用户: Scavenger/Formal Schemes

In these notes I try to figure out the correct definitions of notions concerning formal schemes. I’ll record my progress here until I think these notes is satisfactory enough.

I realized that the “correct” approach should be the sheafy approach, in the sense that formal schemes should be sheaves over a certain site.

[3] tag 0AHY might be a source of materials concerning the classical definition of formal schemes in EGA. [1] and [2] might be a sourse of materials of the sheafy approach.

Convention 0.1. We associate to each partially ordered set a category by the following rule: they have the same set of objects, and $i\to j$ if and only if $i\le j$. We often make no distinctions between a partially ordered set and its associated category.

1Local Story

Definition 1.1. A topological ring is said to be linearly topologized if $0$ has a fundamental system of neighbourhoods consisting of ideals.

If $A$ is a linearly topologized topological ring, an ideal $I\subseteq A$ is said to be an ideal of definition if it is open and every neighbourhood of $0$ contains $I^n$ for some $n\in {\mathbb{Z}}_{\ge 1}$.

An admissible ring is a complete linearly topologized topological ring that has an ideal of definition. In this case the ideals of definition form a fundamental system of neighbourhood of $0$.

If $A$ is an admissible ring then its ideals of definition form a cofiltered partially ordered set, where $I\le J$ if and only if $I\subseteq J$. We denote this partially ordered set by $\mathrm{Ideal}(A)$.

Construction 1.2. Let $\mathrm{AffSch}:={\mathrm{Ring}}^{\mathrm{op}}$ be the opposite category of the category of rings, which is equivalent to the category of affine schemes. Let $A$ be an admissible ring. We define a presheaf $\mathrm{Spf}(A)$ on $\mathrm{AffSch}$ by the formula $(\mathrm{Spf}(A))(R)={\mathrm{Mor}}_{\mathrm{TopRing}}(A,R)$, where $R$ is an arbitrary commutative ring, equipped with the discrete topology.

An affine formal scheme is a presheaf on $\mathrm{AffSch}$ isomorphic to $\mathrm{Spf}(A)$ for some admissible ring $A$.

It is worth noting that there is a natural isomorphism ${\mathrm{Mor}}_{\mathrm{TopRing}}(A,R)\cong {\mathrm{colim}}_{I\in \mathrm{Ideal}(A{)}^{\mathrm{op}}}{\mathrm{Mor}}_{\mathrm{Ring}}(A/I,R)$, and the latter could also be used in the definition of $\mathrm{Spf}(A)$.

We equip $\mathrm{AffSch}$ with the Zariski topology, making it into a site. When we say sheaves on $\mathrm{AffSch}$, we always mean sheaves with respect to the Zariski topology. When we say sheaves with out specifying which site they live on we always mean sheaves on $\mathrm{AffSch}$.

Proposition 1.3. An affine formal scheme is a sheaf.

Proof. This is simply because filtered colimits commutes with finite limits.

$\square$

Proposition 1.4. Let $A,B$ be two admissible rings. A continuous morphism $A\to B$ gives rise to a morphism of sheaves $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$. This construction provides an isomorphism ${\mathrm{Mor}}_{\mathrm{TopRing}}(A,B)\cong {\mathrm{Mor}}_{\mathrm{Sh}(\mathrm{AffSch})}(\mathrm{Spf}(B),\mathrm{Spf}(A))$.

Proof. This can be checked easily, by evaluating on the ring $B/I$, where $I$ goes through elements of $\mathrm{Ideal}(B)$. Note that the proof makes essential use of the completeness of $B$.

$\square$

Thus $\mathrm{Spf}$ gives rise to a fully faithful embedding from the opposite category of the category of admissible rings to the category of sheaves.

2Global Story

Convention 2.1. The category of schemes admits a natural fully faithful embedding into the category of sheaves, and we abuse notation to call a sheaf a scheme if it lies in the essential image of this embedding. We call a sheaf an affine scheme if it is representable.

Definition 2.2. Let $X\to Y$ be a morphism of sheaves, we say that it is representable by schemes if for every morphism of sheaves $Z\to Y$ where $Z$ is an affine scheme (equivalently a scheme) the fiber product $Z{\times }_{Y}X$ is a scheme.

We say that $X\to Y$ is an open immersion if it is representable by schemes and for every morphism of sheaves $Z\to Y$ where $Z$ is an affine scheme (equivalently a scheme) $Z{\times }_{Y}X\to Z$ is an open immersion.

Notice that every open immersion is a monomorphism, this gives us the right to say whether a subsheaf is an open subsheaf (meaning the inclusion is an open immersion).

Let $X$ be a sheaf and $X_i,i\in I$ be a set of open subsheaves. We say that $X_i$ covers $X$ if ${\coprod }_{i\in I}{X}_i\to X$ is an epimorphism of sheaves.

Definition 2.3. We say that a sheaf is a formal scheme if it admits a cover by open subsheaves that are affine formal schemes.

2.4. Let $A$ be an admissible ring. It is easy to see that for all $I\in Ideal(A)$, the topological spaces $\mathrm{Spec}(A/I)$ are the same. It is then easy to see that the open subsheaves of $\mathrm{Spf}(A)$ corresponds bijectively to the open subsets of this topological space. Let $f\in A$ be an element, let $B={\lim }_{I\in \mathrm{Ideal}(A)}{A}_f/I$, equipped with the inverse limit topology, where we give each $A_f/I$ the discrete topology. Then $B$ is an admissible ring, and there is a natural map $A\to B$. It is easy to see that $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$ is an open immersion corresponding to the open subset $\mathrm{Spec}(A_f/I)\subseteq \mathrm{Spec}(A/I)$. Consequently one can show that an open subsheaf of an affine formal scheme is a formal scheme, and going a step forward one can show that an open subsheaf of a formal scheme is a formal scheme.

Definition 2.5. Let $P\in \{\mathrm{flat},\mathrm{faithfullyflat},\mathrm{smooth},\mathrm{etale}\}$. A morphism of formal schemes is said to have property $P$ if it is representable by schemes and all its pullbacks to affine schemes have $P$.

3Finitely Adic Formal Schemes and Adic Morphisms

The notions in this section were invented by myself, and the notations might be nonstandard.

Definition 3.1. An adic ring is an admissible ring $A$ such that for some (equivalently every) ideal of definition $I\subseteq A$ the ideals $I^n,n\in {\mathbb{Z}}_{\ge 1}$ form a fundamental system of neighbourhoods of $0$.

A finitely adic ring is an adic ring which admits a finitely generated ideal of definition.

We say that an affine formal scheme is finitely adic if the admissible ring it corresponds is finitely adic.

Definition 3.2. Let $A\to B$ be a morphism of finitely adic rings. We say it is adic if for some (equivalently every) ideal of definition $I\subseteq A$ the ideal $IB\subseteq B$ is an ideal of definition.

We say that a morphism of finitely adic affine formal schemes is adic if it corresponds to an adic morphism of finitely adic rings.

3.3. Let $A$ be a finitely adic ring. In the construction of 2.4 $B$ is also a finitely adic ring and $A\to B$ is adic. This gives us the right to define a formal scheme to be finitely adic if it admits a cover by open subsheaves that are finitely adic affine formal schemes.

Warning 3.4. A formal scheme that is affine and finitely adic need not be a finitely adic affine formal scheme in the sense of Definition 3.1. When we say a finitely adic affine formal scheme, we always mean the one in Definition 3.1.

Definition 3.5. Let $f:X\to Y$ be a morphism of finitely adic formal schemes. It is called adic if for every pair of open subsheaves $X_0\subseteq X$, $Y_0\subseteq Y$ that are finitely adic affine formal schemes such that $f{|}_{X_0}$ factors through $Y_0\subseteq Y$, the morphism $f:X_0\to Y_0$ is an adic morphism of finitely adic affine formal schemes.

Proposition 3.6. Let $f:X\to Y$ be a morphism of finitely adic formal schemes. Then $f$ is adic if and only if there exists a cover $X={\bigcup }_{i\in I}{X}_i$ of $X$ by open subsheaves that are finitely adic affine formal schemes such that for every $i\in I$ there exists an open subsheaf $Y_0$ of $Y$ that is a finitely adic affine formal scheme such that $f{|}_{X_i}$ factors through $Y_0$ and it induces an adic morphism of finitely adic affine formal schemes $X_i\to Y_0$.

Proof. The only if part is trivial and we check the if part. We may assume $X$ and $Y$ are finitely adic affine formal schemes (Of course this is nontrivial, and I’m shamefully too lazy to write it down. One has to use an argument analogous to the “affine communication lemma” in the first few steps of building algebraic geometry. To make this precise a first step is to define an underlying topological space for affine formal schemes, whose open subspaces corresponds bijectively to the open subsheaves. One then define the notion of “distinguished subspaces (or subsheaves)” as in algebraic geometry and characterize them by a universal property.) and the $X_i$’s are distinguished subsheaves. Now the problem is a simple piece of commutative algebra.

$\square$

Proposition 3.7. An adic morphism $X\to Y$ of finitely adic formal schemes is representable by schemes.

Proof. We may assume $X$ and $Y$ are finitely adic affine formal schemes since schemes glue, and the problem is then a simple commutative algebra.

$\square$

4Examples

We pause for a moment to give some examples.

Definition 4.1. Let $X$ be a finitely adic formal scheme. An $X$-adic formal scheme is a finitely adic formal scheme $Y$ equipped with an adic morphism $Y\to X$. A morphism between two $X$-adic formal schemes is automatically adic, and consequently representable by schemes. A fiber product of $X$-adic formal schemes (in the category of sheaves) is again an $X$-adic formal scheme and consequently the category of $X$-adic formal schemes admits fiber products.

When $X=\mathrm{Spf}(A)$ for some finitely adic ring $A$ we write $A$-adic formal schemes for $\mathrm{Spf}(A)$-adic formal schemes.

Example 4.2. We call a ${\mathbb{Z}}_p$-adic formal scheme a $p$-adic formal scheme, where ${\mathbb{Z}}_p$ is equipped with the $p$-adic topology.

A $\mathbb{Z}$-adic formal scheme is the same as a scheme, where $\mathbb{Z}$ is equipped with the discrete topology.

Example 4.3. Let $A\to B$ be an adic morphism of finitely adic rings, let $I\subseteq A$ be a finitely generated ideal of definition. Let $P\in \{\mathrm{flat},\mathrm{faithfullyflat},\mathrm{smooth},\mathrm{etale}\}$, let’s see what does it mean for $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$ to have property $P$.

For $n\in {\mathbb{Z}}_{\ge 1}$ we can consider the natural morphism $\mathrm{Spec}(A/I^n)\to \mathrm{Spf}(A)$ induced by the continuous morphism $A\to A/I^n$. Every morphism from an affine scheme to $\mathrm{Spf}(A)$ factors through a morphism of this form for some $n\in {\mathbb{Z}}_{\ge 1}$. We can check that $\mathrm{Spec}(B/I^{n}B)\to \mathrm{Spec}(A/I^n)$ is the pullback of $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$ along $\mathrm{Spec}(A/I^n)\to \mathrm{Spf}(A)$, thus $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$ having property $P$ is equivalent to $\mathrm{Spec}(B/I^{n}B)\to \mathrm{Spec}(A/I^n)$ having property $P$ for every $n\in {\mathbb{Z}}_{\ge 1}$.

5Ring of Global Sections of a Formal Scheme

We construct the rings of global sections of formal schemes as a functor $\Gamma :{\mathrm{FSch}}^{\mathrm{op}}\to \mathrm{CTRing}$, where $\mathrm{FSch}$ is the category of formal schemes and $\mathrm{CTRing}$ is the category of complete topological rings.

Construction 5.1. Let $\mathrm{AFSch}$ be the category of affine formal schemes, $\mathrm{AdmRing}$ be the category of admissible rings, then the functor $\mathrm{Spf}:{\mathrm{AdmRing}}^{\mathrm{op}}\to \mathrm{AFSch}$ is an equivalence, we let $\Gamma :\mathrm{AFSch}\to \mathrm{CTRing}$ be induced by its inverse.

Construction 5.2. Let $X$ be a formal scheme, let $\mathrm{AffOpen}(X)$ denote the partially ordered set whose elements are open subsheaves of $X$ that are affine formal schemes, and $Y\le Z$ if and only if $Y\subseteq Z$.

We define $\Gamma (X):={\lim }_{Y\in \mathrm{AffOpen}(X{)}^{\mathrm{op}}}\Gamma (Y)$. This is a complete topological ring as it is a limit of complete topological rings. It is easy to see that this construction is functorial in $X$ and thus we obtain the functor $\Gamma :{\mathrm{FSch}}^{\mathrm{op}}\to \mathrm{CTRing}$ which coincides with Construction 5.1 on affine formal schemes.

Sheafiness of $\Gamma (X)$

I’m too lazy to give the precise statement and argument so I’ll just give a sketch.

5.3. Let $X$ be a finitely adic formal scheme, then we can define the big $\mathrm{fpqc}$-site over $X$, whose underlying category is the category of $X$-adic formal schemes, and the coverings are the $\mathrm{fpqc}$-coverings (that is, a family of maps with common targets, each of which is $\mathrm{fpqc}$, and becomes an $\mathrm{fpqc}$-covering after pulling back to any affine scheme). $Y\to \Gamma (Y)$ gives a presheaf of rings on this site (we forget about the topological structure on $\Gamma (Y)$), and our goal is to prove that this is a sheaf.

We first prove the sheaf axioms for an affine open cover of $X$, which follows from the case $X$ is affine and the covering is a covering by distinguished affine opens, which can be computed directly.

The general case is now reduced to the case where every formal scheme appearing in the covering is affine, which follows from the case of schemes. The trick is to do things at the discrete level (if $A$ is the finitely adic ring at the base, and $I$ is a finitely generated ideal of definition, discrete level means working over $A/I^n$ for $n\in {\mathbb{Z}}_{\ge n}$), and then we can “glue” things together since the rings are complete.

References

[1]	Michael Rapoport, Thomas Zink, “Period spaces for p-divisible groups”, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996.
[2]	Peter Scholze, Jared Weinstein, “Moduli of p-divisible groups”, Cambridge Journal of Mathematics 1 (2013), 145–237.
[3]	The Stacks project authors, “The Stacks project”, https://stacks.math.columbia.edu