Exposé XII. Algebraic Geometry and Analytic Geometry

Mme M. Raynaud. [Translator note: according to unpublished notes of A. Grothendieck.]

Proceeding as in [XII.10], one associates to every scheme $X$ locally of finite type over the field of complex numbers $\mathbb{C}$ an analytic space $X^{an}$, whose underlying set is $X(\mathbb{C})$.

In nos. XII.2 and XII.3 of this exposé, we give a “dictionary” between the usual properties of $X$ and of $X^{an}$, and between the properties of a morphism $f:X\to Y$ and of the associated morphism $f^{an}:X^{an}\to Y^{an}$.

We then show that the comparison theorems between coherent sheaves on $X$ and $X^{an}$, established in [XII.10, no. 12] for a projective variety, are still valid when $X$ is a proper scheme.

Finally, in no. XII.5 we prove the equivalence between the category of finite étale coverings of $X$ and the category of finite étale coverings of $X^{an}$. As a bonus for the reader, we give a new proof of the Grauert-Remmert theorem [XII.6], using resolution of singularities [XII.8].

1. The Analytic Space Associated with a Scheme

Let $X$ be a scheme locally of finite type over $\mathbb{C}$. Let $\Phi$ be the functor from the category of analytic spaces [XII.4, no. 9] to the category of sets which associates to an analytic space $\mathcal{X}$ the set of morphisms of ringed spaces in $\mathbb{C}$-algebras ${\mathrm{Hom}}_{\mathbb{C}}(\mathcal{X},X)$. One has the following theorem:

Theorem-Definition.

The functor $\Phi$ is representable by an analytic space $X^{an}$ and a morphism $\varphi :X^{an}\to X$. One says that $X^{an}$ is the analytic space associated with $X$.

If $|X^{an}|$ is the underlying set of $X^{an}$, $\varphi$ induces a bijection from $|X^{an}|$ to the set $X(\mathbb{C})$ of points of $X$ with values in $\mathbb{C}$. Moreover, for each point $x$ of $X^{an}$, the morphism

$${\varphi }_x:{\mathcal{O}}_X,\varphi (x)\to {\mathcal{O}}_X^{an},x,$$

which is necessarily local, gives after passage to completions an isomorphism

$${\hat{\varphi }}_x:{\hat{\mathcal{O}}}_X,\varphi (x)\simeq {\hat{\mathcal{O}}}_X^{an},x.$$

In particular the morphism $\varphi$ is flat.

Notice that the fact that $\varphi$ induces a bijection from $X^{an}$ to $X(\mathbb{C})$ follows from the universal property of $X^{an}$. On the other hand, one has the following assertions:

a. If the theorem is true for a scheme $Y$, then it is also true for every subscheme $X$ of $Y$. Suppose first that $X$ is an open subscheme of $Y$. If $\psi :Y^{an}\to Y$ is the canonical morphism, ${\psi }^{-1}(X)$ is an open subset

of $Y^{an}$, endowed with the analytic-space structure induced by that of $Y^{an}$. Since every morphism from an analytic space $\mathcal{X}$ to $X$ factors through $Y^{an}$ by the universal property of the latter, and hence through $X^{an}$, which is the fiber product $Y^{an}{\times }_{Y}X$, $X^{an}$ is the analytic space associated with $X$. Finally, the assertion concerning the ${\varphi }_x$ is evident.

It remains only to consider the case where $X$ is a closed subscheme of $Y$. Let $I$ be the coherent ${\mathcal{O}}_Y$-ideal defining $X$. Then $I\cdot {\mathcal{O}}_Y^{an}$ is a coherent sheaf of ideals on ${\mathcal{O}}_Y^{an}$ defining a closed analytic subspace $X^{an}$ of $Y^{an}$. As in the case of an open subscheme, one sees that $X^{an}$ is the analytic space associated with $X$. Let $\varphi :X^{an}\to X$ be the canonical morphism. For every point $x$ of $X^{an}$, the morphism ${\varphi }_x$ is none other than the morphism

𝒪_Y,ψ(x)/I_ψ(x) → 𝒪_Y^an,x / I_ψ(x) · 𝒪_Y^an,x

induced by ${\psi }_x$. Its completion

$${\hat{\mathcal{O}}}_Y,\psi (x)/I_{\psi }(x)\cdot {\hat{\mathcal{O}}}_Y,\psi (x)\to {\hat{\mathcal{O}}}_Y^{an},x/I_{\psi }(x)\cdot {\hat{\mathcal{O}}}_Y^{an},x$$

is an isomorphism, since ${\hat{\psi }}_x$ is one; this proves a.

b. If one has two $\mathbb{C}$-schemes $X_1$, $X_2$, such that $X_1^{an}$ and $X_2^{an}$ exist, then $(X_1\times X_2{)}^{an}$ also exists. Indeed, let ${\varphi }_1:X_1^{an}\to X_1$ and ${\varphi }_2:X_2^{an}\to X_2$ be the canonical morphisms, and let $p_1$, $p_2$ be the two projections from $X_1^{an}\times X_2^{an}$. It follows formally from EGA I 1.8.1 that $X_1\times X_2$ is the product of $X_1$ and $X_2$ in the category of ringed spaces in local rings. Consequently the morphisms ${\varphi }_1\cdot p_1$ and ${\varphi }_2\cdot p_2$ define a

morphism $\varphi :X_1^{an}\times X_2^{an}\to X_1\times X_2$, and the pair $(X_1^{an}\times X_2^{an},\varphi )$ represents the functor $\mathcal{X}\mapsto {\mathrm{Hom}}_{\mathbb{C}}(\mathcal{X},X_1\times X_2)$.

c. If ${\mathcal{E}}^1$ denotes affine space of dimension 1, that is, the topological space $\mathbb{C}$ endowed with the sheaf of holomorphic functions, the functor $\mathcal{X}\mapsto {\mathrm{Hom}}_{\mathbb{C}}(\mathcal{X},E_{\mathbb{C}}^1)$ is representable by ${\mathcal{E}}^1$, the canonical morphism $\varphi :{\mathcal{E}}^1\to E_{\mathbb{C}}^1$ being the evident morphism. [Translator note: the corrected source adds in 2003 that $E_{\mathbb{C}}^1$ denotes the algebraic affine line over $\mathbb{C}$.] Indeed, to give a morphism from an analytic space $\mathcal{X}$ to $E_{\mathbb{C}}^1$ is equivalent to giving an element of $\Gamma (\mathcal{X},{\mathcal{O}}_{\mathcal{X}})$, which is also equivalent to giving a morphism from $\mathcal{X}$ to ${\mathcal{E}}^1$. Plainly one has a bijection $|{\mathcal{E}}^1|\simeq E^1(\mathbb{C})$, and, for each point $x\in {\mathcal{E}}^1$, the morphism ${\hat{\varphi }}_x$ is none other than the identity morphism of a ring of formal power series in one variable over $\mathbb{C}$.

It follows from b and c that the theorem is true for affine space $E_{\mathbb{C}}^n$, $n\ge 0$. Using a, one sees that it is also true for every affine scheme $X$ locally of finite type over $\mathbb{C}$. If $X$ is no longer assumed affine and if $(X_i)$ is a covering of $X$ by affine opens, it follows from the universal property and from a that the $X_i^{an}$ glue and thus define the analytic space $X^{an}$ associated with $X$.

1.2.

Let $f:X\to Y$ be a morphism of $\mathbb{C}$-schemes locally of finite type. If $\varphi :X^{an}\to X$ and $\psi :Y^{an}\to Y$ are the canonical morphisms, it follows from the universal property of $Y^{an}$ that there exists a unique morphism $f^{an}:X^{an}\to Y^{an}$ such that the diagram

X^an → X
 |      |
f^an   f
 |      |
Y^an → Y

is

commutative. We have therefore defined a functor $\Phi$ from the category of $\mathbb{C}$-schemes locally of finite type to the category of analytic spaces.

The functor $\Phi$ commutes with finite projective limits. Indeed it is enough to see that $\Phi$ commutes with fiber products. But if $X$, $Y$, $Z$ are schemes locally of finite type over $\mathbb{C}$, it follows from the fact that $X{\times }_{Z}Y$ is the fiber product of $X$ and $Y$ over $Z$ in the category of ringed spaces in local rings that $X^{an}{\times }_Z^{an}{Y}^{an}$ satisfies the universal property characterizing $(X{\times }_{Z}Y{)}^{an}$.

1.3.

Let $X$ be a $\mathbb{C}$-scheme locally of finite type, let $X^{an}$ be the associated analytic space, and let $\varphi :X^{an}\to X$ be the canonical morphism. If $F$ is an ${\mathcal{O}}_X$-module, the inverse image $\varphi \ast F=F^{an}$ is a sheaf of modules over ${\mathcal{O}}_X^{an}$. This defines a functor from the category of ${\mathcal{O}}_X$-modules to the category of modules on $X^{an}$. This functor commutes with inductive limits (EGA 0 4.3.2). Since the sheaf ${\mathcal{O}}_X^{an}$ is coherent [XII.4, no. 18, §2, th. 2], it sends coherent sheaves to coherent sheaves (EGA 0 5.3.11). Moreover:

Subproposition.

The functor which associates to an ${\mathcal{O}}_X$-module $F$ its inverse image $F^{an}$ on $X^{an}$ is exact, faithful, and conservative.

Exactness follows from the fact that the morphism $\varphi :X^{an}\to X$ is flat (XIII.1.1). Let us prove that the functor $F\mapsto F^{an}$ is faithful. Taking exactness into account, it is enough to show that if $F^{an}$ is zero, then $F$ itself is zero. But for every point $x$ of $X^{an}$ one then has

F_φ(x) ⊗_𝒪_X,φ(x) 𝒪_X^an,x = 0.

Since the morphism ${\mathcal{O}}_X,\varphi (x)\to {\mathcal{O}}_X^{an},x$ is faithfully flat, one has $F_{\varphi }(x)=0$ for every closed point $\varphi (x)$ of $X$; and since $X$ is Jacobson (EGA IV 10.4.8), this implies that $F$ is zero.

The

fact that the functor $F\mapsto F^{an}$ is conservative is formal from exactness and faithfulness.

2. Comparison of Properties of a Scheme and of the Associated Analytic Space

Proposition.

Let $X$ be a $\mathbb{C}$-scheme locally of finite type, let $X^{an}$ be the associated analytic space, and let $n$ be an integer. Consider the property $P$ of being:

(i)    nonempty
(i′)   discrete
(ii)   Cohen-Macaulay
(iii)  (S_n)
(iv)   regular
(v)    (R_n)
(vi)   normal
(vii)  reduced
(viii) of dimension n.

Then $X$ has property $P$ if and only if $X^{an}$ has property $P$.

Let $\varphi :X^{an}\to X$ be the canonical morphism. Assertion (i) follows from the fact that $|X^{an}|=X(\mathbb{C})$ (XIII.1.1) and from the fact that $X$ is Jacobson (EGA IV 10.4.8). To say that $X$, respectively $X^{an}$, is discrete is equivalent to saying that $\dim X=0$, respectively $\dim X^{an}=0$ by [XII.4, no. 19, §4, cor. 6]; hence (i′) follows from (viii).

Let $P$ be one of the properties (ii) through (vii). For $X$ to have property $P$, it is necessary and sufficient that $P$ hold at every closed point of $X$. Indeed, since $X$ is excellent (EGA IV 7.8.6 (iii)), the set of points

where $X$ satisfies $P$ is open, and if this open contains all closed points, it is equal to all of $X$. Thus to say that $X$, respectively $X^{an}$, has property $P$ is equivalent to saying that for every point $x$ of $X^{an}$, the local ring ${\mathcal{O}}_X,\varphi (x)$, respectively ${\mathcal{O}}_X^{an},x$, has property $P$. Since the fact that an excellent local ring has property $P$ can be detected after passage to the completion, the proposition follows from the isomorphisms

$${\hat{\mathcal{O}}}_X,\varphi (x)\simeq {\hat{\mathcal{O}}}_X^{an},x$$

in cases (ii) through (vii). The same holds in case (viii), taking into account the relations

dim X = sup_x dim 𝒪_X,φ(x),     dim X^an = sup_x dim 𝒪_X^an,x,

where $x\in X^{an}$. This completes the proof.

Proposition.

Let $X$ be a $\mathbb{C}$-scheme locally of finite type, let $\varphi :X^{an}\to X$ be the canonical morphism, and let $T$ be a locally constructible subset of $X$. Then one has the relation

$${\varphi }^{-1}(closure(T))=closure({\varphi }^{-1}(T)).$$

We may suppose that $T$ is a dense open subset of $X$. Let $H$ be the reduced closed subscheme of $X$ whose underlying space is $X-T$. The associated space $H^{an}$ is a closed analytic subspace of $X^{an}$ whose underlying space is $X^{an}-{\varphi }^{-1}(T)$. We must show that every point $x$ of $H^{an}$ belongs to $closure({\varphi }^{-1}(T))$. But at such a point $x$, the germ of analytic space $(X^{an},x)$ contains the subgerm $(H^{an},x)$, and this is defined by a non-nilpotent ideal of ${\mathcal{O}}_X^{an},x$. It then follows from the Nullstellensatz [XII.4, no. 19, §4, cor. 3] that every open neighborhood of $x$ contains points of $X^{an}$ which do not belong to $H^{an}$. This proves that $x\in closure({\varphi }^{-1}(T))$.

Corollary.

Let

$X$ be a $\mathbb{C}$-scheme locally of finite type, let $\varphi :X^{an}\to X$ be the canonical morphism, and let $T$ be a locally constructible subset of $X$. For $T$ to be an open subset, respectively a closed subset, respectively a dense subset, it is necessary and sufficient that ${\varphi }^{-1}(T)$ have the corresponding property.

The corollary follows from XII.2.2 and from the fact that, since $X$ is a Jacobson scheme (EGA IV 10.4.8), two locally constructible subsets of $X$ that have the same trace on the very dense set $X(\mathbb{C})$ are equal.

Proposition.

Let $X$ be a $\mathbb{C}$-scheme locally of finite type. For $X$ to be connected, respectively irreducible, it is necessary and sufficient that $X^{an}$ be connected, respectively irreducible.

Suppose $X^{an}$ is connected, respectively irreducible. The image $X(\mathbb{C})$ of $X^{an}$ in $X$ is then connected, respectively irreducible. It follows that $X$ is connected, respectively irreducible, because closed subsets of $X$ and of $X(\mathbb{C})$ correspond bijectively (EGA IV 10.1.2).

Conversely suppose $X$ is connected, respectively irreducible, and let us show that the same is true of $X^{an}$. We may restrict to the case where $X$ is irreducible. Indeed, suppose $X$ is connected. Given a point $x$ of $X$, the set of points $y\in X$ for which there exists a finite sequence of irreducible closed subschemes $X_1,\cdots ,X_n$ of $X$, with $x\in X_1$, $y\in X_n$, and $X_i\cap X_{i+1}\ne \varnothing$ for $1\le i\le n-1$, is both open and closed, hence equal to all of $X$. For a sequence $X_1,\cdots ,X_n$ as above, one also has $X_i^{an}\cap X_{i+1}^{an}\ne \varnothing$ for $1\le i\le n-1$; if the $X_i^{an}$ are known to be connected, then $X^{an}$ is connected as well.

From now on

suppose $X$ is irreducible. We may also suppose $X$ affine. Indeed, if $(U_i{)}_{i\in I}$ is a covering of $X$ by affine opens, any two of these opens have nonempty intersection, and the same property is therefore true for the covering $(U_i^{an}{)}_{i\in I}$ of $X^{an}$. If the $U_i^{an}$ are known to be irreducible, then $X^{an}$ is irreducible as well.

We may further suppose that $X$ is normal. Indeed, let $X$ be the normalization of $X$. Since the morphism $X\to X$ is surjective, so is $X^{an}\to X^{an}$, which proves that if $X^{an}$ is irreducible, then $X^{an}$ is irreducible as well.

From now on suppose $X$ is affine normal. Since the local rings of $X^{an}$ are integral domains, saying that $X^{an}$ is irreducible is equivalent to saying that it is connected. Indeed, if $\mathcal{F}$ is a closed analytic subset of $X^{an}$, the set of points $x$ of $X^{an}$ at which $codi{m}_x(\mathcal{F},X^{an})=0$ is a closed analytic subset of $X^{an}$ [XII.4, no. 20 A, cor. 1] which is also open. If $X^{an}$ is connected, this proves that, whenever $\mathcal{F}\ne X^{an}$, $\mathcal{F}$ is rare; hence $X^{an}$ is irreducible. We are thus reduced to showing that $X^{an}$ is connected.

Let

$$i:X\to P$$

be a compactification of $X$, where $P$ is a normal projective $\mathbb{C}$-scheme and $i$ is a dominant open immersion. It then follows from [XII.10, no. 12, th. 1] that $P^{an}$ is connected. Since $X^{an}$ is obtained by removing from $P^{an}$ a rare closed analytic subset, it follows from XII.2.5 below that $X^{an}$ is also connected.

Lemma.

Let

$\mathcal{P}$ be a connected normal analytic space, and let $\mathcal{Y}$ be a rare closed analytic subset. Then $\mathcal{X}=\mathcal{P}-\mathcal{Y}$ is connected.

When $\mathcal{Y}$ has codimension $\ge 2$, the proposition follows from [XII.11, no. 3, prop. 4]. In the general case one may suppose, after removing from $\mathcal{P}$ a closed analytic subset of codimension $\ge 2$, that $\mathcal{P}$ and $\mathcal{Y}$, regarded as a reduced analytic subspace of $\mathcal{P}$, are regular. By the implicit function theorem, every point $y$ of $\mathcal{Y}$ has a neighborhood $\mathcal{U}$ isomorphic to a ball in an affine space ${\mathcal{E}}^n$, such that $\mathcal{U}\cap \mathcal{Y}$ is defined by the vanishing of a certain number of coordinate functions. This proves that $\mathcal{U}-\mathcal{U}\cap \mathcal{Y}$ is connected, and hence that $\mathcal{X}$ is connected.

Corollary.

Let $X$ be a $\mathbb{C}$-scheme locally of finite type. The morphism

$${\pi }_0(X^{an})\to {\pi }_0(X)$$

induced by the canonical morphism $X^{an}\to X$ is bijective.

3. Comparison of Properties of Morphisms

Proposition.

Let $f:X\to Y$ be a morphism of $\mathbb{C}$-schemes locally of finite type, and let $f^{an}:X^{an}\to Y^{an}$ be the morphism deduced from $f$ on the associated analytic spaces. Let $P$ be the property of being:

(i)    flat
(ii)   net, that is, unramified
(iii)  étale
(iv)   smooth
(v)    normal
(vi)   reduced
(vii)  injective
(viii) separated
(ix)   an isomorphism
(x)    a monomorphism
(xi)   an open immersion.

Then $f$ has property $P$ if and only if $f^{an}$ has property $P$.

Let $\varphi :X^{an}\to X$ and $\psi :Y^{an}\to Y$ be the canonical morphisms. Let $x$ be a point of $X^{an}$, and put $y=f^{an}(x)$. The morphisms ${\mathcal{O}}_Y^{an},y\to {\mathcal{O}}_X^{an},x$ and ${\mathcal{O}}_Y,\psi (y)\to {\mathcal{O}}_X,\varphi (x)$ deduced from $f^{an}$ and $f$ give the same morphism after passage to completions (XII.1.1). By [XII.2, ch. 3, §5, prop. 4], respectively EGA IV 17.4.4, it is therefore equivalent to say that $f^{an}$ satisfies property (i), respectively (ii), and to say that $f$ satisfies (i), respectively (ii), at every closed point of $X$. Since the set of points of $X$ where (i), respectively (ii), holds is open (EGA IV 11.1.1 and I 3.3), this proves (i) and (ii), hence also (iii).

Let $P$ be property (iv), respectively (v), respectively (vi). Taking XII.2.1 ((v), (vi), (vii)) into account, it is equivalent to say that the geometric fibers of $f^{an}$ at the various points $y$ of $Y^{an}$ are regular, respectively normal, respectively reduced, and to say that the same is true of the geometric fibers of $f$ at the various closed points $\psi (y)$ of $Y$. Cases (iv), respectively (v), respectively (vi), then follow from (i) and from the fact that the set of points of $Y$ where the geometric fibers of $f$ are regular is open (EGA IV 12.1.7).

(vii). If $f$ is injective, so is $f^{an}$. Conversely suppose $f^{an}$ is injective and let us show that $f$ is injective. We may suppose

$f$ of finite type. Since $f^{an}$ is injective, the fibers of $f$ at closed points of $Y$ are radicial. Since the set of points of $Y$ whose fiber is radicial is locally constructible (EGA IV 9.6.1), and since $Y$ is a Jacobson scheme, all fibers of $f$ are radicial; hence $f$ is injective.

(viii). Let $\Delta :X\to X{\times }_{Y}X$ and ${\Delta }^{an}:X^{an}\to X^{an}{\times }_{Y^{an}}{X}^{an}$ be the diagonal immersions, and let $\Theta :X^{an}{\times }_{Y^{an}}{X}^{an}\to X{\times }_{Y}X$ be the canonical morphism. By XII.2.3, saying that $\Delta (X)$ is closed in $X{\times }_{Y}X$ is equivalent to saying that ${\Delta }^{an}(X^{an})$ is closed in $X^{an}{\times }_{Y^{an}}{X}^{an}$.

Since an open immersion is nothing other than an étale injective morphism (EGA IV 17.9.1 and [XII.4, no. 13, §1]), (xi) follows from (iii) and (vii). Since an isomorphism is the same thing as a surjective open immersion, (ix) follows from (xi) and XII.3.2 (i) below. Saying that $f$ is a monomorphism is equivalent to saying that the diagonal morphism $\Delta :X\to X{\times }_{Y}X$ is an isomorphism, so (x) follows from (ix).

Proposition.

Let $X$ and $Y$ be two $\mathbb{C}$-schemes locally of finite type, let $f:X\to Y$ be a morphism of finite type, and let $f^{an}:X^{an}\to Y^{an}$ be the morphism deduced from $f$ on the associated analytic spaces. Let $P$ be the property of being:

(i)   surjective
(ii)  dominant
(iii) a closed immersion
(iv)  an immersion
(v)   proper
(vi)  finite.

Then $f$ has property $P$ if and only if $f^{an}$ has property $P$. [Translator note: the source footnote says that a morphism of analytic spaces is called proper if it is proper in the sense of [XII.1, ch. 1, §10, no. 1] and is separated.]

Let $\varphi :X^{an}\to X$ and $\psi :Y^{an}\to Y$ be the canonical morphisms.

(i). If $f$ is surjective, then for every point $y$ of $Y^{an}$, $f^{-1}(\psi (y))$ is a nonempty closed subset of $X$; hence it contains at least one closed point, which proves that $f^{an}$ is surjective. Conversely, if $f^{an}$ is surjective, $f(X)$ is a locally constructible subset of $Y$ (EGA IV 1.8.4) containing all closed points of $Y$; hence $f(X)=Y$.

(ii) follows from XII.2.2.

(iii). If $f$ is a closed immersion, then so is $f^{an}$ by XII.1.1 a. Conversely, if $f^{an}$ is a closed immersion, then so is $f$ by XII.3.1 (x) and XII.3.2 (v), since this is equivalent to saying that $f$ is a proper monomorphism (EGA IV 8.11.5).

(iv). It is clear that if $f$ is an immersion, then so is $f^{an}$. Conversely suppose $f^{an}$ is an immersion, and let $T$ be the image of $X$ in $Y$, and $\bar{T}$ the scheme-theoretic closure of $f$. There is a factorization of $f$

X --i→ T̄ --j→ Y,

where $j$ is a closed immersion and $i$ is the canonical morphism; from it one deduces the following factorization of $f^{an}$:

X^an --i^an→ T̄^an --j^an→ Y^an.

Since

$T=f(X)$ is a locally constructible subset of $Y$ (EGA IV 1.8.4), one has, by XII.2.2, ${\bar{T}}^{an}=closure(f^{an}(X^{an}))$. It follows that $i^{an}(X^{an})$ is open in ${\bar{T}}^{an}$, hence that $i(X)$ is open in $\bar{T}$. Consider the canonical factorization of $i$

X --i₁→ i(X) --i₂→ T̄.

The morphism $i_1^{an}$ is a proper monomorphism, hence so is $i_1$ by XII.3.2 (v) and XII.3.1 (x). This proves that $i_1$, and hence also $f$, is an immersion.

(v). Suppose $f$ is proper and let us show that $f^{an}$ is proper. Since properness of $f^{an}$ is local on $Y^{an}$, we may suppose $Y$ affine. By Chow’s lemma (EGA II 5.6.1), one can find a projective $Y$-scheme $X^{\prime }$ and a projective surjective morphism

$$g:X^{\prime }\to X.$$

The morphism $(fg{)}^{an}=f^{an}{g}^{an}$ is projective, hence proper; $g^{an}$ is surjective; and it follows from [XII.1, ch. 1, §10] that $f^{an}$ is proper.

Conversely suppose $f^{an}$ is proper and let us show that $f$ is proper. By XII.3.1 (viii), $f$ is separated. It remains to prove that $f$ is universally closed, and it is even enough to show that $f$ is closed. Indeed, for every $Y$-scheme $Y^{\prime }$ locally of finite type, the morphism

f_(Y′) = h: X ×_Y Y′ → Y′

will also be closed since $h^{an}$ is proper. Let $T$ be a closed subset of $X$. The set $f(T)$ is locally constructible, and one has

$$f^{an}({\varphi }^{-1}(T))={\psi }^{-1}(f(T)).$$

Since

$f^{an}$ is proper, ${\psi }^{-1}(f(T))$ is a closed subset of $Y^{an}$, and therefore it follows from XII.2.2 that

$${\psi }^{-1}(closure(f(T)))={\psi }^{-1}(f(T)).$$

This implies $f(T)=closure(f(T))$, that is, $f$ is closed; hence $f$ is proper.

(vi). Saying that a morphism is finite is equivalent to saying that it is proper with finite fibers (EGA III 4.4.2 and [XII.4, no. 19, §5]). Since the set of points where the fibers of $f$ are finite is locally constructible (EGA IV 9.7.9), the fibers of $f$ are finite if and only if the fibers of $f^{an}$ are finite. Thus (vi) follows from (v).

Remark.

a. Let $f:X\to Y$ be a morphism of $\mathbb{C}$-schemes locally of finite type. The fact that $f^{an}$ is a local isomorphism does not imply that $f$ is one. Indeed, if $f$ is étale, $f^{an}$ is étale and hence is a local isomorphism [XII.4, no. 13, §1], but this need not be true of $f$.

b. The statement XII.3.2 is not true if $f$ is not assumed of finite type. For example, $f^{an}$ can be a closed immersion without $f$ being one. It is enough to take $X$ to be the sum of $\mathbb{Z}$ copies of $\mathrm{Spec}\mathbb{C}$, $Y$ to be the affine line, and $f$ the morphism obtained by sending the points of $X$ to distinct points of $Y$ forming a discrete subset.

4. Cohomological Comparison Theorems and Existence Theorems

The purpose of this number is to reprove the results of [XII.3, no. 2, ths. 5 and 6]. These generalize to the case of a proper scheme the theorems established in [XII.10, no. 12] when $X$ is projective, and extend them to the relative case. More general results, concerning relative proper schemes over an analytic space, are proved in [XII.7, ch. VIII, no. 3].

Recall that the Čech cohomology used in [XII.10, no. 12] coincides with the usual cohomology in the algebraic case as well as in the analytic case (EGA III 1.4.1 and [XII.5, II 5.10]).

4.1.

Let $f:X\to Y$ be a morphism of $\mathbb{C}$-schemes locally of finite type, and consider the commutative diagram

X^an --φ→ X
 |        |
f^an     f
 |        |
Y^an --ψ→ Y.

If $F$ is an ${\mathcal{O}}_X$-module, then for every integer $p\ge 0$ one has morphisms

Rᵖf_*F --i→ Rᵖf_*(φ_*F^an) --j→ Rᵖ(f·φ)_*F^an --k→ ψ_*(Rᵖf_*^an F^an),

where $i$ is deduced from the canonical morphism $F\to {\varphi }_{\ast }{F}^{an}$, and $j$, $k$ are “edge homomorphisms” of Leray spectral sequences. With the composite $k\cdot j\cdot i$ there is associated a canonical morphism

$$(\mathrm{4.1.1}){\theta }_p:(R^p{f}_{\ast }F{)}^{an}\to R^p{f}_{\ast }^{an}(F^{an}).$$

Theorem.

Let

$f:X\to Y$ be a proper morphism of $\mathbb{C}$-schemes locally of finite type, and let $F$ be a coherent ${\mathcal{O}}_X$-module. Then, for every integer $p\ge 0$, the morphism (4.1.1)

$${\theta }_p:(R^p{f}_{\ast }F{)}^{an}\to R^p{f}_{\ast }^{an}(F^{an})$$

is an isomorphism.

1. The case where $f$ is projective. The proof is analogous to that of [XII.10, no. 13]. Let us recall it briefly. One reduces to the case where $X$ is a projective space of type ${\mathbb{P}}_Y^r$ over $Y$. Let $\mathcal{Y}=Y^{an}$ and $\mathcal{P}={\mathbb{P}}_{\mathcal{Y}}^r$. One first proves that

f_*^an 𝒪_𝓟 = 𝒪_𝓨,     Rᵖf_*^an(𝒪_𝓟) = 0 for p > 0.

To verify the preceding relations, one may reduce to the case where $\mathcal{Y}$ is a ball $\mathcal{B}$ in an affine space ${\mathcal{E}}^n$. One considers the “standard covering” ${\mathcal{U}}_i$ of $\mathcal{P}$ by $r+1$ open subsets isomorphic to $\mathcal{B}\times {\mathcal{E}}^r$. Since these opens are Stein, one has, for every integer $p\ge 0$, isomorphisms

$$H^p({\mathcal{U}}_i,{\mathcal{O}}_{\mathcal{P}})\simeq H^p(\mathcal{P},{\mathcal{O}}_{\mathcal{P}}).$$

One can then express the sections of the structural sheaf ${\mathcal{O}}_{\mathcal{P}}$ on the opens ${\mathcal{U}}_i$ and on their intersections in terms of Laurent series. An easy calculation proves that

H⁰(𝓟,𝒪_𝓟) ≃ H⁰(𝓨,𝒪_𝓨),     Hᵖ(𝓟,𝒪_𝓟) = 0 for p > 0.

The proof is then completed by copying [XII.10, no. 12, lemma 5], with cohomology groups replaced by cohomology sheaves.

The case where $f$ is proper. One uses EGA III 3.1.2 to reduce to the projective case. Let $\mathcal{K}$ be the category of coherent ${\mathcal{O}}_X$-modules such that ${\theta }_p$ is an isomorphism for every $p\ge 0$. It is enough to prove that, for every exact sequence $0\to F^{\prime }\to F\to F^{\prime \prime }\to 0$ whose two terms are in $\mathcal{K}$, the third is also in $\mathcal{K}$; that a direct factor of an object of $\mathcal{K}$ is in $\mathcal{K}$; and that, for every point $x$ of $X$, one can find an object $F$ of $\mathcal{K}$ such that $F_x\ne 0$.

The first condition follows by applying the five lemma to the following commutative diagram, whose rows are exact:

… → (Rᵖf_*F′)^an → (Rᵖf_*F)^an → (Rᵖf_*F″)^an → (Rᵖ⁺¹f_*F′)^an → …
      ↓                ↓                 ↓                   ↓
… → Rᵖf_*^an F′^an → Rᵖf_*^an F^an → Rᵖf_*^an F″^an → Rᵖ⁺¹f_*^an F′^an → …

The second condition is verified analogously.

To verify the third condition, one may restrict to the case where $X$ is an irreducible scheme with generic point $x$. We could have supposed $Y$ noetherian from the beginning. By Chow’s lemma (EGA II 5.6.1), one can find a projective $Y$-scheme $X^{\prime }$ and a projective surjective morphism $g:X^{\prime }\to X$. On the other hand, there exists an integer $n$ such that $R^p{g}_{\ast }({\mathcal{O}}_X^{\prime }(n))=0$ for all $p>0$ and such that the canonical morphism $g\ast g_{\ast }({\mathcal{O}}_X^{\prime }(n))\to {\mathcal{O}}_X^{\prime }(n)$ is surjective (EGA III 2.2.1). If one puts $F=g_{\ast }({\mathcal{O}}_X^{\prime }(n))$, the sheaf $F$ answers the question. Indeed $F_x\ne 0$; moreover, the Leray spectral sequence

$$R^p{f}_{\ast }(R_{\ast }^{qg}({\mathcal{O}}_X^{\prime }(n)))\Rightarrow R^{p+q}(f\cdot g{)}_{\ast }({\mathcal{O}}_X^{\prime }(n))$$

is degenerate, so one has an isomorphism

$$R^p{f}_{\ast }F\simeq R^p(f\cdot g{)}_{\ast }({\mathcal{O}}_X^{\prime }(n)).$$

As in the algebraic case, one has a canonical isomorphism

$$R^p{f}_{\ast }^{an}{F}^{an}\simeq R^p(f\cdot g{)}_{\ast }^{an}({\mathcal{O}}_X^{\prime }(n{)}^{an}),$$

and the diagram

$$(R^p{f}_{\ast }F{)}^{an}\simeq (R^p(f\cdot g{)}_{\ast }({\mathcal{O}}_X^{\prime }(n)){)}^{an}\downarrow {\theta }_p\downarrow {\psi }_p{R}^p{f}_{\ast }^{an}{F}^{an}\simeq R^p(f\cdot g{)}_{\ast }^{an}({\mathcal{O}}_X^{\prime }(n{)}^{an})$$

is commutative. By 1, ${\psi }_p$ is an isomorphism; hence ${\theta }_p$ is also an isomorphism. This completes the proof.

Corollary.

Let $X$ be a proper $\mathbb{C}$-scheme, and let $F$ be a coherent ${\mathcal{O}}_X$-module. Then, for every integer $p\ge 0$, the canonical morphism

$$H^p(X,F)\to H^p(X^{an},F^{an})$$

is an isomorphism.

Theorem.

Let $X$ be a proper $\mathbb{C}$-scheme. The functor which associates to every coherent ${\mathcal{O}}_X$-module $F$ its inverse image $F^{an}$ on $X^{an}$ is an equivalence of categories.

1. The functor is fully faithful. Indeed, let $F$ and $G$ be two coherent ${\mathcal{O}}_X$-modules. The canonical morphism

Hom_𝒪_X(F,G) → Hom_𝒪_X^an(F^an,G^an)

identifies with the canonical morphism

H⁰(X,SheafHom_𝒪_X(F,G)) → H⁰(X^an,SheafHom_𝒪_X(F,G))

(EGA 0_I 6.7.6). Since SheafHom_𝒪_X(F,G) is coherent, it follows from XII.4.3 that this morphism is bijective.

1. The functor is essentially surjective. When $X$ is projective, the assertion follows from [XII.10, no. 12, th. 3]. The general case reduces to the preceding one by using Chow’s lemma (EGA II 5.6.1). Indeed, let $X^{\prime }$ be a projective $\mathbb{C}$-scheme, let $f:X^{\prime }\to X$ be a projective surjective morphism, and let $U$ be a dense open subset of $X$ such that $f$ induces an isomorphism $f^{-1}(U)\simeq U$. We reason by noetherian induction on $X$; hence we may suppose that for every coherent sheaf $\mathcal{G}$ on $X^{an}$ for which one can find a closed subset $Y$ of $X$, distinct from $X$, satisfying $Y^{an}\supset Supp\mathcal{G}$, there exists a coherent sheaf $G$ on $X$ such that $G^{an}\simeq \mathcal{G}$.

Let $\mathcal{F}$ be a coherent sheaf of modules over ${\mathcal{O}}_X^{an}$, and let $\mathcal{K}$ and $\mathcal{L}$ be the coherent sheaves defined by requiring the sequence

0 → 𝓚 → 𝓕 → f_*^an f^an*𝓕 → 𝓛 → 0

to be exact. Since $X^{\prime }$ is projective, there exists a coherent ${\mathcal{O}}_X^{\prime }$-module $F^{\prime }$ such that $F^{\prime an}\simeq f^{an}\ast \mathcal{F}$. From XII.4.2 one then deduces an isomorphism $(f_{\ast }{F}^{\prime }{)}^{an}\simeq f_{\ast }^{an}{f}^{an}\ast \mathcal{F}$. Since $\mathcal{K}|U^{an}$ and $\mathcal{L}|U^{an}$ are zero, there exist coherent ${\mathcal{O}}_X$-modules $K$ and $L$ such that $K^{an}\simeq \mathcal{K}$ and $L^{an}\simeq \mathcal{L}$. By 1, the morphism $f_{\ast }^{an}{f}^{an}\ast \mathcal{F}\to \mathcal{L}$ comes from a unique morphism $f_{\ast }{F}^{\prime }\to L$. Let $I=Ker(f_{\ast }{F}^{\prime }\to L)$. The sheaf $\mathcal{F}$ is then an extension of $I^{an}$ by $K^{an}$, and it remains only to see that this extension comes

by inverse image from an extension of $I$ by $K$. It is therefore enough to prove that the canonical morphism

(*)   Ext^q_𝒪_X(I,K)^an ≃ Ext^q_𝒪_X^an(I^an,K^an),     q = 1,

is bijective. [Translator note: the source prints “$q\ne 1$,” but the preceding sentence shows that the needed case is $q=1$; this is a mathematical correction rather than a change of argument.] Now one has isomorphisms

ExtSheaf^q_𝒪_X(I,K)^an ≃ ExtSheaf^q_𝒪_X^an(I^an,K^an)

for every integer $q\ge 0$ (EGA $0_{III}$ 12.3.5), and a morphism of spectral sequences

Hᵖ(X,ExtSheaf^q_𝒪_X(I,K))              ⇒ Ext^(p+q)_𝒪_X(I,K)
     ↓                                             ↓
Hᵖ(X^an,ExtSheaf^q_𝒪_X^an(I^an,K^an)) ⇒ Ext^(p+q)_𝒪_X^an(I^an,K^an).

This morphism is an isomorphism because, by XII.4.3, it is so on the $E_2^{p,q}$-terms; this proves the bijectivity of (*).

Corollary.

The functor which associates $X^{an}$ to every proper $\mathbb{C}$-scheme $X$ is fully faithful.

We must show that, if $X$ and $Y$ are two proper $\mathbb{C}$-schemes, the canonical map

$${\mathrm{Hom}}_{\mathbb{C}}(X,Y)\to \mathrm{Hom}(X^{an},Y^{an})$$

is bijective. But to give a morphism from $X$ to $Y$, respectively from $X^{an}$ to $Y^{an}$, is equivalent to giving its graph, that is, a closed subscheme $Z$ of $X\times Y$, respectively a closed analytic subspace $\mathcal{Z}$ of $X^{an}\times Y^{an}$, such that the restriction of the first projection $X\times Y\to X$ to $Z$, respectively of $X^{an}\times Y^{an}\to X^{an}$ to $\mathcal{Z}$, is

an isomorphism. Since giving a closed subscheme of $X\times Y$, respectively a closed analytic subspace of $X^{an}\times Y^{an}$, is equivalent to giving a coherent sheaf of ideals on ${\mathcal{O}}_{X\times Y}$, respectively on ${\mathcal{O}}_{X^{an}\times Y^{an}}$, the corollary follows from XII.4.4.

Corollary.

Let $X$ be a proper $\mathbb{C}$-scheme. The functor which associates $X^{\prime an}$ to every finite, respectively finite étale, scheme $X^{\prime }$ over $X$ is an equivalence from the category of finite, respectively finite étale, schemes over $X$ to the category of finite, respectively finite étale, analytic spaces over $X^{an}$.

Indeed, to give a finite morphism $X^{\prime }\to X$, respectively $X^{\prime an}\to X^{an}$, is equivalent to giving a coherent sheaf of algebras over ${\mathcal{O}}_X$, respectively over ${\mathcal{O}}_X^{an}$ [XII.4, no. 19, §5, th. 2]. The corollary therefore follows from XII.4.4 in the non-respective case, and the respective case follows from it in view of XII.3.1 (iii).

5. Comparison Theorems for Étale Coverings

5.0.

Let us make precise the notion of a finite covering of an analytic space. If $\mathcal{X}$ is an analytic space, an analytic space ${\mathcal{X}}^{\prime }$ finite over $\mathcal{X}$ is called a finite covering of $\mathcal{X}$ if every irreducible component of ${\mathcal{X}}^{\prime }$ dominates an irreducible component of $\mathcal{X}$.

Theorem (“Riemann existence theorem”).

Let $X$ be a $\mathbb{C}$-scheme locally of finite type, and let $X^{an}$ be the analytic space associated with $X$. The functor $\Psi$ which associates $X^{\prime an}$ to every finite étale covering $X^{\prime }$ of $X$ is an equivalence

from the category of finite étale coverings of $X$ to the category of finite étale coverings of $X^{an}$.

1. The functor $\Psi$ is fully faithful. Let $X^{\prime }$ and $X^{\prime \prime }$ be two finite étale coverings of $X$, and let us prove that the canonical map

$$(\ast ){\mathrm{Hom}}_X(X^{\prime },X^{\prime \prime })\to {\mathrm{Hom}}_X^{an}(X^{\prime an},X^{\prime \prime an})$$

is bijective. We may suppose $X^{\prime }$ connected. To give an $X$-morphism from $X^{\prime }$ to $X^{\prime \prime }$ is equivalent to giving a connected component $X_i$ of $X^{\prime }{\times }_X{X}^{\prime \prime }$ such that the morphism $X_i\to X^{\prime }$ induced by the first projection is an isomorphism. Since the connected components of $X^{\prime }{\times }_X{X}^{\prime \prime }$ correspond bijectively to the connected components of $X^{\prime an}{\times }_{X^{an}}{X}^{\prime \prime an}$ (XII.2.6), and since a morphism $X_i\to X^{\prime }$ is an isomorphism if and only if $X_i^{an}\to X^{\prime an}$ is one, this proves the bijectivity of (*).

1. The functor $\Psi$ is essentially surjective. Let ${\mathcal{X}}^{\prime }$ be a finite étale covering of $X^{an}$, and let us prove that there exists an étale covering $X^{\prime }$ of $X$ such that one has an isomorphism $X^{\prime an}\simeq {\mathcal{X}}^{\prime }$. In view of 1, the question is local on $X$, so we may suppose $X$ affine.

a. Reduction to the case where $X$ is normal. We may suppose $X$ reduced. Indeed, suppose the theorem proved for $X_{red}$. The functor which associates to a finite étale covering $X^{\prime }$ of $X$ the finite étale covering $X_{red}^{\prime an}$ of $X_{red}^{an}$ is then an equivalence. Since it is obtained by composing $\Psi$ with the functor $\Theta$ which associates to a finite étale covering of $X^{an}$ its inverse image on $X_{red}^{an}$, and since $\Theta$ is fully faithful, this shows that $\Psi$ is an equivalence of categories.

We

may suppose $X$ normal. Indeed, let $X$ be the normalization of $X$, and let $p:X\to X$ be the canonical morphism. Since $p$ is finite, $p$ is a morphism of effective descent for the category of étale coverings (IX.4.7). Supposing the theorem proved for $X$, put ${\mathcal{X}}^{\prime }={\mathcal{X}}^{\prime }{\times }_{X^{an}}{X}^{an}$. There exists an étale covering $X^{\prime }$ of $X$ and an isomorphism $X^{\prime an}\simeq {\mathcal{X}}^{\prime }$. It then follows from 1 that the natural descent datum on ${\mathcal{X}}^{\prime }$ lifts to a descent datum on $X^{\prime }$ relative to $X\to X$. This proves the existence of an étale covering $X^{\prime }$ of $X$ such that one has an isomorphism

i: X′^an ×_{X^an} X̃^an ≃ 𝓧̃′,

whose inverse images by the two projections from $X^{an}{\times }_{X^{an}}{X}^{an}$ are the same. By IX.3.2, whose proof is valid in the analytic case, the morphism ${\tilde{X}}^{an}\to X^{an}$ is a morphism of descent for the category of étale coverings, and consequently $i$ comes from an isomorphism $X^{\prime an}\simeq {\mathcal{X}}^{\prime }$.

b. Reduction to the case where $X$ is regular. Let $U$ be the open subset of regular points of $X$, and let $i:U\to X$ and $i^{an}:U^{an}\to X^{an}$ be the canonical morphisms. Since $X$ is normal, one has $codim(X-U,X)\ge 2$. Suppose that there exists an étale covering $U^{\prime }$ of $U$ such that $U^{\prime an}\simeq {\mathcal{X}}^{\prime }|U^{an}$, and let us show that $U^{\prime }$ then extends to an étale covering $X^{\prime }$ of $X$ such that $X^{\prime an}\simeq {\mathcal{X}}^{\prime }$. It is enough to see that $U^{\prime }$ extends to an étale covering $X^{\prime }$ of $X$. Indeed, one will then have an isomorphism $X^{\prime an}|U^{an}\simeq {\mathcal{X}}^{\prime }|U^{an}$; but if $\mathcal{F}$ and $\mathcal{G}$ are the coherent sheaves of algebras on ${\mathcal{O}}_X^{an}$ defining respectively ${\mathcal{X}}^{\prime }$ and $X^{\prime an}$, the fact that $X$ is normal and that $codim(X-U,X)\ge 2$ implies that the canonical morphisms

𝓕 → i_*^an(𝓕|U^an),     𝓖 → i_*^an(𝓖|U^an)

are

isomorphisms [XII.11, no. 3, prop. 4]. It follows that $\mathcal{F}$ and $\mathcal{G}$, and hence also $X^{\prime an}$ and ${\mathcal{X}}^{\prime }$, are isomorphic.

Let $\varphi :X^{an}\to X$ be the canonical morphism. Since the problem of extending $U^{\prime }$ to $X$ is local on $X$, it is enough to prove that, for every point $y$ of $X^{an}-U^{an}$, the étale covering

U′_φ(y) = U′ ×_X Spec 𝒪_X,φ(y)

of

U_φ(y) = U ×_X Spec 𝒪_X,φ(y)

extends to $\mathrm{Spec}{\mathcal{O}}_X,\varphi (y)$. Let $H$ be the coherent ${\mathcal{O}}_U$-algebra defining $U^{\prime }$. The canonical morphism

α: (i_*H)^an → i_*^an(H^an) = 𝓕

defines a morphism of sheaves of modules on $\mathrm{Spec}{\mathcal{O}}_X^{an},y$:

$${\alpha }_y:(i_{\ast }H{)}_y^{an}\to {\mathcal{F}}_y,$$

whose restriction to

U_y = U_φ(y) ×_{Spec 𝒪_X,φ(y)} Spec 𝒪_X^an,y

is an isomorphism. But this proves that $H|U_y$ is trivial, hence that $U_{\varphi }^{\prime }(y)$ extends to $\mathrm{Spec}{\mathcal{O}}_X,\varphi (y)$.

c. The case where $X$ is affine regular. Let

$$j:X\to P$$

be a compactification of $X$, where $P$ is a projective $\mathbb{C}$-scheme and $j$ is a dominant open immersion. By the resolution of singularities theorem [XII.8], one can find a regular scheme $R$ and a projective morphism $r:R\to P$, such that $r$ induces an isomorphism $r^{-1}(X)\simeq X$ and such that $r^{-1}(X)$ is the complement in $R$ of a divisor with normal crossings. Let

$$k:X\to R$$

be the canonical immersion. We shall show that there exists a finite normal covering ${\mathcal{R}}^{\prime }$ of $R^{an}$, in the sense of XII.5.0, which extends the étale covering $X^{\prime an}$. By Proposition XII.5.3 below, such a covering is unique; the problem of extending $X^{\prime an}$ is therefore local on $R^{an}$ near $R^{an}-X^{an}$. But each point of $R^{an}-X^{an}$ has an open neighborhood $\mathcal{V}$ isomorphic to a ball in an affine space ${\mathcal{E}}^n$, such that $\mathcal{V}-\mathcal{V}\cap X^{an}$ is defined by the vanishing of the first $p$ coordinate functions $z_1,\cdots ,z_p$, with $0\le p\le n$. The fundamental group of $\mathcal{U}=\mathcal{V}\cap X^{an}$ is isomorphic to ${\mathbb{Z}}^p$, and every étale covering of $\mathcal{U}$ is a quotient of a covering of the form

𝓤″ = 𝓤[T₁,…,T_p]/(T₁ⁿ¹ − z₁, …, T_pⁿᵖ − z_p),

where the $n_i$ are integers > 0, by a subgroup $H$ of the Galois group $\mathbb{Z}/n_1\mathbb{Z}\times \cdots \times \mathbb{Z}/n_p\mathbb{Z}$ of ${\mathcal{U}}^{\prime \prime }$. But ${\mathcal{U}}^{\prime \prime }$ extends to the regular covering

𝓥″ = 𝓥[T₁,…,T_p]/(T₁ⁿ¹ − z₁, …, T_pⁿᵖ − z_p)

of $\mathcal{V}$ on which $H$ acts, and the quotient of ${\mathcal{V}}^{\prime \prime }$ by $H$ is the desired extension.

The proof is then completed by XII.4.6. The covering ${\mathcal{R}}^{\prime }$ comes from a finite covering $R^{\prime }$ of $R$; the restriction of $R^{\prime }$ to $X$ is a covering $X^{\prime }$ of $X$ such that $X^{\prime an}\simeq {\mathcal{X}}^{\prime }$, and by XII.3.1 (iii), $X^{\prime }$ is an étale covering of $X$.

Corollary.

Let

$X$ be a connected $\mathbb{C}$-scheme locally of finite type, let $\varphi :X^{an}\to X$ be the canonical morphism, and let $x$ be a point of $X^{an}$. Let ${\pi }_1(X^{an},x)$ be the fundamental group of the topological space $X^{an}$ at $x$, and let ${\pi }_1(X,\varphi (x))$ be the fundamental group of the scheme $X$ at $\varphi (x)$ (V.7). Then ${\pi }_1(X,\varphi (x))$ is canonically isomorphic to the completion of ${\pi }_1(X^{an},x)$ for the topology of subgroups of finite index.

Indeed, let $\mathcal{C}$ be the category of finite étale coverings of $X^{an}$, let $F$ be the functor from $\mathcal{C}$ to Sets which associates to every finite étale covering ${\mathcal{X}}^{\prime }$ of $X^{an}$ the set of points of ${\mathcal{X}}^{\prime }$ above $x$, and let ${\hat{\pi }}_1(X^{an},x)$ be the profinite group associated with $\mathcal{C}$ and $F$ as in V.4. Since every finite étale covering of $X^{an}$ is a quotient of the universal covering by a subgroup of finite index, ${\hat{\pi }}_1(X^{an},x)$ is nothing other than the completion of ${\pi }_1(X^{an},x)$ for the topology of subgroups of finite index. The corollary therefore follows from XII.5.1 and V.6.10.

Proposition.

Let $\mathcal{X}$ be a normal analytic space, and let $\mathcal{Y}$ be a closed analytic subset such that $\mathcal{U}=\mathcal{X}-\mathcal{Y}$ is dense in $\mathcal{X}$. Then the functor which associates to every normal finite covering ${\mathcal{X}}^{\prime }$ of $\mathcal{X}$, in the sense of XII.5.0, its restriction to $\mathcal{U}$ is fully faithful.

Let ${\mathcal{X}}^{\prime }$ and ${\mathcal{X}}^{\prime \prime }$ be two finite normal coverings of $\mathcal{X}$. We must show that the canonical map

$${\mathrm{Hom}}_{\mathcal{X}}({\mathcal{X}}^{\prime },{\mathcal{X}}^{\prime \prime })\to {\mathrm{Hom}}_{\mathcal{U}}({\mathcal{X}}^{\prime }|\mathcal{U},{\mathcal{X}}^{\prime \prime }|\mathcal{U})$$

is bijective. Let $u$, $v$ be two $\mathcal{X}$-morphisms from ${\mathcal{X}}^{\prime }$ to ${\mathcal{X}}^{\prime \prime }$ whose restrictions to $\mathcal{U}$ are the same, and let us prove that $u=v$. The morphisms $u$ and

$v$ coincide on the dense open $\mathcal{U}{\times }_{\mathcal{X}}{\mathcal{X}}^{\prime }$, hence on the underlying topological spaces. By [XII.4, no. 19, §4, cor. 5], this proves $u=v$.

Let now $u$ be a $\mathcal{U}$-morphism from ${\mathcal{X}}^{\prime }|\mathcal{U}$ to ${\mathcal{X}}^{\prime \prime }|\mathcal{U}$, and let us show that it extends to all of ${\mathcal{X}}^{\prime }$. We may suppose ${\mathcal{X}}^{\prime }$ regular. Indeed, since ${\mathcal{X}}^{\prime }$ is normal, one can find an open subset $\mathcal{V}$ of $\mathcal{X}$ whose complement is an analytic subset of codimension $\ge 2$, such that ${\mathcal{X}}^{\prime }{\times }_{\mathcal{X}}\mathcal{V}={\mathcal{V}}^{\prime }$ is regular. Let ${\mathcal{V}}^{\prime \prime }={\mathcal{X}}^{\prime \prime }{\times }_{\mathcal{X}}\mathcal{V}$, and suppose the proposition proved for $\mathcal{V}$. Consider the commutative diagram

𝓥′  → 𝓧′
 ↓      ↓
𝓥   → 𝓧

and similarly 𝓥″ → 𝓧″ over 𝓥 → 𝓧.

With $u$ there is associated a morphism of ${\mathcal{O}}_{\mathcal{V}}$-algebras

$$g_{\ast }^{\prime \prime }{\mathcal{O}}_{\mathcal{V}}^{\prime \prime }\to g_{\ast }^{\prime }{\mathcal{O}}_{\mathcal{V}}^{\prime },$$

from which one deduces a morphism

$$i_{\ast }{g}_{\ast }^{\prime \prime }{\mathcal{O}}_{\mathcal{V}}^{\prime \prime }\to i_{\ast }{g}_{\ast }^{\prime }{\mathcal{O}}_{\mathcal{V}}^{\prime }.$$

Taking into account the isomorphisms $i_{\ast }^{\prime }{\mathcal{O}}_{\mathcal{V}}^{\prime }\simeq {\mathcal{O}}_{\mathcal{X}}^{\prime }$ and $i_{\ast }^{\prime \prime }{\mathcal{O}}_{\mathcal{V}}^{\prime \prime }\simeq {\mathcal{O}}_{\mathcal{X}}^{\prime \prime }$ [XII.11, no. 3, prop. 4], one deduces a morphism of ${\mathcal{O}}_{\mathcal{X}}$-algebras

$$f_{\ast }^{\prime \prime }{\mathcal{O}}_{\mathcal{X}}^{\prime \prime }\to f_{\ast }^{\prime }{\mathcal{O}}_{\mathcal{X}}^{\prime },$$

hence the desired morphism ${\mathcal{X}}^{\prime }\to {\mathcal{X}}^{\prime \prime }$.

We

now suppose ${\mathcal{X}}^{\prime }$ regular. Let ${\mathcal{U}}^{\prime }=\mathcal{U}{\times }_{\mathcal{X}}{\mathcal{X}}^{\prime }$ and ${\mathcal{Y}}^{\prime }={\mathcal{X}}^{\prime }-{\mathcal{U}}^{\prime }$. We regard ${\mathcal{Y}}^{\prime }$ as a reduced analytic subspace of ${\mathcal{X}}^{\prime }$. If ${\mathcal{Y}}_1^{\prime }$ is the singular closed subset of ${\mathcal{Y}}^{\prime }$, then $\dim {\mathcal{Y}}_1^{\prime }<\dim {\mathcal{Y}}^{\prime }$ [XII.4, no. 20 D, th. 3]. Thus, by induction on the dimension of ${\mathcal{Y}}^{\prime }$, one may suppose ${\mathcal{Y}}^{\prime }$ smooth. Since it is enough to extend $u$ to an open neighborhood of each point of ${\mathcal{Y}}^{\prime }$, the implicit function theorem lets us suppose that ${\mathcal{X}}^{\prime }$ is a ball in an affine space ${\mathcal{E}}^n$ and that ${\mathcal{Y}}^{\prime }$ is the closed subset defined by the vanishing of the first $p$ coordinate functions $z_1,\cdots ,z_p$, with $0\le p\le n$.

To $u$ one associates a section $s$ of

p: 𝓧′ ×_𝓧 𝓧″ → 𝓧′

above ${\mathcal{U}}^{\prime }$. After restricting ${\mathcal{X}}^{\prime }$ if necessary, one may suppose that $p_{\ast }({\mathcal{O}}_{{\mathcal{X}}^{\prime }{\times }_{\mathcal{XX}}^{\prime \prime }})$ is generated by elements $x_1,\cdots ,x_q$ of $\Gamma ({\mathcal{X}}^{\prime },p_{\ast }{\mathcal{O}}_{{\mathcal{X}}^{\prime }{\times }_{\mathcal{XX}}^{\prime \prime }})$. Let $u_1,\cdots ,u_q\in \Gamma ({\mathcal{U}}^{\prime },{\mathcal{O}}_{\mathcal{X}}^{\prime })$ be the images by $s$ of $x_1|{\mathcal{U}}^{\prime },\cdots ,x_q|{\mathcal{U}}^{\prime }$. Saying that $s$ extends to ${\mathcal{X}}^{\prime }$ is equivalent to saying that $u_1,\cdots ,u_q$ extend to sections of $\Gamma ({\mathcal{X}}^{\prime },{\mathcal{O}}_{\mathcal{X}}^{\prime })$. But, since $f$ is finite, each $u_i$ is a Laurent series in $z_1,\cdots ,z_p$ with coefficients that are convergent power series in $z_{p+1},\cdots ,z_n$, and satisfies integral-dependence relations. It follows that $u_i$ is bounded, hence is a convergent power series in $z_1,\cdots ,z_n$, and therefore extends to ${\mathcal{X}}^{\prime }$.

One may ask whether the functor introduced in XII.5.3 is an equivalence of categories. This question has an answer by the theorem of Grauert-Remmert [XII.6], for which we give below a proof using resolution of singularities. One could also have used the Grauert-Remmert theorem to prove XII.5.1; that was what was done before [XII.8] was available.

Theorem (Grauert-Remmert theorem).

Let

$\mathcal{X}$ be a normal analytic space, and let $\mathcal{Y}$ be a closed analytic subset such that $\mathcal{U}=\mathcal{X}-\mathcal{Y}$ is dense in $\mathcal{X}$. Let ${\mathcal{U}}^{\prime }$ be a normal finite covering of $\mathcal{U}$. Suppose that there exists a rare closed analytic subset $\mathcal{S}$ of $\mathcal{X}$ such that the restriction of ${\mathcal{U}}^{\prime }$ to $\mathcal{U}-\mathcal{U}\cap \mathcal{S}$ is étale. Then there exists a normal finite covering ${\mathcal{X}}^{\prime }$ of $\mathcal{X}$ extending ${\mathcal{U}}^{\prime }$, and ${\mathcal{X}}^{\prime }$ is unique up to isomorphism.

Uniqueness follows from XII.5.3. The problem of extending ${\mathcal{U}}^{\prime }$ is therefore local on $\mathcal{X}$. We may suppose $\mathcal{U}$ regular and ${\mathcal{U}}^{\prime }$ étale over $\mathcal{U}$. Indeed, the set of regular points of $\mathcal{U}$ is a dense open subset $\mathcal{V}$ of $\mathcal{X}$ whose complement is an analytic subset [XII.4, no. 20 D, th. 2], and it is enough to replace $\mathcal{U}$ by the open subset $\mathcal{V}-\mathcal{V}\cap \mathcal{S}$.

Let $y$ be a point of $\mathcal{X}-\mathcal{U}$ and let us show that one can extend ${\mathcal{U}}^{\prime }$ to a neighborhood of $y$. After restricting $\mathcal{X}$ to an open neighborhood of $y$, it follows from the resolution of singularities theorem [XII.8] that one can find a regular analytic space ${\mathcal{X}}_1$ and a projective morphism $f:{\mathcal{X}}_1\to \mathcal{X}$ inducing by restriction to $\mathcal{U}$ an isomorphism ${\mathcal{U}}_1=f^{-1}(\mathcal{U})\simeq \mathcal{U}$, such that ${\mathcal{U}}_1$ is the complement in ${\mathcal{X}}_1$ of a divisor with normal crossings. Let us show that ${\mathcal{U}}^{\prime }$ extends to a normal finite covering of ${\mathcal{X}}_1$. Since the question is local on ${\mathcal{X}}_1$, one may suppose that ${\mathcal{X}}_1$ is a ball in an affine space ${\mathcal{E}}^n$ and that ${\mathcal{X}}_1-{\mathcal{U}}_1$ is defined by the vanishing of the first $p$ coordinate functions $z_1,\cdots ,z_p$, with $0\le p\le n$. [Translator note: the corrected source fixes $z_q$ to $z_p$ in this list.] The étale covering ${\mathcal{U}}^{\prime }$ of ${\mathcal{U}}_1$ is a quotient of a covering of the form

𝓤₂ = 𝓤₁[T₁,…,T_p]/(T₁ⁿ¹ − z₁, …, T_pⁿᵖ − z_p)

by a subgroup $H$ of the Galois group of ${\mathcal{U}}_2$. The covering ${\mathcal{U}}_2$ extends

to the covering

𝓧₂ = 𝓧₁[T₁,…,T_p]/(T₁ⁿ¹ − z₁, …, T_pⁿᵖ − z_p)

of ${\mathcal{X}}_1$ on which $H$ acts, and ${\mathcal{X}}_2/H$ extends ${\mathcal{U}}^{\prime }$ to ${\mathcal{X}}_1$.

Let ${\mathcal{X}}_1^{\prime }$ denote the normal finite covering of ${\mathcal{X}}_1$ extending ${\mathcal{U}}^{\prime }$, and let ${\mathcal{F}}_1$ be the coherent 𝒪_𝓧₁-algebra defining ${\mathcal{X}}_1^{\prime }$. By the finiteness theorem of Grauert-Remmert [XII.4, no. 15, th. 1.1], $f_{\ast }{\mathcal{F}}_1$ is a coherent ${\mathcal{O}}_{\mathcal{X}}$-algebra. It therefore corresponds to a finite covering ${\mathcal{X}}^{\prime }$ of $\mathcal{X}$, which is normal since ${\mathcal{X}}_1^{\prime }$ is, and ${\mathcal{X}}^{\prime }$ is the desired extension of ${\mathcal{U}}^{\prime }$.

Remark.

In the statement XII.5.4, one cannot remove the hypothesis on the locus of points where the morphism ${\mathcal{U}}^{\prime }\to \mathcal{U}$ is not étale. For example, let $\mathcal{X}$ be the unit disk in the complex plane, let $\mathcal{U}$ be the complement of the origin in $\mathcal{X}$, and let

$${\mathcal{U}}^{\prime }=\mathcal{U}[T]/(T^2-\sin (1/z)),$$

where $z$ is the coordinate function on $\mathcal{X}$. Then ${\mathcal{U}}^{\prime }$ is a normal finite covering of $\mathcal{U}$ which does not extend to $\mathcal{X}$. Indeed, suppose ${\mathcal{U}}^{\prime }$ extended to a finite covering ${\mathcal{X}}^{\prime }$ of $\mathcal{X}$. The locus of points of $\mathcal{X}$ where the morphism ${\mathcal{X}}^{\prime }\to \mathcal{X}$ is not étale would then be a closed analytic subset containing all points $z$ such that $\sin (1/z)=0$, which is absurd.

One can, however, remove the hypothesis on the singular locus of the morphism ${\mathcal{U}}^{\prime }\to \mathcal{U}$ when $codim(\mathcal{X}-\mathcal{U},\mathcal{X})\ge 2$. Indeed, one may suppose $\mathcal{U}$ regular. The locus of points of $\mathcal{U}$ where ${\mathcal{U}}^{\prime }\to \mathcal{U}$ is not étale is a divisor of $\mathcal{U}$, and it follows from the Remmert-Stein theorem [XII.9, th. 3] that it is

the trace on $\mathcal{U}$ of a divisor of $\mathcal{X}$. In this case, if $\mathcal{A}$ is a coherent ${\mathcal{O}}_{\mathcal{U}}$-algebra such that ${\mathcal{U}}^{\prime }={\mathrm{Spec}}_{an}(\mathcal{A})$, and if $i:\mathcal{U}\to \mathcal{X}$ is the canonical morphism, it is enough to take ${\mathcal{X}}^{\prime }={\mathrm{Spec}}_{an}(i_{\ast }\mathcal{A})$; indeed one knows that $i_{\ast }\mathcal{A}$ is coherent [XII.11, no. 1, th. 1]. [Translator note: the corrected source fixes the adjective “coherent” to agree with i_*𝓐.]

Remark (M. Raynaud, added in 2003).

There exist nontrivial finitely presented groups $G$ which have no subgroups of finite index distinct from $G$, for example G. Higman’s group; cf. J.-P. Serre, Arbres et amalgames, Astérisque no. 46, prop. 6, ch. I, §1. Consequently, if such a group is realized as the topological fundamental group of a scheme $V$ over $\mathbb{C}$, say smooth and projective, the topological space $V_{top}$ underlying $V$ is not simply connected, but $V$ is algebraically simply connected. At present no such $V$ is known. Let us nevertheless mention that D. Toledo constructed smooth projective schemes $V$ over $\mathbb{C}$ whose topological fundamental group is not separated for the topology of subgroups of finite index; the natural morphism ${\pi }_1(V_{top})\to {\pi }_1(V_{alg})$ is not injective. [D. Toledo, “Projective varieties with non-residually finite fundamental group,” Publ. Math. IHÉS 77 (1993), pp. 103-119.]

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