ideal of definition. A prime ideal is open if and only if it contains . The set of open prime ideals of A, or equivalently the set of prime ideals of, is the underlying topological space of the formal spectrum of A, denoted Spf A. Spf A has a structure sheaf which is defined using the structure sheaf of the spectrum of a ring. Let be a neighborhood basis for zero consisting of ideals of definition. All the spectra of have the same underlying topological space but a different structure sheaf. The structure sheaf of Spf A is the projective limit .

It can be shown that if f ∈ A and D_f is the set of all open prime ideals of A not containing f, then, where is the completion of the localization A_f.

Finally, a locally noetherian formal scheme is a topologically ringed space (that is, a ringed space whose sheaf of rings is a sheaf of topological rings) such that each point of admits an open neighborhood isomorphic (as topologically ringed spaces) to the formal spectrum of a noetherian ring.