Chapter I — The Language of Schemes

§10. Formal Schemes

Translation status. Skeleton with definitions and principal statements; full proofs reference .

Note. The results of §10 are not used before §3 of Chapter III. Readers are encouraged to skip this section on first reading.

10.1. Formal affine schemes

(10.1.1) Let $A$ be an admissible topological ring (0.7.1). The formal spectrum $Spf(A)$ is the topological space of open prime ideals of $A$, with topology induced from $\mathrm{Spec}(A)$ (the discrete-topology spectrum). For each ideal of definition $\mathfrak{J}$, $Spf(A)$ is identified with $\mathrm{Spec}(A/\mathfrak{J})$ (with topology, but with structure sheaf inheriting the adic completion).

Definition (10.1.2). A formal affine scheme is a topologically ringed space $(X,{\mathcal{O}}_X)$ isomorphic to $(Spf(A),{\mathcal{O}}_{Spf(A)})$ for some admissible ring $A$. Here ${\mathcal{O}}_{Spf(A)}$ is the sheaf of topological rings whose sections over $D(f)$ are the completed rings of fractions $A_{\{f\}}$ (0.7.6.15).

Proposition (10.1.3). $\Gamma (Spf(A),{\mathcal{O}}_{Spf(A)})=A$ topologically.

Proposition (10.1.4). For $A$ adic Noetherian, $Spf(A)$ is canonically ${\underrightarrow{\lim }}_n\mathrm{Spec}(A/{\mathfrak{J}}^{n+1})$ (filtered inductive limit in the category of ringed spaces).

Proposition (10.1.6). Spf is a functor from admissible rings to topologically ringed spaces, contravariant in the ring.

10.2. Morphisms of formal affine schemes

(10.2.1) A morphism of formal affine schemes $(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ is a morphism of topologically ringed spaces with continuous stalk homomorphisms; equivalently (for $X=Spf(A)$, $Y=Spf(B)$), a continuous ring homomorphism $B\to A$.

Proposition (10.2.2). Hom-sets are bijective with continuous ring homomorphisms:

Hom(Spf(A), Spf(B)) ≅ Hom_{cont}(B, A).

10.3. Ideals of definition for a formal affine scheme

Definition (10.3.3). A sheaf of ideals of definition for a formal prescheme $X$ is a quasi-coherent sheaf of ideals $\mathcal{I}\subset {\mathcal{O}}_X$ such that locally $\mathcal{I}$ corresponds to an ideal of definition.

Proposition (10.3.4)–(10.3.6). Existence and uniqueness of sheaves of ideals of definition for formal affine schemes; the fundamental system of ideals of definition corresponds to a fundamental system of neighborhoods of 0 in $A$.

10.4. Formal preschemes and morphisms

Definition (10.4.2). A formal prescheme is a topologically ringed space $(X,{\mathcal{O}}_X)$ covered by open subsets each isomorphic to a formal affine scheme.

Proposition (10.4.3). Formal preschemes form a category, with morphisms locally those of formal affine schemes.

Corollary (10.4.4). The functor Spf extends to admissible topological algebras over a fixed admissible base.

Definition (10.4.5). Adic morphism: a morphism of formal preschemes $f:X\to Y$ is adic if for every affine open $Spf(B)\subset Y$ with ideal of definition $\mathfrak{J}\subset B$, the preimage $f^{-1}(Spf(B))$ admits an open cover by formal affine opens $Spf(A_{\alpha })$ whose topologies are $\mathfrak{J}{A}_{\alpha }$-adic.

Proposition (10.4.6). Adic morphisms are stable under composition and base change (when fiber products exist; see (10.7)).

(10.4.7) Formal $S$-preschemes: for a formal prescheme $S$, a formal $S$-prescheme is a formal prescheme $X$ together with a morphism $X\to S$.

10.5. Sheaves of ideals of definition for formal preschemes

(10.5.1) A formal prescheme $X$ carries a largest sheaf of ideals of definition $\mathcal{I}\subset {\mathcal{O}}_X$ such that locally $\mathcal{I}$ corresponds to an ideal of definition. Quotient ${\mathcal{O}}_X/{\mathcal{I}}^n$ defines a sheaf of rings making $(X,{\mathcal{O}}_X/{\mathcal{I}}^n)$ a prescheme for each $n$.

Proposition (10.5.3). Existence of the largest sheaf of ideals of definition.

Proposition (10.5.4)–(10.5.6). Comparison with sheaves of nilpotents, with adic completion, and with quotient-sheaf constructions.

10.6. Formal preschemes as inductive limits

Proposition (10.6.2)–(10.6.3). For an adic Noetherian formal prescheme $X$ with sheaf of ideals of definition $\mathcal{I}$, the system $X_n=(X,{\mathcal{O}}_X/{\mathcal{I}}^{n+1})$ ($n\ge 0$) is an adic inductive system of preschemes, and $X={\underrightarrow{\lim }}_n{X}_n$ (in a suitable sense).

Corollaries (10.6.4)–(10.6.5). Properties carried from the $X_n$ to $X$ (and conversely).

Proposition (10.6.9). Sheafification of inductive systems.

10.7. Products of formal preschemes

Proposition (10.7.2)–(10.7.3). For $S$ a formal prescheme and X, Y formal $S$-preschemes, the fiber product $X{\hat{\times }}_{S}Y$ exists in the category of formal preschemes, computed via completed tensor products of admissible algebras (0.7.7).

10.8. Formal completion of a prescheme along a closed subset

Lemma (10.8.2). For a prescheme $Y$ and a closed subset $Y^{\prime }\subset Y$, the formal completion ${\hat{Y}}_{/Y^{\prime }}$ exists as a formal prescheme.

Definition (10.8.4). For a quasi-coherent ideal $\mathcal{J}\subset {\mathcal{O}}_Y$ defining $Y^{\prime }$, the formal completion of $Y$ along $Y^{\prime }$ is the topologically ringed space $(Y^{\prime },{\underleftarrow{\lim }}_n{\mathcal{O}}_Y/{\mathcal{J}}^n)$. As a formal prescheme, this is the inductive limit of $(Y^{\prime },{\mathcal{O}}_Y/{\mathcal{J}}^n)$.

Proposition (10.8.5). The formal completion is a formal prescheme; functorial in the pair $(Y,Y^{\prime })$.

Corollary (10.8.6). Coherent sheaves on $Y$ give rise canonically to coherent sheaves on ${\hat{Y}}_{/Y^{\prime }}$ by base change.

Proposition (10.8.8)–(10.8.11). Behavior of formal completion under products, fiber products, and morphisms; commutation with operations on coherent sheaves.

Corollaries (10.8.12)–(10.8.14). Affineness, Noetherianness, and topological completeness of formal completions.

10.9. Extension of morphisms to completions

Proposition (10.9.4)–(10.9.5). Morphisms $Y\to Z$ of preschemes extend functorially to morphisms ${\hat{Y}}_{/Y^{\prime }}\to {\hat{Z}}_{/Z^{\prime }}$ of formal completions, provided closed subsets are compatible.

Proposition (10.9.7). Existence and uniqueness of extension under standard hypotheses.

10.10. Coherent sheaves on formal affine schemes

Proposition (10.10.2)–(10.10.8). For an adic Noetherian formal affine scheme $Spf(A)$: every coherent sheaf is the formal completion of a coherent sheaf on $\mathrm{Spec}(A_n)$ for some $n$; the category of coherent sheaves is equivalent to the category of $A$-modules of finite type.

10.11. Coherent sheaves on formal preschemes

Proposition (10.11.1). Coherent sheaves on locally Noetherian formal preschemes admit local descriptions in terms of coherent sheaves on associated preschemes.

Theorem (10.11.3). Comparison of coherent sheaves with formal completion: the formal completion of a coherent sheaf on an algebraic prescheme along a closed subset is again coherent.

Corollaries (10.11.4)–(10.11.5). Specialization, restriction, and stalk computations for coherent sheaves on formal preschemes.

Proposition (10.11.7)–(10.11.9). Functoriality of coherence and projective limits of coherent sheaves.

10.12. Adic morphisms of formal preschemes

Definition (10.12.1). An adic morphism (from (10.4.5)) preserves ideals of definition: $f^{♯}({\mathcal{I}}_Y)\cdot {\mathcal{O}}_X={\mathcal{I}}_X$ locally.

Theorem (10.12.3). Adic morphisms of locally Noetherian formal preschemes are stable under composition and base change.

Proposition (10.12.3.1). Characterization of adic morphisms via inductive systems.

10.13. Morphisms of finite type

Proposition (10.13.1). A morphism of locally Noetherian formal preschemes $f:X\to Y$ is of finite type iff locally on $Y=Spf(B)$, $X$ is $Spf(C)$ with $C$ a topologically finitely generated $B$-algebra (quotient of $B{T}_1,\cdots ,T_r$).

Definition (10.13.3). A formal prescheme $X$ of finite type over a formal prescheme $S$ is one whose structure morphism is of finite type.

Corollaries (10.13.2), (10.13.4)–(10.13.6). Composition, base change, and base change to ordinary preschemes preserve finite-type morphisms.

10.14. Closed subpreschemes of formal preschemes

Proposition (10.14.1). Closed formal subpreschemes of a formal prescheme correspond bijectively to quasi-coherent sheaves of ideals.

Definition (10.14.2). A closed immersion of formal preschemes $Y\to X$ is a morphism realizing $Y$ as a closed formal subprescheme of $X$.

Proposition (10.14.3)–(10.14.4). Stability properties of closed immersions: composition, base change, fiber products.

10.15. Separated formal preschemes

Definition (10.15.1). A morphism of formal preschemes $f:X\to S$ is separated if the diagonal $\Delta :X\to X{\hat{\times }}_{S}X$ is a closed immersion of formal preschemes. Formal $S$-schemes are separated formal $S$-preschemes.

Composition, base change, and standard valuative criteria carry over from the prescheme setting.