In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings.
For this reason, formal schemes frequently appear in topics such as deformation theory.
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.
Algebraic geometry based on formal schemes is called formal algebraic geometry.
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.
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.
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.
, 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.
be a neighborhood basis for zero consisting of ideals of definition.
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 Df is the set of all open prime ideals of A not containing f, then
Finally, a locally noetherian formal scheme is a topologically ringed space
admits an open neighborhood isomorphic (as topologically ringed spaces) to the formal spectrum of a noetherian ring.
of locally noetherian formal schemes is a morphism of them as locally ringed spaces such that the induced map
is a continuous homomorphism of topological rings for any affine open subset U. f is said to be adic or
-adic formal scheme if there exists an ideal of definition
If f is adic, then this property holds for any ideal of definition.
For any ideal I and ring A we can define the I-adic topology on A, defined by its basis consisting of sets of the form a+In.
In this case Spf A is the topological space Spec A/I with sheaf of rings