Formal spectrum - Knowlpedia

Definition

Let $A$ be a complete and separated adic ring, and choose a finitely generated ideal of definition $I$ . The formal spectrum $\operatorname{Spf}(A)$ is the locally topologically ringed space whose underlying topological space is

\{\mathfrak p\in\operatorname{Spec}(A):\mathfrak p\text{ is open}\} =V(I)\cong\operatorname{Spec}(A/I),

and whose structure sheaf retains the compatible structure sheaves of all quotients $A/I^n$ . It may be viewed as the system of infinitesimal thickenings

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

Functions and morphisms

Global functions recover the topological ring:

\Gamma(\operatorname{Spf}(A),\mathcal O)=A.

For complete adic rings $A$ and $B$ in this convention, continuous ring homomorphisms give morphisms in the opposite direction:

\operatorname{Hom}\bigl(\operatorname{Spf}(B),\operatorname{Spf}(A)\bigr) \cong \operatorname{Hom}_{\mathrm{cont}}(A,B).

Continuity is essential because the sheaves remember the adic topologies.

Relation to ordinary spectra

If $A$ has the discrete topology, so that $0$ may be chosen as an ideal of definition, then $\operatorname{Spf}(A)$ is the ordinary $\operatorname{Spec}(A)$ . For a general ideal of definition, $\operatorname{Spf}(A)$ has the topological space of the scheme of definition $\operatorname{Spec}(A/I)$ , while its structure sheaf retains functions to every infinitesimal order. The scheme of definition need not be reduced.

For example,

\operatorname{Spf}(k[[X_1,\ldots,X_n]])

has one underlying point when $k$ is a field, yet its structure sheaf contains all formal directions around that point.

Convention

Several related notions of formal spectrum occur in the literature. This knowl uses the classical adic/EGA convention with complete separated adic rings and finitely generated ideals of definition, which is sufficient for the finite-dimensional formal discs used in formal Lie theory.

References

The Stacks Project Authors, “Formal schemes à la EGA.” Section 87.2, Tag 0AHY. Relevant: the construction of $\operatorname{Spf}(A)$ and continuous 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, formal schemes.