Éléments de Géométrie Algébrique — English

§8. Projective limits of preschemes

8.1. Introduction

(8.1.1) In this section we shall systematically study the following situation. Let $I$ be a filtered (increasing) preordered set, $(A_{\alpha },{\varphi }_{\beta \alpha })$ an inductive system of rings indexed by $I$, and $A=\lim A_{\alpha }$ its inductive limit. For every $\alpha \in I$ and every $A_{\alpha }$-prescheme $X_{\alpha }$, consider the $A_{\lambda }$-preschemes $X_{\lambda }=X_{\alpha }{\otimes }_{A_{\alpha }}{A}_{\lambda }$ for $\lambda \ge \alpha$, and the $A$-prescheme $X=X_{\alpha }{\otimes }_{A_{\alpha }}A$; it is clear that the preschemes $X_{\lambda }$ (for $\lambda \ge \alpha$) form a projective system, and one will see (8.2.5) that $X$ is a projective limit of this system in the category of preschemes. We propose to find conditions on $X_{\alpha }$ or on the $A_{\lambda }$ allowing us to prove properties of the following type: in order that $X$ possess a property $P$ (for example, the property of being proper over $S=\mathrm{Spec}(A)$, or irreducible, or connected, etc.), it is necessary and sufficient that there exist an index $\mu \ge \alpha$ such that, for every $\lambda \ge \mu$, $X_{\lambda }$ has (with respect to $S_{\lambda }=\mathrm{Spec}(A_{\lambda })$, if applicable) the same property $P$. We shall obtain analogous statements for properties of $\mathcal{O}$-Modules, of $A$-morphisms of $A$-preschemes, etc. We shall also show (8.9.1) that giving an $A$-prescheme of finite presentation (1.6.1) is essentially equivalent to giving an $A_{\lambda }$-prescheme of finite presentation $X_{\lambda }$ for $\lambda$ large enough, $X$ being then isomorphic to $X_{\lambda }{\otimes }_{A_{\lambda }}A$. One has analogous statements for ${\mathcal{O}}_X$-Modules of finite presentation, their homomorphisms, the $A$-morphisms of $A$-preschemes of finite presentation, etc.

(8.1.2) The utility of such results will appear, for example, in the following questions:

a) Let $Y$ be a prescheme, $y$ a point of $Y$, $(U_{\alpha })$ the filtered (decreasing) projective system of affine open neighbourhoods of $y$ in $Y$; if $A_{\alpha }$ is the ring of $U_{\alpha }$, the $A_{\alpha }$ form a filtered (increasing) inductive system whose inductive limit $A$ is the local ring ${\mathcal{O}}_y$. Moreover, if one denotes by ${\mathfrak{p}}_{\alpha }$ the prime ideal of $y$ in the ring $A_{\alpha }$, the inductive system $(A_{\alpha })$ is cofinal with every inductive system $(A_{\alpha ,f})$, where $f$ runs through $A_{\alpha }-{\mathfrak{p}}_{\alpha }$ (for a fixed $\alpha$), since the $D(f)$ form a fundamental system of neighbourhoods of $y$ in $U_{\alpha }$, hence in $Y$. The results of the present section will imply that the algebraic geometry of ${\mathcal{O}}_y$-preschemes of finite presentation (and the theory of Modules of finite presentation on these preschemes) is essentially equivalent to the algebraic geometry of preschemes of finite presentation on "sufficiently small" open neighbourhoods of $y$. Thus, the statement (8.10.5, (xiii)) implies that if a morphism $f:X\to Y$ is of finite presentation, then, in order that $X{\times }_Y\mathrm{Spec}({\mathcal{O}}_y)$ be proper over $\mathrm{Spec}({\mathcal{O}}_y)$, it is necessary and sufficient that there exist an open neighbourhood $U$ of $y$ in $Y$ such that $f^{-1}(U)$ be proper over $U$.

A particularly important case, and to a certain extent classical, is that in which $Y$ is integral and $y$ is its generic point, so that ${\mathcal{O}}_y$ is none other than the field $R(Y)=K$ of rational functions on $Y$. The results of the present section then amount to interpreting the algebraic geometry over $K$ in terms of the algebraic geometry above non-empty "sufficiently small" open sets of $Y$, that is to say, intuitively, in terms of "families" of geometric objects indexed by the points of such an open set. This point of view has moreover been commonly used for a long time, not only in algebraic geometry over algebraically closed fields, but also in the arithmetic study of varieties defined over a number field $K$ (finite extension of $\mathbb{Q}$), by considering this latter as the field of fractions of its ring of integers $A$ ("theory of reduction modulo $\mathfrak{p}$", $\mathfrak{p}$ a prime ideal of $A$; cf. (I, 3.7)). The results of §§8 and 9 thus furnish among other things foundations of the language of "reduction modulo $\mathfrak{p}$" in arithmetic.

One will note that in the example envisaged here, the morphisms $S_{\mu }\to S_{\lambda }$ (for $\lambda \le \mu$) are the canonical open immersions $U_{\mu }\to U_{\lambda }$, and a fortiori are flat morphisms (but not faithfully flat in general), which explains the interest of statements that appeal to such a restriction.

b) Suppose that the $A_{\lambda }$ are fields, so that $A=\lim A_{\lambda }$ is also a field. This case generally arises when one starts from geometric data above an arbitrary field $K$, which one considers as an extension of a field $k$ (for example the prime subfield of $K$). It is then advantageous to consider $K$ as the inductive limit of its sub-extensions that are of finite type over $k$, which permits in many questions to reduce to the case where $K$ is an extension of finite type of $k$. Using also the method sketched in a), one can then generally reduce to the case of a base ring $A$ that is an integral algebra of finite type over $k$.

One will note that in this example, the morphisms $S_{\mu }\to S_{\lambda }$ are faithfully flat.

c) Suppose one is interested in the properties, local on $Y$, of preschemes of finite presentation above an arbitrary prescheme $Y$, which one may therefore assume affine with ring $A$. It is then advantageous to consider $A$ as the inductive limit of its sub-rings that are $\mathbb{Z}$-algebras of finite type, which permits to reduce many questions to the case where $Y$ is the spectrum of such an algebra. This is the explicit form of the "Kroneckerian point of view", according to which algebraic geometry reduces to the algebraic geometry of preschemes of finite type over $\mathbb{Z}$ (which is sometimes called "absolute algebraic geometry"). This example shows us in particular that in most "relative" questions over a base prescheme $Y$, one can reduce to the case where $Y$ is Noetherian.

One will note that in this example, contrary to the preceding ones, the morphisms $S_{\mu }\to S_{\lambda }$ have in general no particular regularity property.

In what follows, when we apply the results that follow to any one of the three particular situations just described, we shall dispense with redescribing in detail the procedure that permits these applications, contenting ourselves with referring back to the foregoing.

(8.1.3) In example a) of (8.1.2), we saw that if $Y$ is an integral prescheme with generic point $y$, and $f:X\to Y$ a morphism of finite presentation, then, if the generic fibre $f^{-1}(y)$ is proper over $k(y)$, there is an open neighbourhood $U$ of $y$ such that $f^{-1}(U)$ is proper over $U$; a fortiori, for every $s\in U$, $f^{-1}(s)$ is proper over $k(s)$. There are occasions when one needs a converse, asserting that if $f^{-1}(s)$ is proper over $k(s)$ for "sufficiently many" points $s\in U$, then $f^{-1}(y)$ is proper over $k(y)$. For example, suppose that $X$ and $Y$ are algebraic preschemes over an algebraically closed field $k$ (one can take for $k$ the field $\mathbb{C}$ of complex numbers, to fix the ideas); one sometimes needs to know that if, for every $s\in Y$ rational over $k$, the fibre $f^{-1}(s)$ is proper over $k(s)$, then $f^{-1}(y)$ is proper over $k(y)$, and consequently $f^{-1}(U)$ is proper over $U$ for some neighbourhood $U$ of $y$ (¹). Now this statement will follow easily from the following: the set $E$ of points $s\in Y$ such that $f^{-1}(s)$ is proper over $k(s)$ is constructible (and consequently identical to all of $Y$ if it contains the closed points of $Y$, thanks to Hilbert's Nullstellensatz (10.4.8)); this also amounts to saying that if $f^{-1}(y)$ is not proper over $k(y)$, then there exists an open neighbourhood $U$ of $y$ such that $f^{-1}(s)$ is not proper over $k(s)$ for every $s\in U$ (cf. (9.6.1, (iv))). This example illustrates the interest of systematically developing constructibility criteria for the most important notions: this is what will be done in §9.

(¹) One will note that such a statement is in the end purely geometric, in the sense that it only appeals to points rational over $k$, and not to generic points; for example, when $k=\mathbb{C}$, this statement has an obvious topological meaning for the analyst, when one interprets "proper" in the topological sense of the term, for the spaces underlying the analytic spaces formed by the points of $X$ and $Y$ rational over $\mathbb{C}$.

8.2. Projective limits of preschemes

(8.2.1) Let S_0 be a ringed space, $L$ a filtered (increasing) preordered set, $({\mathcal{A}}_{\lambda },{\varphi }_{\mu \lambda })$ an inductive system of ${\mathcal{O}}_{S_0}$-Algebras (not necessarily commutative) indexed by $L$. One knows that, considered as an inductive system of ${\mathcal{O}}_{S_0}$-Modules, $({\mathcal{A}}_{\lambda },{\varphi }_{\mu \lambda })$ admits an inductive limit $\mathcal{A}$; let us denote by ${\varphi }_{\lambda }:{\mathcal{A}}_{\lambda }\to \mathcal{A}$ the canonical homomorphism (of ${\mathcal{O}}_{S_0}$-Modules). Let $m_{\lambda }:{\mathcal{A}}_{\lambda }\otimes {\mathcal{A}}_{\lambda }\to {\mathcal{A}}_{\lambda }$ be the homomorphism of ${\mathcal{O}}_{S_0}$-Modules that defines the multiplication in the ${\mathcal{O}}_{S_0}$-Algebra ${\mathcal{A}}_{\lambda }$; the hypothesis on the ${\varphi }_{\mu \lambda }$ entails that the $m_{\lambda }$ form an inductive system of homomorphisms, and since the functor lim commutes with tensor product, $m=\lim m_{\lambda }$ is a homomorphism $\mathcal{A}\otimes \mathcal{A}\to \mathcal{A}$ of ${\mathcal{O}}_{S_0}$-Modules; by passage to the limit on the commutative diagrams expressing the associativity of $m_{\lambda }$ and the existence of a unit section in ${\mathcal{A}}_{\lambda }$, one sees that $m$ defines on $\mathcal{A}$ a structure of ${\mathcal{O}}_{S_0}$-Algebra and that ${\varphi }_{\lambda }$ is a homomorphism of ${\mathcal{O}}_{S_0}$-Algebras for every $\lambda \in L$. Moreover $\mathcal{A}$ is the inductive limit of the system $({\mathcal{A}}_{\lambda },{\varphi }_{\mu \lambda })$ in the category of ${\mathcal{O}}_{S_0}$-Algebras; in other words, for every ${\mathcal{O}}_{S_0}$-Algebra $\mathcal{B}$, the canonical map

  (8.2.1.1)    Hom_{S_0-Alg.}(𝒜, ℬ) → lim Hom_{S_0-Alg.}(𝒜_λ, ℬ)

which to every homomorphism $f:\mathcal{A}\to \mathcal{B}$ of ${\mathcal{O}}_{S_0}$-Algebras associates the family $(f\circ {\varphi }_{\lambda })$, is bijective. Indeed, one already knows that it is injective and identifies ${\mathrm{Hom}}_{S_0-Alg.}(\mathcal{A},\mathcal{B})$ with a part of lim Hom_{S_0-Mod.}(𝒜_λ, ℬ); everything comes down to seeing that if $(f_{\lambda })$ is an inductive system of homomorphisms of ${\mathcal{O}}_{S_0}$-Algebras, $f_{\lambda }:{\mathcal{A}}_{\lambda }\to \mathcal{B}$, its inductive limit $f:\mathcal{A}\to \mathcal{B}$, which by definition is a homomorphism of ${\mathcal{O}}_{S_0}$-Modules, is also a homomorphism of ${\mathcal{O}}_{S_0}$-Algebras; but this results from passage to the inductive limit in the commutative diagram of homomorphisms of ${\mathcal{O}}_{S_0}$-Modules expressing that the $f_{\lambda }$ are Algebra homomorphisms, and from the fact that the functor lim commutes with tensor products.

One will note finally that if the ${\mathcal{A}}_{\lambda }$ are commutative ${\mathcal{O}}_{S_0}$-Algebras, the same is true of $\mathcal{A}$.

(8.2.2) Suppose now that S_0 is a prescheme and that the ${\mathcal{A}}_{\lambda }$ are quasi-coherent (commutative) ${\mathcal{O}}_{S_0}$-Algebras; one knows then that $\mathcal{A}=\lim {\mathcal{A}}_{\lambda }$ is a quasi-coherent ${\mathcal{O}}_{S_0}$-Algebra (I, 4.1.1). Let us denote by $S_{\lambda }$ (resp. $S$) the spectrum of the ${\mathcal{O}}_{S_0}$-Algebra ${\mathcal{A}}_{\lambda }$ (resp. $\mathcal{A}$) (II, 1.3.1), and let $u_{\mu \lambda }:S_{\mu }\to S_{\lambda }$ (for $\lambda \le \mu$) and $u_{\lambda }:S\to S_{\lambda }$ be the S_0-morphisms corresponding to the homomorphisms ${\varphi }_{\mu \lambda }$ and ${\varphi }_{\lambda }$ respectively (II, 1.2.7); it is clear that $(S_{\lambda },u_{\mu \lambda })$ is a projective system in the category of S_0-preschemes. One will note that the $u_{\mu \lambda }$ and $u_{\lambda }$ are affine morphisms (II, 1.6.2), hence quasi-compact and separated.

Proposition (8.2.3).

With the notations of (8.2.2), the morphisms $u_{\lambda }:S\to S_{\lambda }$ make $S$ a projective limit of the projective system $(S_{\lambda },u_{\mu \lambda })$ in the category of preschemes. Moreover, if $h:S_0\to T$ is a morphism, making every S_0-prescheme a $T$-prescheme, $S$ is also a projective limit of the system $(S_{\lambda },u_{\mu \lambda })$ in the category of $T$-preschemes.

Let us first prove the second assertion of the statement in the case $T=S_0$.

Everything comes down to showing that if $X$ is an arbitrary S_0-prescheme, the canonical map

  (8.2.3.1)    Hom_{S_0}(X, S) → lim Hom_{S_0}(X, S_λ)

which to every S_0-morphism $v:X\to S$ associates the family $(u_{\lambda }\circ v)$, is bijective. Now, if $g:X\to S_0$ is the structure morphism and if one sets $\mathcal{B}=g_{\ast }({\mathcal{O}}_X)$, which is an ${\mathcal{O}}_{S_0}$-Algebra, the map (8.2.3.1) is canonically identified with (8.2.1.1) (II, 1.2.7), and the conclusion therefore results from what was seen in (8.2.1).

The other assertions of (8.2.3) are consequences of the following general lemma:

Lemma (8.2.4).

Let $\mathcal{C}$ be a category, $T$ an object of $\mathcal{C}$, ${\mathcal{C}}_T$ the subcategory of objects of $\mathcal{C}$ above $T$. Let $(S_{\lambda },u_{\mu \lambda })$ be a projective system in ${\mathcal{C}}_T$; then every projective limit of this system in ${\mathcal{C}}_T$ is also a projective limit in $\mathcal{C}$, and conversely.

Let $f_{\lambda }:S_{\lambda }\to T$ be the structure morphism. Suppose that $S$ is a projective limit of $(S_{\lambda },u_{\mu \lambda })$ in $\mathcal{C}$, and denote by $u_{\lambda }:S\to S_{\lambda }$ the corresponding canonical morphisms. Consider then a projective system of $T$-morphisms $w_{\lambda }:Y\to S_{\lambda }$, where $Y\in {\mathcal{C}}_T$. There exists by hypothesis a unique morphism $w:Y\to S$ (in $\mathcal{C}$) such that $w_{\lambda }=u_{\lambda }\circ w$ for every $\lambda$. The hypothesis that the $u_{\lambda }$ are $T$-morphisms entails that the morphisms $f_{\lambda }\circ u_{\lambda }:S\to T$ are all equal, and this morphism $f$ therefore makes $S$ a $T$-object. If $g:Y\to T$ is the structure morphism of $Y$, one has then $f\circ w=f_{\lambda }\circ u_{\lambda }\circ w=f_{\lambda }\circ w_{\lambda }=g$ for every $\lambda$, which proves that $w$ is a $T$-morphism. Conversely, suppose (with the same notations) that $S$ is a projective limit of $(S_{\lambda },u_{\mu \lambda })$ in ${\mathcal{C}}_T$, and consider now a projective system of morphisms (of $\mathcal{C}$) $w_{\lambda }:Y\to S_{\lambda }$. The composite morphisms $f_{\lambda }\circ w_{\lambda }:Y\to T$ are then all equal: indeed, for any two indices $\lambda$, $\mu$, there is an index $\nu$ such that $\lambda \le \nu$ and $\mu \le \nu$, whence $f_{\nu }=f_{\lambda }\circ u_{\lambda \nu }=f_{\mu }\circ u_{\mu \nu }$; since $w_{\lambda }=u_{\lambda \nu }\circ w_{\nu }$ and $w_{\mu }=u_{\mu \nu }\circ w_{\nu }$, one has $f_{\lambda }\circ w_{\lambda }=f_{\lambda }\circ u_{\lambda \nu }\circ w_{\nu }=f_{\nu }\circ w_{\nu }$ and one sees in the same way that $f_{\mu }\circ w_{\mu }=f_{\nu }\circ w_{\nu }$. If $g:Y\to T$ is the unique morphism thus defined, $g$ makes $Y$ a $T$-object, and the $w_{\lambda }$ are then $T$-morphisms; they consequently have a projective limit $w:Y\to S$ which is a $T$-morphism, and a fortiori a morphism of $\mathcal{C}$; moreover the first part of the reasoning shows that every projective limit $w^{\prime }$ (in $\mathcal{C}$) of the projective system $(w_{\lambda })$ is necessarily also a $T$-morphism, hence equal to $w$, which completes the proof of the lemma.

Proposition (8.2.5).

Under the conditions of (8.2.2), let $\alpha$ be an element of $L$, $X_{\alpha }$ an $S_{\alpha }$-prescheme. For every $\lambda \ge \alpha$, set $X_{\lambda }=X_{\alpha }{\times }_{S_{\alpha }}{S}_{\lambda }$, and for $\alpha \le \lambda \le \mu$, set $v_{\mu \lambda }=1_{X_{\alpha }}\times u_{\mu \lambda }$, so that $(X_{\lambda },v_{\mu \lambda })$ is a projective system of $X_{\alpha }$-preschemes, whose index set is formed of the $\lambda \ge \alpha$ in $L$. Set likewise $X=X_{\alpha }{\times }_{S_{\alpha }}S$ and $v_{\lambda }=1_{X_{\alpha }}\times u_{\lambda }$. Then the $X_{\alpha }$-morphisms $v_{\lambda }:X\to X_{\lambda }$ make $X$ a projective limit of the projective system $(X_{\lambda },v_{\mu \lambda })$ in the category of $X_{\alpha }$-preschemes, or in the category of all preschemes.

This will again result from the following general lemma:

Lemma (8.2.6).

Let $\mathcal{C}$ be a category in which fibre products exist, $q:T^{\prime }\to T$ a morphism of $\mathcal{C}$, ${\mathcal{C}}_T$ (resp. ${\mathcal{C}}_{T^{\prime }}$) the category of objects of $\mathcal{C}$ above $T$ (resp. $T^{\prime }$).

Let $(S_{\lambda },u_{\mu \lambda })$ be a projective system (not necessarily filtered) in ${\mathcal{C}}_T$, and set $S_{\lambda }^{\prime }=S_{\lambda }{\times }_T{T}^{\prime }$, $u_{\mu \lambda }^{\prime }=u_{\mu \lambda }\times 1_{T^{\prime }}$, so that $(S_{\lambda }^{\prime },u_{\mu \lambda }^{\prime })$ is a projective system in ${\mathcal{C}}_{T^{\prime }}$. Then, if $(S,u_{\lambda })$ is a projective limit of $(S_{\lambda },u_{\mu \lambda })$ in ${\mathcal{C}}_T$, $(S{\times }_T{T}^{\prime },u_{\lambda }\times 1_{T^{\prime }})$ is a projective limit of $(S_{\lambda }^{\prime },u_{\mu \lambda }^{\prime })$ in ${\mathcal{C}}_{T^{\prime }}$.

One has by hypothesis, for every $\lambda$, a commutative diagram

  S'  ──u'_λ──→  S'_λ  ──h'_λ──→  T'
   │              │                │
   p│            p_λ│               q
   ↓              ↓                ↓
   S   ──u_λ───→  S_λ  ───f_λ───→  T

where one has set $S^{\prime }=S{\times }_T{T}^{\prime }$, $u_{\lambda }^{\prime }=u_{\lambda }\times 1_{T^{\prime }}$, $h_{\lambda }^{\prime }=f_{\lambda }\times 1_{T^{\prime }}$. Let $Y$ be a $T^{\prime }$-object, $g^{\prime }:Y\to T^{\prime }$ the corresponding morphism, and consider a projective system of $T^{\prime }$-morphisms $w_{\lambda }^{\prime }:Y\to S_{\lambda }^{\prime }$. Then $Y$ is a $T$-object via the morphism $g=q\circ g^{\prime }$, and the $w_{\lambda }=p_{\lambda }\circ w_{\lambda }^{\prime }$ are $T$-morphisms, since $h_{\lambda }\circ w_{\lambda }=h_{\lambda }\circ p_{\lambda }\circ w_{\lambda }^{\prime }=q\circ h_{\lambda }^{\prime }\circ w_{\lambda }^{\prime }=q\circ g^{\prime }$ by hypothesis. Moreover, one verifies at once that $(w_{\lambda })$ is a projective system. There exists therefore by hypothesis a unique $T$-morphism $w:Y\to S$ such that $u_{\lambda }\circ w=w_{\lambda }$ for every $\lambda$. By definition of the fibre product, there is a unique $T^{\prime }$-morphism $w^{\prime }:Y\to S^{\prime }$ such that $p\circ w^{\prime }=w$. One has then $u_{\lambda }\circ p\circ w^{\prime }=u_{\lambda }\circ w=w_{\lambda }=p_{\lambda }\circ w_{\lambda }^{\prime }$, which can also be written $p_{\lambda }\circ u_{\lambda }^{\prime }\circ w^{\prime }=p_{\lambda }\circ w_{\lambda }^{\prime }$; on the other hand, by writing that $w^{\prime }$ is a $T^{\prime }$-morphism, one gets $h_{\lambda }^{\prime }\circ u_{\lambda }^{\prime }\circ w^{\prime }=g^{\prime }=h_{\lambda }^{\prime }\circ w_{\lambda }^{\prime }$. The definition of $S_{\lambda }^{\prime }$ as fibre product $S_{\lambda }{\times }_T{T}^{\prime }$ thus gives $u_{\lambda }^{\prime }\circ w^{\prime }=w_{\lambda }^{\prime }$, and it is immediate that $w^{\prime }$ is the unique $T^{\prime }$-morphism verifying these relations, whence the lemma.

Remark (8.2.7).

Given an arbitrary ringed space $S$, the inductive limits with respect to an arbitrary preordered set $L$ (not necessarily filtered) exist in the category of commutative ${\mathcal{O}}_S$-Algebras, since the filtered inductive limit exists by (8.2.1) and on the other hand, for two homomorphisms of ${\mathcal{O}}_S$-Algebras $\mathcal{A}\to \mathcal{B}$, $\mathcal{A}\to \mathcal{C}$, the tensor product $\mathcal{B}{\otimes }_{\mathcal{A}}\mathcal{C}$ is the corresponding "amalgamated sum" in this category. When $S$ is a prescheme, one knows that the tensor product $\mathcal{B}{\otimes }_{\mathcal{A}}\mathcal{C}$ is a quasi-coherent ${\mathcal{O}}_S$-Algebra when this is so of $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ (I, 1.3.13); one concludes that, in the category of quasi-coherent ${\mathcal{O}}_S$-Algebras, the inductive limits for an arbitrary preordered index set always exist. This permits one to generalize the definition of a projective limit of preschemes and Propositions (8.2.3) and (8.2.5) to the case where the preordered set $L$ is not necessarily filtered.

(8.2.8) With the notations of (8.2.2), set $u_{\mu \lambda }=({\psi }_{\mu \lambda },{\theta }_{\mu \lambda })$ and $u_{\lambda }=({\psi }_{\lambda },{\theta }_{\lambda })$; thus $(S_{\lambda },{\psi }_{\mu \lambda })$ is a projective system of topological spaces and $({\psi }_{\lambda })$ an inductive system of continuous maps $S\to S_{\lambda }$ of the spaces underlying the preschemes $S$ and $S_{\lambda }$ respectively.

Proposition (8.2.9).

With the notations of (8.2.8), the projective limit of the projective system $({\psi }_{\lambda })$ of continuous maps is a homeomorphism of the space underlying $S$ onto the projective limit of the projective system $(S_{\lambda },{\psi }_{\mu \lambda })$ of topological spaces.

Let $T$ be the topological space limit of the system $(S_{\lambda },{\psi }_{\mu \lambda })$ and set $\psi =\lim {\psi }_{\lambda }:S\to T$. One may restrict to the case where $S_0=S_{\alpha }$ for some $\alpha \in L$, and $\lambda \ge \alpha$.

(i) Let us show first that the topology of $S$ is the inverse image under $\psi$ of the topology of $T$; in other words, if ${\pi }_{\lambda }:T\to S_{\lambda }$ is the canonical map, one must show that every open set of $S$ is a union of open sets of the form ${\psi }^{-1}({\pi }_{\lambda }^{-1}(U_{\lambda }))$, where $U_{\lambda }$ is open in $S_{\lambda }$. Now every open set of $S$ is by definition a union of open sets obtained as follows: one considers an affine open set U_0 of S_0, with ring A_0, so that $u_0^{-1}(U_0)$ is the affine open set of $S$ with ring $A=\Gamma (U_0,\mathcal{A})$, then one takes an element $f\in A$ and one considers in $\mathrm{Spec}(A)$, identified with $u_0^{-1}(U_0)$, the open set $D(f)$. It is these open sets $D(f)$ that form a base of the topology of $S$ (II, 1.3.1). Now, if one sets $A_{\lambda }=\Gamma (U_0,{\mathcal{A}}_{\lambda })$, one has $A=\lim A_{\lambda }$ (I, 1.3.9), so there exists an index $\lambda$ such that $f$ is the canonical image of an element $f_{\lambda }\in A_{\lambda }$; one has then $D(f)={\psi }_{\lambda }^{-1}(D(f_{\lambda }))$ (I, 1.2.2), and since ${\psi }_{\lambda }={\pi }_{\lambda }\circ \psi$, our assertion is proved.

(ii) Let us now prove that $\psi$ is bijective, which will complete the proof. Since $S$ is a Kolmogorov space, it already follows from (i) that $\psi$ is injective, and it therefore remains to show that $\psi$ is surjective. One can evidently replace $T$ for this purpose by an open set ${\pi }_0^{-1}(U_0)$, where U_0 is an affine open set in $S_{\alpha }=S_0$, so one can limit oneself to the case where the $S_{\lambda }$ and $S$ are affine, in other words ${\mathcal{A}}_{\lambda }$ is the sheaf associated with an A_0-algebra $A_{\lambda }$, and $\mathcal{A}$ the sheaf of algebras associated with $A=\lim A_{\lambda }$; we shall again denote by ${\varphi }_{\mu \lambda }:A_{\lambda }\to A_{\mu }$ and ${\varphi }_{\lambda }:A_{\lambda }\to A$ the canonical homomorphisms. By definition, an element of $T$ is a family $({\mathfrak{p}}_{\lambda }{)}_{\lambda \in L}$, where ${\mathfrak{p}}_{\lambda }$ is a prime ideal of $A_{\lambda }$ and where one has ${\mathfrak{p}}_{\lambda }={\varphi }_{\mu \lambda }^{-1}({\mathfrak{p}}_{\mu })$ for $\lambda \le \mu$. One knows then ((5.13.3) and (5.13.1)) that there is a prime ideal $\mathfrak{p}$ of $A$ such that ${\mathfrak{p}}_{\lambda }={\varphi }_{\lambda }^{-1}(\mathfrak{p})$ for every $\lambda \in L$, which completes the proof.

In particular, we have thus proved the

Corollary (8.2.10).

Let $(A_{\lambda }{)}_{\lambda \in L}$ be a filtered inductive system of rings, and let $A=\lim A_{\lambda }$, ${\varphi }_{\lambda }:A_{\lambda }\to A$ the canonical homomorphisms. The canonical map $\mathfrak{p}\mapsto ({\varphi }_{\lambda }^{-1}(\mathfrak{p}))$ is a homeomorphism of $\mathrm{Spec}(A)$ onto the topological space $\lim \mathrm{Spec}(A_{\lambda })$.

Corollary (8.2.11).

With the notations of (8.2.8), for every quasi-compact open set $U$ of $S$, there exist an index $\lambda$ and a quasi-compact open set $U_{\lambda }$ of $S_{\lambda }$ such that $U={\psi }_{\lambda }^{-1}(U_{\lambda })$.

This results from the fact that, by definition of the projective limit topology, the ${\psi }_{\lambda }^{-1}(U_{\lambda })$ ($U_{\lambda }$ quasi-compact open in $S_{\lambda }$) form a base of the topology of $S$, and from the fact that the index set $L$ is filtered.

Corollary (8.2.12).

With the notations of (8.2.8), the inductive limit of the inductive system of homomorphisms ${\theta }_{\lambda }^{♯}:{\psi }_{\lambda }^{\ast }({\mathcal{O}}_{S_{\lambda }})\to {\mathcal{O}}_S$ of sheaves of rings on $S$ is an isomorphism

$$(\mathrm{8.2.12.1})\lim {\psi }_{\lambda }^{\ast }({\mathcal{O}}_{S_{\lambda }})\stackrel{\sim }{\to }{\mathcal{O}}_S.$$

One can evidently suppose the $S_{\lambda }$ affine; with the notations of the proof of (8.2.9), everything comes down to seeing that the inductive limit of the system of canonical maps $(A_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}\to A_{\mathfrak{p}}$ is an isomorphism, which is none other than (5.13.3, (ii)).

Proposition (8.2.13).

Suppose that the morphisms $u_{\mu \lambda }:S_{\mu }\to S_{\lambda }$ are open immersions, so that $S_{\mu }$ is identified with the sub-prescheme induced on an open set of $S_{\lambda }$ for $\lambda \le \mu$. Then, for every $\alpha \in L$, the space underlying the prescheme $S$ is identified with the subspace of $S_{\alpha }$ intersection of the $S_{\lambda }$ for $\lambda \ge \alpha$, and the structure sheaf ${\mathcal{O}}_S$ with the sheaf induced (G, II, 1.5) by ${\mathcal{O}}_{S_{\alpha }}$ on this intersection; more generally, for every ${\mathcal{O}}_{S_{\alpha }}$-Module ${\mathcal{F}}_{\alpha }$, $u_{\alpha }^{\ast }({\mathcal{F}}_{\alpha })$ is identified with the ${\mathcal{O}}_S$-Module induced by ${\mathcal{F}}_{\alpha }$ on $S$.

The first assertion results from (8.2.9), in view of the definition of a projective limit of topological spaces; in addition all the ${\psi }_{\lambda }^{\ast }({\mathcal{O}}_{S_{\lambda }})$ are equal to the sheaf induced by ${\mathcal{O}}_{S_{\alpha }}$ on $S$ by definition $(0_I,\mathrm{3.7.1})$ and, with the notations of the proof of (8.2.9), the homomorphisms $(A_{\lambda }{)}_{{\mathfrak{p}}_{\lambda }}\to (A_{\alpha }{)}_{\mathfrak{p}}$ are bijective for a system $({\mathfrak{p}}_{\lambda })$ of prime ideals corresponding to a single point of $S$; the assertion relative to ${\mathcal{O}}_S$ therefore follows from (8.2.12). The last assertion results then from the definition of $u_{\alpha }^{\ast }({\mathcal{F}}_{\alpha })$ $(0_I,\mathrm{4.3.1})$.

Remark (8.2.14).

The results of (8.2.9) and (8.2.12) show that $S$ is the projective limit of the projective system $(S_{\lambda },u_{\mu \lambda })$ in the category of all ringed spaces (or of all ringed spaces in local rings). Indeed, let $Y$ be a ringed space, and consider a projective system of morphisms of ringed spaces $w_{\lambda }:Y\to S_{\lambda }$. If one sets $w_{\lambda }=(p_{\lambda },{\omega }_{\lambda })$, the $p_{\lambda }$ form a projective system of continuous maps and, by virtue of (8.2.9), their projective limit $p$ is identified with a continuous map $Y\to S$ such that $p_{\lambda }={\psi }_{\lambda }\circ p$. On the other hand, the ${\omega }_{\lambda }^{♯}:p_{\lambda }^{\ast }({\mathcal{O}}_{S_{\lambda }})\to {\mathcal{O}}_Y$ form an inductive system of homomorphisms of sheaves of rings; since one may write $p_{\lambda }^{\ast }({\mathcal{O}}_{S_{\lambda }})=p^{\ast }({\psi }_{\lambda }^{\ast }({\mathcal{O}}_{S_{\lambda }}))$ and the functor $p^{\ast }$ is exact, the inductive limit of the $p_{\lambda }^{\ast }({\mathcal{O}}_{S_{\lambda }})$ is $p^{\ast }({\mathcal{O}}_S)$ by virtue of (8.2.12), and there is therefore a unique homomorphism ${\omega }^{♯}:p^{\ast }({\mathcal{O}}_S)\to {\mathcal{O}}_Y$ such that ${\omega }_{\lambda }^{♯}={\omega }^{♯}\circ p^{\ast }({\theta }_{\lambda }^{♯})$, which proves our assertion.

8.3. Constructible parts in a projective limit of preschemes

(8.3.1) In all that follows in this section, we suppose the conditions of (8.2.2) to be satisfied, and we preserve its notations.

Theorem (8.3.2).

For every $\lambda$, let $E_{\lambda }$, $F_{\lambda }$ be two parts of $S_{\lambda }$. Set

  (8.3.2.1)    E = ⋂_λ u_λ⁻¹(E_λ),    F = ⋃_λ u_λ⁻¹(F_λ).

Assume the following conditions:

(i) For every $\lambda$, $E_{\lambda }$ is pro-constructible and $F_{\lambda }$ is ind-constructible (1.9.4).

(ii) For $\lambda \le \mu$, one has $u_{\mu \lambda }^{-1}(E_{\lambda })\supset E_{\mu }$ and $u_{\mu \lambda }^{-1}(F_{\lambda })\subset F_{\mu }$.

(iii) There exists $\alpha \in L$ such that $S_{\alpha }$ is quasi-compact (which entails that $S_{\lambda }$ is quasi-compact for $\lambda \ge \alpha$).

Then the following properties are equivalent:

a) $E\subset F$.

b) There exists $\lambda \ge \alpha$ such that $u_{\lambda }^{-1}(E_{\lambda })\subset u_{\lambda }^{-1}(F_{\lambda })$ (and one then has $u_{\mu }^{-1}(E_{\mu })\subset u_{\mu }^{-1}(F_{\mu })$ for $\mu \ge \lambda$).

c) There exists $\lambda \ge \alpha$ such that $E_{\lambda }\subset F_{\lambda }$ (and one then has $E_{\mu }\subset F_{\mu }$ for $\mu \ge \lambda$).

The remarks in parentheses in b) and c) result from (ii). Set

  G_λ = E_λ ∩ (S_λ − F_λ),    G = E ∩ (S − F).

Then $G_{\lambda }$ is a pro-constructible part of $S_{\lambda }$ (1.9.5, (i)), and by virtue of (8.3.2.1) and (ii), one has $G={\bigcap }_{\lambda }{u}_{\lambda }^{-1}(G_{\lambda })$.

One is thus reduced to proving the particular case of (8.3.2) corresponding to $F_{\lambda }=\varnothing$ for every $\lambda$:

Corollary (8.3.3).

For every $\lambda$, let $E_{\lambda }$ be a pro-constructible part of $S_{\lambda }$ such that, for $\lambda \le \mu$, one has $E_{\mu }\subset u_{\mu \lambda }^{-1}(E_{\lambda })$. Suppose there exists $\alpha \in L$ such that $S_{\alpha }$ is quasi-compact. Then the following conditions are equivalent:

a) $E={\bigcap }_{\lambda }{u}_{\lambda }^{-1}(E_{\lambda })=\varnothing$.

b) There exists $\lambda$ such that $u_{\lambda }^{-1}(E_{\lambda })=\varnothing$ (and then $u_{\mu }^{-1}(E_{\mu })=\varnothing$ for $\mu \ge \lambda$).

c) There exists $\lambda$ such that $E_{\lambda }=\varnothing$ (and then $E_{\mu }=\varnothing$ for $\mu \ge \lambda$).

It is clear that c) implies a). Let us prove that a) entails b): since $S_{\alpha }$ is quasi-compact, so is $S$ (8.2.2); $E_{\lambda }$ being pro-constructible, so is $u_{\lambda }^{-1}(E_{\lambda })$ (1.9.5, (vi)); the filtered decreasing family of pro-constructible sets $u_{\lambda }^{-1}(E_{\lambda })$ then has empty intersection, hence (1.9.9) one of them is empty.

Finally, let us show that b) entails c). Since $S_{\alpha }$ is quasi-compact and $L$ filtered, one can replace $S_{\alpha }$ by an affine open set, so one can suppose (8.2.2) that $S$ and the $S_{\lambda }$ for $\lambda \ge \alpha$ are affine; one has then (1.9.2.1), for $\mu \ge \lambda$,

  u_λ⁻¹(E_λ) = ⋂_{μ ≥ λ} (E_λ ∩ u_{μλ}(S_μ)),

whence $E_{\lambda }\cap u_{\lambda }(S)={\bigcap }_{\mu \ge \lambda }(E_{\lambda }\cap u_{\mu \lambda }(S_{\mu }))$.

Now, since $u_{\lambda }$ and the $u_{\mu \lambda }$ are quasi-compact, $u_{\lambda }(S)$ and $u_{\mu \lambda }(S_{\mu })$ are pro-constructible in $S_{\lambda }$ (1.9.5, (vii)), so the sets $E_{\lambda }\cap u_{\mu \lambda }(S_{\mu })$ for $\mu \ge \lambda$ form a filtered decreasing family of pro-constructible parts of $S_{\lambda }$. Since $S_{\lambda }$ is quasi-compact, hypothesis b) entails that the intersection of this family is empty, hence (1.9.9) one of the sets $E_{\lambda }\cap u_{\mu \lambda }(S_{\mu })$ is empty, hence $E_{\mu }\subset u_{\mu \lambda }^{-1}(E_{\lambda })$ is empty. Q.E.D.

Corollary (8.3.4).

For every $\lambda$, let $F_{\lambda }$ be an ind-constructible part of $S_{\lambda }$ such that for $\lambda \le \mu$ one has $u_{\mu \lambda }^{-1}(F_{\lambda })\subset F_{\mu }$. Suppose there exists $\alpha \in L$ such that $S_{\alpha }$ is quasi-compact. Then the following conditions are equivalent:

a) The set $F={\bigcup }_{\lambda }{u}_{\lambda }^{-1}(F_{\lambda })$ is equal to $S$.

b) There exists $\lambda$ such that $u_{\lambda }^{-1}(F_{\lambda })=S$ (and then $u_{\mu }^{-1}(F_{\mu })=S$ for $\mu \ge \lambda$).

c) There exists $\lambda$ such that $F_{\lambda }=S_{\lambda }$ (and then $F_{\mu }=S_{\mu }$ for $\mu \ge \lambda$).

This follows at once from (8.3.3) by passage to complements.

Corollary (8.3.5).

For every $\lambda$, let $E_{\lambda }$, $F_{\lambda }$ be two constructible parts of $S_{\lambda }$ such that, for $\lambda \le \mu$, one has $E_{\mu }\subset u_{\mu \lambda }^{-1}(E_{\lambda })$ and $F_{\mu }\supset u_{\mu \lambda }^{-1}(F_{\lambda })$. Suppose there exists an index $\alpha$ such that $S_{\alpha }$ is quasi-compact. Then, in order that $E\subset F$ (resp. $E=F$), it is necessary and sufficient that there exist $\lambda$ such that $E_{\lambda }\subset F_{\lambda }$ (resp. $E_{\lambda }=F_{\lambda }$), in which case one also has $E_{\mu }\subset F_{\mu }$ (resp. $E_{\mu }=F_{\mu }$) for $\mu \ge \lambda$.

This is a particular case of (8.3.2).

In particular:

Corollary (8.3.6).

Suppose there exists an $\alpha$ such that $S_{\alpha }$ is quasi-compact. In order that $S=\varnothing$, it is necessary and sufficient that there exist $\lambda$ such that $S_{\lambda }=\varnothing$.

Corollary (8.3.7).

One has, for every $\lambda$,

  (8.3.7.1)    u_λ(S) = ⋂_{μ ≥ λ} u_{μλ}(S_μ).

It is clear that the first member of (8.3.7.1) is contained in the second. Let $s$ be a point of $S_{\lambda }$ and set $X_{\lambda }=\mathrm{Spec}(k(s))$; consider the projective system $(X_{\mu },z_{\nu \mu })$ where $X_{\mu }=X_{\lambda }{\times }_{S_{\lambda }}{S}_{\mu }$ and $z_{\nu \mu }=1\times u_{\nu \mu }$ for $\lambda \le \mu \le \nu$; its projective limit is $X=X_{\lambda }{\times }_{S_{\lambda }}S$ and $p_{\lambda }=1\times u_{\lambda }$ is the canonical map $X\to X_{\lambda }$ (8.2.5). If $s\in {\bigcap }_{\mu \ge \lambda }{u}_{\mu \lambda }(S_{\mu })$, this entails that $X_{\mu }\ne \varnothing$ for every $\mu \ge \lambda$ (I, 3.4.8); it follows then from (8.3.6) that $X\ne \varnothing$, hence $s\in u_{\lambda }(S)$ by (I, 3.4.8).

Proposition (8.3.8).

(i) In order that the morphism $u_{\lambda }:S\to S_{\lambda }$ be dominant (resp. surjective), it is necessary and sufficient that for $\mu \ge \lambda$ the morphism $u_{\mu \lambda }:S_{\mu }\to S_{\lambda }$ be dominant (resp. surjective).

(ii) If, for some index $\lambda$, the morphisms $u_{\mu \lambda }:S_{\mu }\to S_{\lambda }$ are flat (resp. faithfully flat) for all $\mu \ge \lambda$, then the morphism $u_{\lambda }:S\to S_{\lambda }$ is flat (resp. faithfully flat).

(iii) Suppose that the morphisms $u_{\mu \lambda }:S_{\mu }\to S_{\lambda }$ are surjective for $\mu \ge \lambda$. In order that $u_{\lambda }$ be an open morphism (resp. universally open), it is necessary and sufficient that, for every $\mu \ge \lambda$, $u_{\mu \lambda }$ be an open morphism (resp. universally open).

(i) Since $u_{\lambda }(S)\subset u_{\mu \lambda }(S_{\mu })$ for $\mu \ge \lambda$, the necessity of the conditions is trivial, and it follows at once from (8.3.7.1) that if the $u_{\mu \lambda }$ are surjective, so is $u_{\lambda }$. Suppose now the $u_{\mu \lambda }$ dominant for $\mu \ge \lambda$, and consider in $S_{\lambda }$ a non-empty quasi-compact open set $U_{\lambda }$; then the $U_{\mu }=u_{\mu \lambda }^{-1}(U_{\lambda })$ for $\mu \ge \lambda$ form a projective system whose projective limit is $U=u_{\lambda }^{-1}(U_{\lambda })$ (8.2.5). By hypothesis the $U_{\mu }$ are all non-empty, so the same is true of $U$ by (8.3.6), which proves that $u_{\lambda }$ is dominant.

(ii) By virtue of (i), it suffices to consider the case where the $u_{\mu \lambda }$ are flat; one can then restrict to the case where $S_{\lambda }$ is affine, so also the $S_{\mu }$ for $\mu \ge \lambda$ and $S$, and the assertion follows then from (2.1.2) and $(0_I,\mathrm{6.2.3})$.

(iii) By virtue of (8.2.5) and (I, 3.5.2), it suffices to treat the case of open morphisms. Since $u_{\lambda }=u_{\mu \lambda }\circ u_{\mu }$ and $u_{\mu }$ is surjective, one knows that if $u_{\lambda }$ is open so is $u_{\mu \lambda }$ for $\mu \ge \lambda$ (Bourbaki, Top. gén., chap. I, 3rd ed., §5, n° 1, prop. 1).

Conversely, to show that $u_{\lambda }$ is open when all the $u_{\mu \lambda }$ are open for $\mu \ge \lambda$, it suffices to see that for every quasi-compact open set $V$ of $S$, $u_{\lambda }(V)$ is open in $S_{\lambda }$; but there exists then $\mu \ge \lambda$ such that $V=u_{\mu }^{-1}(V_{\mu })$, where $V_{\mu }$ is open in $S_{\mu }$ (8.2.11); since $u_{\mu }$ is surjective, one has $V_{\mu }=u_{\mu }(V)$ and $u_{\lambda }(V)=u_{\mu \lambda }(u_{\mu }(V))$ is therefore open by hypothesis.

One will note that it may happen that all the $u_{\mu \lambda }$ are open without $u_{\lambda }$ being so when the $u_{\mu \lambda }$ are not surjective. One has an example by considering an integral ring $A$ which is not a field, and its field of fractions $K$, which is the inductive limit of the $A_f$, where $f$ runs through $A-0$; if one sets $S_1=\mathrm{Spec}(A)$, $S_f=\mathrm{Spec}(A_f)$, one has $S=\lim S_f=\mathrm{Spec}(K)$, and the morphism $S\to S_1$ is not open, although the morphisms $S_f\to S_1$ are.

(8.3.9) For every prescheme $X$, we shall denote as usual by $\mathfrak{P}(X)$ the set of parts of the underlying set of $X$, by $\mathfrak{E}(X)$ (resp. $\mathfrak{Oc}(X)$, $\mathfrak{Fc}(X)$, $\mathfrak{LFc}(X)$) the set of constructible (resp. constructible and open, resp. constructible and closed, resp. constructible and locally closed) parts of $X$. It is clear that $(\mathfrak{P}(S_{\lambda }),u_{\mu \lambda }^{-1})$ is an inductive system of sets and that the maps $u_{\lambda }^{-1}:\mathfrak{P}(S_{\lambda })\to \mathfrak{P}(S)$ form an inductive system of maps, whence, by passage to the inductive limit, a canonical map

$$(\mathrm{8.3.9.1})\lim \mathfrak{P}(S_{\lambda })\to \mathfrak{P}(S).$$

Moreover, it follows from (1.8.2) that $u_{\mu \lambda }^{-1}$ carries $\mathfrak{E}(S_{\lambda })$ (resp. $\mathfrak{Oc}(S_{\lambda })$, $\mathfrak{Fc}(S_{\lambda })$, $\mathfrak{LFc}(S_{\lambda })$) into $\mathfrak{E}(S_{\mu })$ (resp. $\mathfrak{Oc}(S_{\mu })$, $\mathfrak{Fc}(S_{\mu })$, $\mathfrak{LFc}(S_{\mu })$) and that $u_{\lambda }^{-1}$ carries $\mathfrak{E}(S_{\lambda })$ (resp. $\mathfrak{Oc}(S_{\lambda })$, $\mathfrak{Fc}(S_{\lambda })$, $\mathfrak{LFc}(S_{\lambda })$) into $\mathfrak{E}(S)$ (resp. $\mathfrak{Oc}(S)$, $\mathfrak{Fc}(S)$, $\mathfrak{LFc}(S)$). One therefore has by restriction of (8.3.9.1) canonical maps

$$(\mathrm{8.3.9.2})\lim \mathfrak{E}(S_{\lambda })\to \mathfrak{E}(S)(\mathrm{8.3.9.3})\lim \mathfrak{Oc}(S_{\lambda })\to \mathfrak{Oc}(S)(\mathrm{8.3.9.4})\lim \mathfrak{Fc}(S_{\lambda })\to \mathfrak{Fc}(S)(\mathrm{8.3.9.5})\lim \mathfrak{LFc}(S_{\lambda })\to \mathfrak{LFc}(S).$$

(8.3.10) Let $g_{\alpha }:X_{\alpha }\to S_{\alpha }$ be a morphism; with the notations of (8.2.5) one has as above a canonical map $v_{\alpha }^{-1}:\lim \mathfrak{P}(X_{\lambda })\to \mathfrak{P}(X)$; on the other hand, one has projection morphisms $g_{\lambda }:X_{\lambda }\to S_{\lambda }$ for every $\lambda \ge \alpha$ and a projection morphism $g:X\to S$. It is clear that the $g_{\lambda }^{-1}:\mathfrak{P}(S_{\lambda })\to \mathfrak{P}(X_{\lambda })$ form an inductive system of maps, and that the diagrams

  𝔓(S_λ)  ──g_λ⁻¹──→  𝔓(X_λ)
    │                    │
    u_{μλ}⁻¹            v_{μλ}⁻¹
    ↓                    ↓
  𝔓(S_μ)  ──g_μ⁻¹──→  𝔓(X_μ)

are commutative; one therefore deduces by passage to the inductive limit a commutative diagram

  (8.3.10.1)    lim 𝔓(S_λ)  ───→  lim 𝔓(X_λ)
                    │                  │
                    ↓                  ↓
                  𝔓(S)    ──g⁻¹──→  𝔓(X)

and it follows from (1.8.2) that one has analogous diagrams on replacing $\mathfrak{P}$ by $\mathfrak{E}$, $\mathfrak{Oc}$, $\mathfrak{Fc}$ or $\mathfrak{LFc}$.

It results from (8.3.5) that under the hypothesis that for some $\alpha \in L$, $S_{\alpha }$ is quasi-compact, the canonical map (8.3.9.2) is injective. Moreover:

Theorem (8.3.11).

Suppose there exists $\alpha \in L$ such that $S_{\alpha }$ is quasi-compact and quasi-separated. Then the canonical maps (8.3.9.2), (8.3.9.3), (8.3.9.4) and (8.3.9.5) are bijective.

By virtue of the preceding remark, it remains to prove that these maps are surjective; since every constructible part of $S$ is a finite union of sets of the form $U\cap \complement V$, where $U$ and $V$ are open and constructible, it will suffice to prove that (8.3.9.3) is surjective for the same to hold of (8.3.9.2) (and also of (8.3.9.4), by passage to complements). Now, since the morphisms $S_{\lambda }\to S_{\alpha }$ and $S\to S_{\alpha }$ are affine, hence separated, the $S_{\lambda }$ for $\lambda \ge \alpha$ and $S$ are quasi-compact and quasi-separated (1.2.2), and one knows that the constructible open parts in such a prescheme are none other than the quasi-compact open parts (1.8.1). The conclusion therefore follows from (8.2.11) except for the map (8.3.9.5). To prove that this last map is surjective, consider a part $Z$ locally closed and constructible in $S$; $Z$ is therefore quasi-compact $(0_{III},\mathrm{9.1.4})$. Since every point $x\in Z$ admits by hypothesis a quasi-compact open neighbourhood $V_x$ in $S$ such that $Z\cap V_x$ is closed in $V_x$, one can cover $Z$ by a finite number of the $V_x$; in other words, there is a quasi-compact open set $U$ containing $Z$ and such that $Z$ is closed in $U$; since $Z$ is constructible in $S$, it is so also in $U$ $(0_{III},\mathrm{9.1.8})$. One knows (8.2.11) that there is an index $\lambda$ and a quasi-compact open set $U_{\lambda }$ in $S_{\lambda }$ such that $U=u_{\lambda }^{-1}(U_{\lambda })$. Applying to $U$ (which is the projective limit of the $U_{\mu }=u_{\mu \lambda }^{-1}(U_{\lambda })$ for $\mu \ge \lambda$) the fact that the map (8.3.9.4) is surjective, one sees that there exists $\mu \ge \lambda$ and a constructible closed set $Z_{\mu }$ in $U_{\mu }$ such that $Z=u_{\mu }^{-1}(Z_{\mu })$. But since the canonical immersion $U_{\mu }\to S_{\mu }$ is quasi-compact by hypothesis (1.2.7), it is of finite presentation (1.6.2, (i)), and $Z_{\mu }$ is also a constructible part of $S_{\mu }$ by virtue of (1.8.4) and (1.8.1); since $Z_{\mu }$ is evidently locally closed in $S_{\mu }$, this completes the proof.

Corollary (8.3.12).

Suppose there exists $\alpha$ such that $S_{\alpha }$ is quasi-compact, and let, for every $\lambda$, $Z_{\lambda }$ be a constructible part of $S_{\lambda }$ such that $Z_{\mu }=u_{\mu \lambda }^{-1}(Z_{\lambda })$ for $\mu \ge \lambda$. If $Z=u_{\lambda }^{-1}(Z_{\lambda })$,

then, in order that $Z$ be open (resp. closed, resp. locally closed) in $S$, it is necessary and sufficient that there exist $\lambda \ge \alpha$ such that $Z_{\lambda }$ be so in $S_{\lambda }$.

Cover $S_{\alpha }$ by a finite number of affine open sets $U_{\alpha }^{(i)}$; then the $U^{(i)}=u_{\alpha }^{-1}(U_{\alpha }^{(i)})$ form an open affine cover of $S$, and for $Z$ to be open (resp. closed, resp. locally closed) in $S$, it is necessary and sufficient that each of the $Z\cap U^{(i)}$ be so in $U^{(i)}$. Since $L$ is filtered and each of the $Z\cap U^{(i)}$ is constructible in $U^{(i)}$ $(0_{III},\mathrm{9.1.8})$, one can restrict to proving the corollary when $S_{\alpha }$ is affine, hence quasi-compact and quasi-separated; but in that case it follows from (8.3.11).

Proposition (8.3.13).

Suppose that the morphisms $u_{\mu \lambda }:S_{\mu }\to S_{\lambda }$ are flat for $\lambda \le \mu$, and that there exists $\alpha$ such that $S_{\alpha }$ is quasi-compact. For every $\lambda$, let $Z_{\lambda }$, $Z_{\lambda }^{\prime }$ be two pro-constructible parts of $S_{\lambda }$, such that $Z_{\mu }=u_{\mu \lambda }^{-1}(Z_{\lambda })$ and $Z_{\mu }^{\prime }=u_{\mu \lambda }^{-1}(Z_{\lambda }^{\prime })$ for $\mu \ge \lambda$; suppose moreover that $Z_{\alpha }$ is constructible in $S_{\alpha }$. Let $Z^{\prime }=u_{\lambda }^{-1}(Z_{\lambda })$, $Z^{\prime \prime }=u_{\lambda }^{-1}(Z_{\lambda }^{\prime })$; in order that ${\overline{Z}}^{\prime \prime }\subset {\overline{Z}}^{\prime }$, it is necessary and sufficient that there exist $\lambda \ge \alpha$ such that ${\overline{Z}}_{\lambda }^{\prime }\subset {\overline{Z}}_{\lambda }$.

Indeed, one knows that $u_{\lambda }:S\to S_{\lambda }$ is also a flat morphism for every $\lambda$ (8.3.8); since $Z_{\lambda }$ is pro-constructible, the closure of $Z_{\mu }$ for $\mu \ge \lambda$ (resp. of $Z^{\prime }$) in $S_{\mu }$ (resp. $S$) is equal to $u_{\mu \lambda }^{-1}({\overline{Z}}_{\lambda })$ (resp. $u_{\lambda }^{-1}({\overline{Z}}_{\lambda })$) (2.3.10). Since the $u_{\mu \lambda }^{-1}({\overline{Z}}_{\lambda })$ and $u_{\lambda }^{-1}({\overline{Z}}_{\lambda })$ are constructible (1.8.2), the conclusion follows from (8.3.2).

8.4. Irreducibility and connectedness criteria for projective limits of preschemes

Proposition (8.4.1).

Suppose there exists an index $\alpha$ such that $S_{\alpha }$ is quasi-compact.

(i) If $S$ is not irreducible and if in addition the space underlying $S$ is Noetherian and $S_{\alpha }$ quasi-separated, there exists $\lambda \ge \alpha$ such that, for $\mu \ge \lambda$, $S_{\mu }$ is not irreducible.

(ii) If $S$ is not connected, there exists $\lambda \ge \alpha$ such that, for $\mu \ge \lambda$, $S_{\mu }$ is not connected.

Suppose that $S$ is the union of two closed sets $S^{\prime }$, S'' distinct from $S$ (resp. disjoint non-empty closed sets). In case (i), $S^{\prime }$ and S'' are constructible since the space $S$ is Noetherian. By virtue of (8.3.11), there exist therefore $\lambda \ge \alpha$ and two constructible closed sets $S_{\lambda }^{\prime }$, $S_{\lambda }^{\prime \prime }$ of $S_{\lambda }$ such that $S^{\prime }=u_{\lambda }^{-1}(S_{\lambda }^{\prime })$, $S^{\prime \prime }=u_{\lambda }^{-1}(S_{\lambda }^{\prime \prime })$; since $S=S^{\prime }\cup S^{\prime \prime }$, it follows also from (8.3.11) that one can suppose that $S_{\lambda }=S_{\lambda }^{\prime }\cup S_{\lambda }^{\prime \prime }$; since $S_{\lambda }^{\prime }$ and $S_{\lambda }^{\prime \prime }$ are distinct from $S_{\lambda }$, this proves that $S_{\lambda }$ is not irreducible.

In case (ii), $S^{\prime }$ and S'' are quasi-compact open sets, hence, by virtue of (8.2.11), there exist $\lambda \ge \alpha$ and two quasi-compact open sets $S_{\lambda }^{\prime }$, $S_{\lambda }^{\prime \prime }$ of $S_{\lambda }$ such that $S^{\prime }=u_{\lambda }^{-1}(S_{\lambda }^{\prime })$, $S^{\prime \prime }=u_{\lambda }^{-1}(S_{\lambda }^{\prime \prime })$. Moreover, since $S^{\prime }$ and S'' are open and closed in $S$, they are at once pro-constructible and ind-constructible (1.9.6), hence constructible (1.9.11), and it follows therefore from (8.3.5) that one can suppose $\lambda$ taken such that $S_{\lambda }=S_{\lambda }^{\prime }\cup S_{\lambda }^{\prime \prime }$ and $S_{\lambda }^{\prime }\cap S_{\lambda }^{\prime \prime }=\varnothing$, which shows that $S_{\lambda }$ is not connected.

Proposition (8.4.2).

Suppose that the space underlying $S$ is Noetherian and that one of the following two conditions is satisfied:

a) For $\lambda \le \mu$, $u_{\mu \lambda }:S_{\mu }\to S_{\lambda }$ is dominant, and there exists $\alpha$ such that $S_{\alpha }$ is quasi-compact.

b) There exists $\alpha$ such that the space underlying $S_{\alpha }$ is Noetherian, and for $\mu \ge \lambda$, $u_{\mu \lambda }$ is a homeomorphism of $S_{\mu }$ onto a subspace of $S_{\lambda }$.

Under these conditions, there exists $\lambda$ such that, for every $\mu \ge \lambda$:

(i) For every irreducible component $Y_i$ of $S$ ($1\le i\le m$), $\overline{u_{\mu }(Y_i)}$ is an irreducible component of $S_{\mu }$, and the map $Y_i\mapsto \overline{u_{\mu }(Y_i)}$ is a bijection of the set of irreducible components of $S$ onto the set of irreducible components of $S_{\mu }$.

(ii) For every connected component $C_j$ of $S$ ($1\le j\le n$), $u_{\mu }(C_j)$ is a connected component of $S_{\mu }$, and the map $C_j\mapsto u_{\mu }(C_j)$ is a bijection of the set of connected components of $S$ onto the set of connected components of $S_{\mu }$.

We shall first establish the

Lemma (8.4.2.1).

Under condition a) or b) of (8.4.2), there exists $\lambda$ such that, for $\mu \ge \lambda$, $u_{\mu }:S\to S_{\mu }$ is dominant.

In case a), this has already been proved without supposing the space $S$ Noetherian (8.3.8, (i)). In case b), set $Z_{\alpha }=\overline{u_{\alpha }(S)}$; as a closed part of the Noetherian space $S_{\alpha }$, $Z_{\alpha }$ is constructible, and since $u_{\alpha }^{-1}(Z_{\alpha })=S$, it follows from (8.3.5) that there exists $\lambda \ge \alpha$ such that, for $\mu \ge \lambda$, one has $Z_{\mu }=u_{\mu \alpha }^{-1}(Z_{\alpha })=S_{\mu }$. But since $u_{\alpha \mu }$ is a homeomorphism of $S_{\mu }$ onto a subspace of $S_{\alpha }$, and since the composite map $S\to Z_{\mu }\to Z_{\alpha }$ is dominant, the same is true of $S\to Z_{\mu }=S_{\mu }$.