Formal scheme - Knowlpedia

Definition

An adic formal scheme is a locally topologically ringed space locally isomorphic to a formal spectrum

\operatorname{Spf}(A),

where $A$ is a complete and separated adic ring. A morphism of formal schemes is a morphism of locally topologically ringed spaces whose maps on local sections are continuous.

Infinitesimal presentation

Locally, if $A$ has an ideal of definition $I$ , the formal scheme packages the compatible tower

\operatorname{Spec}(A/I) \hookrightarrow \operatorname{Spec}(A/I^2) \hookrightarrow\cdots .

The first term is a scheme of definition, while the higher terms retain progressively thicker infinitesimal neighborhoods. It is not generally the reduction: $\operatorname{Spec}(A/I)$ can itself be nonreduced. Formal schemes can therefore remember an entire completed neighborhood even when their underlying topological spaces are small.

Ordinary schemes and formal completions

An ordinary scheme becomes a formal scheme by giving its structure sheaf the discrete topology. More importantly, completing a scheme along a closed subscheme produces a genuinely formal scheme. For a group scheme, completion along the identity produces a formal group.

Group objects

Finite products exist in the category of formal schemes over a base. Hence one can form group objects in this category. This coordinate-free definition is the natural home of formal groups; a formal group law appears only after choosing parameters on the underlying formal scheme.

Convention

There are broader frameworks of formal schemes and formal algebraic spaces, with weaker hypotheses on the topology. This knowl fixes the classical adic/EGA setting needed for complete power-series rings. Statements using a broader convention should say so explicitly.

References

The Stacks Project Authors, “Formal schemes à la EGA.” Section 87.2, Tag 0AHY. Relevant: definitions of affine formal schemes, formal schemes, and morphisms.
Alexander Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique I: Le langage des schémas, Publications Mathématiques de l’IHÉS 4 (1960). Relevant: Chapter 0, §7.