Exposé X. Theory of Specialization of the Fundamental Group

In the present exposé, we restrict ourselves to the study of the fundamental group of geometric fibers in a proper morphism, that is, of the fundamental group of a variable proper algebraic scheme. In a later exposé, we shall generalize the technique used here to étale coverings tamely ramified “at infinity.” This will give, for example, a solution of the “three point problem” in the case of Galois coverings of order prime to the characteristic, that is, a determination of the Galois coverings of the line ${\mathbb{P}}_k^1$ ramified at most at three given points and tamely ramified at those points, together with its evident variants.

1. The Homotopy Exact Sequence for a Proper and Separable Morphism

Definition.

A prescheme $X$ over a field $k$ is called separable, or separable over $k$, if for every extension $K$ of $k$, $X{\otimes }_{k}K$ is reduced. If $f:X\to Y$ is a morphism of preschemes, one says that $f$ is separable, or that $X$ is separable over $Y$, if $X$ is flat over $Y$ and if for every $y\in Y$, the fiber $X{\otimes }_Y\kappa (y)$ is separable over $\kappa (y)$.

If $X$ is a prescheme over a field $k$, to say that it is separable also means that it is reduced, and that the fields $\kappa (x)$, for $x$ the generic point of an irreducible component of $X$, are separable extensions of $k$. If $k$ is perfect, this is therefore the same as saying that $X$ is reduced.

Notice that if $X$ is separable over $Y$, then for every base change $Y^{\prime }\to Y$, $X^{\prime }=X{\times }_Y{Y}^{\prime }$ is separable over $Y^{\prime }$. One can also prove, under suitable finiteness hypotheses, that the composite of two separable morphisms is

separable. We shall need this only in the following form: if $X$ is separable over $Y$ and $X^{\prime }$ is étale over $X$, then $X^{\prime }$ is separable over $Y$. This is an immediate consequence of the definitions and I.9.2. Moreover, the hypothesis “separable morphism” will be used through the following proposition:

Proposition.

Let $f:X\to Y$ be a proper and separable morphism, with $Y$ locally noetherian, and consider its Stein factorization $X\to Y^{\prime }\to Y$, where $f_{\ast }^{\prime }({\mathcal{O}}_X)={\mathcal{O}}_{Y^{\prime }}$, with $Y^{\prime }$ finite over $Y$ and isomorphic to the spectrum of the algebra $f_{\ast }({\mathcal{O}}_X)$. Then $Y^{\prime }$ is an étale covering of $Y$.

This proposition will appear in EGA III 7. [Translator note: the source footnote cites EGA III 7.8.10(i).] Let us indicate the principle of the proof. One reduces easily to the case where $Y$ is the spectrum of a complete local ring $A$, and, after making a suitable finite flat extension of the latter corresponding to a suitable residue extension, one may suppose that the connected components of the fiber over the closed point $y$ are geometrically connected. This also means that $H^0(X_y,{\mathcal{O}}_{X_y})$ decomposes as a product of fields identical with $k=\kappa (y)$. Supposing then that $X$ is connected, as one may, one has $H^0(X_y,{\mathcal{O}}_{X_y})=k$, hence the homomorphism $A\to H^0(X_y,{\mathcal{O}}_{X_y})$ is surjective. By a general proposition of Künneth type, one concludes that $f_{\ast }({\mathcal{O}}_X)$ is defined by a module $B$ over $A$ that is free over $A$, and that $B/\mathfrak{m}B\to H^0(X_y,{\mathcal{O}}_{X_y})=k$ is bijective. Thus in the present case $B$ is an étale algebra over $A$, completing the proof.

Theorem.

Let $f:X\to Y$ be a proper and separable morphism, with $Y$ locally noetherian and connected, and suppose $f_{\ast }({\mathcal{O}}_X)={\mathcal{O}}_Y$. This implies that the fibers of $X$ over $Y$ are geometrically connected, and conversely by X.1.2. Let $y$ be a point of $Y$, let $\kappa \bar{y}$ be an algebraic closure of $\kappa (y)$, and let ${\bar{X}}_y=X_y{\otimes }_{\kappa (y)}\kappa \bar{y}$. Finally, let $X^{\prime }$ be a connected étale covering of $X$, and let ${\bar{X}}_y^{\prime }=X_y^{\prime }{\otimes }_{\kappa (y)}\kappa \bar{y}$. Then there exists an étale covering $Y^{\prime }$ of $Y$ and an $X$-isomorphism

X′ ≃ X ×_Y Y′

if and only if ${\bar{X}}_y^{\prime }$ admits a section over ${\bar{X}}_y$.

Putting $Y^{\prime }=\mathrm{Spec}(h_{\ast }({\mathcal{O}}_{X^{\prime }}))$, where $h:X^{\prime }\to Y$ is the composite $X^{\prime }\to X\to Y$, it is enough to prove that the canonical $Y$-morphism

X′ → X ×_Y Y′

is an isomorphism, and that $Y^{\prime }$ is étale over $Y$. We already know by X.1.2 that $Y^{\prime }$ is étale over $Y$, hence $X{\times }_Y{Y}^{\prime }$ is étale over $X$, and therefore the morphism $X^{\prime }\to X{\times }_Y{Y}^{\prime }$ is also étale (I.4.8). Moreover, $Y^{\prime }$ is connected as the image of $X^{\prime }$, which is connected; hence $X{\times }_Y{Y}^{\prime }$ is connected, since $X$ has connected fibers over $Y$ (IX.3.4 and V.6.9(iii)). Thus to prove that $X^{\prime }\to X{\times }_Y{Y}^{\prime }$ is an isomorphism, it is enough to see that its projection degree at one point of $X{\times }_Y{Y}^{\prime }$ is equal to 1. This follows easily from the hypothesis that ${\bar{X}}_y^{\prime }$ admits a section over ${\bar{X}}_y$, either by using IX.6.6 or more simply by noting that it is enough to prove the existence of such a point in $X{\times }_Y{Y}^{\prime }$ after the base change $\mathrm{Spec}(\kappa \bar{y})\to Y$, where this is evident. This proves X.1.3.

Taking IX.3.4 and the dictionary V.6.9 and V.6.11 into account, one can put X.1.3 in the following equivalent form:

Corollary.

With the preceding notation for $f:X\to Y$ and ${\bar{X}}_y$, let ā be a geometric point of ${\bar{X}}_y$, let $a$ be its image in $X$, and let $b$ be its image in $Y$. Then the following sequence of group homomorphisms is exact:

π₁(X̄_y,ā) → π₁(X,a) → π₁(Y,b) → e.

Remarks.

Notice that the proof of X.1.3 uses X.1.2 in an essential way and hence the “first comparison theorem” in algebraic-formal geometry. By contrast, the descent theory of Exposé IX entered only through IX.3.4, for which a direct proof is easy in the case of a proper morphism $f:X\to Y$ such that $f_{\ast }({\mathcal{O}}_X)={\mathcal{O}}_Y$.

Indeed, let $Y^{\prime }$ be étale over $Y$ and suppose $X^{\prime }=X{\times }_Y{Y}^{\prime }$ is the disjoint sum of two nonempty open subsets; we prove that the same is true of $Y^{\prime }$. One has $Y^{\prime }=\mathrm{Spec}(\mathcal{A})$, hence $X^{\prime }=\mathrm{Spec}(\mathcal{B})$, with $\mathcal{B}=\mathcal{A}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_X$, and the decomposition of $X^{\prime }$ as a direct sum corresponds to a decomposition of $\mathcal{B}$ as a product of two nonzero Algebras ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$. Since $f_{\ast }({\mathcal{O}}_X)={\mathcal{O}}_Y$, one easily concludes $f_{\ast }(\mathcal{B})=\mathcal{A}$, so $\mathcal{A}$ is a sum of two Algebras, also nonzero because their unit sections are nonzero, namely $f_{\ast }({\mathcal{B}}_1)$ and $f_{\ast }({\mathcal{B}}_2)$.

1.6.

Suppose again that $f$ is proper and separable, but no longer make any hypothesis on $f_{\ast }({\mathcal{O}}_X)$, which will correspond to a well-determined étale covering $Y^{\prime }$ of $Y$, pointed above $b$ by the image $b^{\prime }$ of $a$. Applying X.1.4 to the canonical morphism $X\to Y^{\prime }$, and supposing $f$ surjective, the exact sequence X.1.4 is replaced by the following, analogous to the homotopy exact sequence of fiber spaces in algebraic topology:

π₁(X̄_y,ā) → π₁(X,a) → π₁(Y,b) →
π₀(X̄_y,ā) → π₀(X,a) → π₀(Y,b) → e.

Of course, in X.1.4 one cannot in general assert that the homomorphism ${\pi }_1({\bar{X}}_y,\bar{a})\to {\pi }_1(X,a)$ is injective; in algebraic topology its kernel is the image of ${\pi }_2(Y,b)$, and in algebraic geometry as well there would be reason to introduce homotopy groups in all dimensions, and the complete homotopy exact sequence for a proper morphism satisfying suitable hypotheses, for example being smooth. At present no result in this direction is available, except for a reasonable, though perhaps not definitive, definition of higher homotopy groups.

Corollary.

Let $k$ be an algebraically closed field, and let $X$ and $Y$ be two connected preschemes over $k$. Suppose $X$ proper over $k$ and $Y$ locally

noetherian. Let $a$ be a geometric point of $X$, and let $b$ be a geometric point of $Y$ with values in the same algebraically closed extension $K$ of $k$. Consider the geometric point $c=(a,b)$ of $X{\times }_{k}Y$, and the homomorphism

π₁(X ×_k Y,c) → π₁(X,a) × π₁(Y,b)

deduced from the homomorphisms on fundamental groups associated with the two projections $X{\times }_{k}Y\to X$ and $X{\times }_{k}Y\to Y$. This homomorphism is an isomorphism.

First suppose $K=k$. Put $Z=X{\times }_{k}Y$, consider the projection $f:Z\to Y$ and the locality $y$ of the geometric point $b$ of $Y$, and apply X.1.4 to this situation. Notice that, after replacing $X$ by $X_{red}$ (which does not change the fundamental groups in question), one may assume $X$ reduced, hence separable over $k$; therefore $Z$ is separable over $k$, and plainly has geometrically connected fibers, since $X$ is connected. The geometric fiber of $Z$ at $b$ is canonically isomorphic to $X{\otimes }_{k}K=X$.

On the other hand, since the composite $X\to Z\to X$ is the identity, one finds that ${\pi }_1(X,a)\to {\pi }_1(Z,c)$ is injective, and X.1.4 gives an exact sequence

e → π₁(X,a) → π₁(Z,c) → π₁(Y,b) → e.

There is also the canonical exact sequence

e → π₁(X,a) → π₁(X,a) × π₁(Y,b) → π₁(Y,b) → e,

where the maps written are the canonical injection and projection. Finally, the canonical homomorphism ${\pi }_1(Z,c)\to {\pi }_1(X,a)\times {\pi }_1(Y,b)$, together with the identity maps on the two end terms, gives a homomorphism from the first exact sequence to the second. The commutativity of the resulting diagram is immediate. Since the homomorphisms on the end terms are isomorphisms, the same is true for the middle terms; this proves X.1.7 in this case.

When $K$ is no longer assumed equal to $k$, one obtains only an isomorphism

π₁(Z,c) → π₁(X ⊗_k K,a) × π₁(Y,b),

and X.1.7 is then equivalent to the following special case:

Corollary.

Let $X$ be a proper connected scheme over an algebraically closed field $k$, let $k^{\prime }$ be an algebraically closed extension of $k$, let $a^{\prime }$ be a geometric point of $X{\otimes }_k{k}^{\prime }$, and let $a$ be its image in $X$. Then the canonical homomorphism

π₁(X ⊗_k k′,a′) → π₁(X,a)

is an isomorphism.

The fact that this homomorphism is surjective is equivalent to saying that if $X^{\prime }$ is a connected étale covering of $X$, then $X^{\prime }{\otimes }_k{k}^{\prime }$ is also connected; this follows at once from the fact that $k$ is algebraically closed, and is also a special case of IX.3.4. The properness hypothesis on $X$ has not yet been used.

It remains to say that injectivity of the homomorphism under consideration means: every étale covering of $X{\otimes }_k{k}^{\prime }$ is isomorphic to the inverse image of an étale covering of $X$. It is essentially sorital that one can find a sub-$k$-algebra $A$ of $k^{\prime }$, of finite type over $k$, and an étale covering of $X{\otimes }_{k}A$ whose inverse image on $X{\otimes }_k{k}^{\prime }$ is isomorphic to the given covering. Let $Y=\mathrm{Spec}(A)$, an integral $k$-scheme of finite type, hence having $k$-rational points. Applying X.1.7 to the fundamental group of $X\times Y$ at a point $(a,b)$ rational over $k$, one finds that every connected étale covering of $X\times Y$ is isomorphic to a quotient of a covering $X^{\prime }\times Y^{\prime }$, where $X^{\prime }$ and $Y^{\prime }$ are Galois étale coverings of $X$ and $Y$ with groups $G$ and $G^{\prime }$, by a subgroup $H$ of $G\times G^{\prime }$.

It follows that the inverse image of this covering of $X\times Y$ on $X\times Y^{\prime }$ is isomorphic to a covering of the form $X_1^{\prime }\times Y^{\prime }$, where $X_1^{\prime }$ is an étale covering of $X$. If $L$ is the function field of $Y$, equal to the fraction field of $A$ in $k^{\prime }$, the étale covering of $X{\otimes }_{k}L$ induced by the given covering of $X{\times }_{k}Y$ is such that there exists a finite separable extension $L^{\prime }$ of $L$ for which the inverse image of that covering on $X{\otimes }_k{L}^{\prime }$ is isomorphic to $X_1^{\prime }{\otimes }_k{L}^{\prime }$. Since $k^{\prime }$ is algebraically closed, one may suppose the extension $L^{\prime }$ of $L$ is contained in $k^{\prime }$. This proves that the given étale covering of $X{\otimes }_k{k}^{\prime }$ is isomorphic to $X_1^{\prime }{\otimes }_k{k}^{\prime }$.

The explicit form just mentioned for étale coverings of a product

$X{\times }_{k}Y$ immediately implies:

Corollary.

Let $k$ be an algebraically closed field, let $X$ and $Y$ be two locally noetherian preschemes over $k$, let $Z=X{\times }_{k}Y$ be their product, and let $Z^{\prime }$ be an étale covering of $Z$. For every point $y\in Y$ rational over $k$, let $i_y:\mathrm{Spec}(k)\to Y$ be the associated canonical morphism, and let $j_y=i{d}_X{\times }_k{i}_y$ be the corresponding morphism $X\to Z$. Finally, let $X_y^{\prime }$ be the étale covering of $X$ obtained as inverse image of $Z^{\prime }$ by $j_y$. Suppose $Y$ connected, and suppose $X$ or $Y$ proper over $k$. Then the coverings $X_y^{\prime }$ of $X$ are all isomorphic.

Figuratively, one may say that a family of étale coverings of $X$, parametrized by a connected prescheme $Y$, is constant if $X$ or the parameter prescheme $Y$ is proper over $k$.

Remarks.

Corollaries X.1.7 to X.1.9 are due to Lang and Serre [X.2] in the case of normal algebraic schemes. Their work was the initial motivation for the theory of the fundamental group developed in this seminar. As these authors observed, these results become false if the properness hypothesis is dropped, at least in characteristic $p>0$. Taking for example $X$ to be the affine line $X=\mathrm{Spec}(k[t])$, it is not difficult to see that the coverings of $X$, parametrized by the affine line $Y=\mathrm{Spec}(k[s])$, defined by the equations

$$x^p-x=st,$$

are étale and pairwise non-isomorphic. This contradicts X.1.9 and a fortiori X.1.7; similarly, if $s$ is regarded as a transcendental element over $k$ in an algebraically closed extension $K$ of $k$, one obtains an étale covering $X^{\prime }$ of $X{\otimes }_{k}K$ that does not come from an étale covering of $X$.

2. Application of the Existence Theorem for Sheaves: Semicontinuity Theorem for Fundamental Groups of Fibers of a Proper and Separable Morphism

Theorem.

Let $Y$ be the spectrum of a complete noetherian local ring, with residue field $k$; let $X$ be a proper $Y$-scheme; let $X_0=X{\otimes }_{A}k$; let $a_0$ be a geometric point of $X_0$; and let $a$ be the corresponding geometric point of $X$. Then the canonical homomorphism

$${\pi }_1(X_0,a_0)\to {\pi }_1(X,a)$$

is an isomorphism.

This is only a translation, into the language of the fundamental group, of the result recalled in IX.1.10. It is here that the existence theorem for sheaves in algebraic-formal geometry enters essentially into the theory of the fundamental group.

Now introduce an algebraic closure $\bar{k}$ of the residue field $k$, and the geometric fiber ${\bar{X}}_0=X_0{\otimes }_k\bar{k}$. We have the exact sequence (IX.6.1)

e → π₁(X̄₀,ā) → π₁(X₀,a₀) → π₁(k,k̄) → e.

On the other hand, we have the isomorphism X.2.1 and the analogous, more elementary isomorphism

$${\pi }_1(k,\bar{k})\to {\pi }_1(Y,b),$$

where $b$ is the image of $a$ in $Y$. Thus one obtains:

Corollary.

With the preceding notation, suppose ${\bar{X}}_0=X_0{\otimes }_k\bar{k}$ connected, and let ${\bar{a}}_0$ be a geometric point of ${\bar{X}}_0$, let $a_0$ be its image in $X$, and let $b_0$ be its image in $Y$. Then the following sequence of canonical homomorphisms is exact:

e → π₁(X̄₀,ā) → π₁(X,a₀) → π₁(Y,b₀) → e.

Compare this sequence with the exact sequence X.1.4, but note that: a) no flatness or fiberwise separability hypothesis has had to be made for $X\to Y$; b) one has the important supplement that the morphism ${\pi }_1({\bar{X}}_0,{\bar{a}}_0)\to {\pi }_1(X,a_0)$ is injective.

This last fact will allow us to compare the fundamental group of the other

geometric fibers of $X$ over $Y$ with that of ${\bar{X}}_0$. Indeed, let $y_1$ be any point of $Y$, let $X_1$ be its fiber and ${\bar{X}}_1$ its geometric fiber relative to an algebraically closed extension of $\kappa (y_1)$, let ${\bar{a}}_1$ be a geometric point of ${\bar{X}}_1$, and let $a_1$ and $b_1$ be its images in $X$ and $Y$. Choose a “path class” from $a_1$ to $a_0$, whence a path class from $b_1$ to $b_0$. This gives a commutative diagram of homomorphisms

π₁(X̄₁,ā₁) → π₁(X,a₁) → π₁(Y,b₁) → e
     ↓            ↓≃           ↓≃
e → π₁(X̄₀,ā₀) → π₁(X,a₀) → π₁(Y,b₀) → e,

where the two displayed vertical arrows in the middle and on the right are isomorphisms. Since the second row is exact, one obtains a canonical homomorphism, which we shall call the specialization homomorphism for the fundamental group. It depends only on the chosen path class from $a_1$ to $a_0$, and is therefore defined modulo inner automorphism of ${\pi }_1(X,a_0)$:

$${\pi }_1({\bar{X}}_1,{\bar{a}}_1)\to {\pi }_1({\bar{X}}_0,{\bar{a}}_0).$$

When the first row above is also exact, it follows at once that the specialization homomorphism is surjective. Thus, taking X.1.4 into account:

Corollary.

Under the conditions of X.2.1, suppose in addition that the morphism $f:X\to Y$ is separable (X.1.1) and that ${\bar{X}}_0$ is connected. Then, by X.1.2, $f_{\ast }({\mathcal{O}}_X)={\mathcal{O}}_Y$. For every geometric fiber ${\bar{X}}_1$ of $X$ over $Y$, endowed with a geometric point ${\bar{a}}_1$, the specialization homomorphism defined above is surjective.

This is a semicontinuity result for the fundamental group, and it does not yet seem to have an analogue in algebraic topology. It can also be stated under more general conditions:

Corollary.

Let $f:X\to Y$ be a proper morphism with geometrically connected fibers, with $Y$ locally noetherian. Let $y_0$ and $y_1$ be two points of $Y$ such that $y_0\in cl(y_1)$, let ${\bar{X}}_0$ and ${\bar{X}}_1$ be the geometric fibers of $X$ corresponding to given algebraically closed extensions of $\kappa (y_0)$ and $\kappa (y_1)$, and let ${\bar{a}}_0$, respectively ${\bar{a}}_1$, be a geometric point of ${\bar{X}}_0$, respectively ${\bar{X}}_1$. Then one can define naturally a specialization homomorphism

$${\pi }_1({\bar{X}}_1,{\bar{a}}_1)\to {\pi }_1({\bar{X}}_0,{\bar{a}}_0),$$

defined up to inner automorphism, and it is surjective if $f$ is separable (X.1.1).

Indeed, X.1.8 first implies that X.2.4 is essentially independent of the chosen algebraically closed extensions of the residue fields $\kappa (y_0)$ and $\kappa (y_1)$. This allows us to replace $Y$ by a prescheme $Y^{\prime }$ over $Y$ having a point $y_0^{\prime }$, respectively $y_1^{\prime }$, above $y_0$, respectively $y_1$. We then take $Y^{\prime }$ to be the spectrum of the completion of the local ring of $y_0$ in $Y$, and apply X.2.3.

Remarks.

The final conclusion of X.2.4 on surjectivity of the specialization homomorphism, and a fortiori the results X.1.3 and X.1.4 of which it is a consequence, become false if one no longer assumes $f:X\to Y$ to be separable, even for projective schemes over an algebraically closed field of characteristic 0. We shall see examples later, both in the case where $f$ is flat but has a nonseparable fiber (with $X$ and $Y$ nevertheless smooth over $k$), and in the case where the fibers of $f$ are separable but $f$ is not flat, for instance with $f:X\to Y$ a birational morphism of normal integral schemes; cf. XI.3. In these examples it can happen that the fundamental group of the generic geometric fiber is trivial, while that of a suitable special geometric fiber is not.

On the other hand, even if $f:X\to Y$ is a proper separable morphism as in X.2.4, the specialization morphism often fails to be an isomorphism.

For instance, it is easy to give examples where ${\bar{X}}_1$ is a nonsingular elliptic curve, so its fundamental group is commutative and its $\ell$-primary component for a prime $\ell$ different from the characteristic is isomorphic to ${\mathbb{Z}}_{\ell }^2$ (cf. XI), while ${\bar{X}}_0$ is formed either of two nonsingular rational curves meeting in two points, or of two rational curves tangent at one point, or finally of a rational curve with a singularity that is a cusp. For the complete classification of degenerate elliptic curves, see the recent work of Kodaira [X.1] and Néron. In these cases the fundamental group of ${\bar{X}}_0$ is respectively $\hat{\mathbb{Z}}$, $e$, $e$, hence “strictly smaller” than that of ${\bar{X}}_1$.

We shall see later, however, that when $f$ is a smooth morphism there is an upper bound on the kernel of the specialization homomorphism, implying in particular that if $\kappa (y_0)$ has characteristic 0, the specialization homomorphism is an isomorphism. But even for a smooth morphism, if the characteristic of $\kappa (y_0)$ is > 0, the specialization homomorphism may fail to be an isomorphism, as one sees for example when $X$ is an abelian scheme over $Y$ (of relative dimension 1, if desired); cf. XI.2.

A satisfactory theory of specialization of the fundamental group must take into account the “continuous component” of the “true” fundamental group, corresponding to the classification of principal coverings with structural group an infinitesimal group. Once this is done, one would be entitled to expect that the “true” fundamental groups of the geometric fibers of a smooth and proper morphism $f:X\to Y$ form a nice local system on $X$, an inverse limit of finite flat group schemes over $X$. [Translator note: the source footnote says this very attractive conjecture is unfortunately contradicted by an unpublished example of M. Artin, already for fibers that are algebraic curves of fixed genus $g\ge 2$.] We shall return later to this viewpoint; our present aim is, on the contrary, to push as far as possible the phenomena common to the topological theory and the schematic theory of the fundamental group.

Let now $X_0$ be a proper, smooth, connected curve of genus $g$ over an algebraically closed field $k$. If $k$ has characteristic zero, its fundamental group can be determined by transcendental methods as follows. One knows that $X_0$ is obtained by base extension from a curve defined over an algebraically closed extension of finite transcendence degree of the prime field $\mathbb{Q}$; taking X.1.8 into account, one may suppose $k$ itself has finite transcendence degree over $\mathbb{Q}$. Hence one may suppose $k$ is a subfield of the complex numbers $\mathbb{C}$, and another application of X.1.8 allows one to suppose $k=\mathbb{C}$.

It is then not difficult to verify that the fundamental group of $X$ is isomorphic to the profinite completion of the fundamental group of the associated topological space $\tilde{X}$, a compact oriented surface of genus $g$, for the topology defined by subgroups of finite index. [Translator note: the source footnote says this deduction was explained in one of the oral expos\acute{e}s that were not written up.] Classically, the topological fundamental group is generated by 2g generators $s_i,t_i$, $1\le i\le g$, subject to the single relation

(s₁t₁s₁⁻¹t₁⁻¹)⋯(s_gt_gs_g⁻¹t_g⁻¹) = 1.

Thus the fundamental group of $X$ admits 2g topological generators $s_i,t_i$, $1\le i\le g$, bound by the preceding single relation.

If now $k$ has characteristic $p>0$, let $A$ be the ring of Witt vectors built from $k$, and let $K$ be an algebraically closed extension of its fraction field. We saw in III.7.4 that there exists a scheme $X$ over $Y=\mathrm{Spec}(A)$, proper and smooth over $Y$, reducing to $X_0$. Applying X.2.3 to it, one obtains a surjective morphism

$${\pi }_1(X_1)\to {\pi }_1(X_0),$$

where $X_1=X{\otimes }_{A}K$. It is immediate (cf. EGA IV 12.2) that $X_1$ is smooth over $K$, connected (X.1.2), of dimension 1, and that its genus is equal to $g$, by invariance of the Euler-Poincaré characteristic (cf. EGA III 7). Since $K$ has characteristic 0, the preceding result applies to it. We have thus proved, by transcendental methods:

Theorem.

Let $X_0$ be a smooth, proper, connected algebraic curve over an algebraically closed field $k$, and let $g$ be its genus. Then ${\pi }_1(X_0)$ admits a system of 2g topological generators, bound by the relation written above. When the characteristic of $k$ is 0, ${\pi }_1(X_0)$ is even the group of galoisian type freely generated by the preceding generators and relation.

Remarks.

At present, to the editor’s knowledge, there is no purely algebraic proof of the preceding result, except in genera 0 and 1. To begin with, it is hardly clear how to distinguish 2g elements in ${\pi }_1(X_0)$ which one could then expect to form a system of topological generators. In this respect, the situation of the rational line punctured at $n$ points, and the study of its coverings tamely ramified at those points, is more sympathetic, since the ramification groups at these $n$ points provide $n$ elements of the fundamental group to be studied, which one indeed shows generate it topologically, as we shall see later. [Translator note: the source footnote refers to Expos\acute{e} XII and notes that these elements are really determined only up to conjugacy, so a judicious simultaneous choice of representatives is required.] But even in this particularly concrete case, there does not seem to be a purely algebraic proof. Such a proof would plainly be extremely interesting. The only fact concerning the fundamental group of a curve that one knows how to prove by purely algebraic methods, apart from the weak finiteness theorem X.2.12 below proved algebraically by Lang and Serre [X.2], seems to be the determination of the abelianized fundamental group via the Jacobian, mentioned at the end of IX.5.8.

2.8.

The last assertion of X.2.6 is no longer valid in characteristic $p>0$, as one already sees for elliptic curves. As we have already pointed out, we do not know whether the fundamental group of $X_0$ is topologically finitely presented; this seems quite improbable.

Theorem.

Let $k$ be an algebraically closed field, and let $X$ be a proper connected scheme over $k$. Then the fundamental group of $X$ is topologically finitely generated.

We proceed by induction on $n=\dim X$, the assertion being trivial for $n\le 0$. Suppose $n>0$ and the theorem proved in dimensions $n^{\prime }<n$. By Chow’s lemma (EGA II 5.6.2), there exists a projective scheme $X^{\prime }$ over $k$ and a surjective morphism $X^{\prime }\to X$. One may plainly suppose $X^{\prime }$ reduced, and after passing to the normalization, normal. By descent theory, it is enough to prove that the fundamental groups of the connected components of $X^{\prime }$ are topologically finitely generated (IX.5.2). This reduces us to the case where $X$ is projective and normal. If $n=1$, it is enough to apply X.2.6. If $n\ge 2$, consider a projective immersion $X\to {\mathbb{P}}_k^r$ and a hyperplane section $Y=X\cdot H$, endowed with the induced reduced structure, such that $Y\ne X$, that is, $H$ does not contain $X$. Then $\dim Y<n$, and by the induction hypothesis it is enough to prove that ${\pi }_1(Y)\to {\pi }_1(X)$ is surjective. More generally:

Lemma.

Let $X$ be a prescheme proper over an algebraically closed field $k$, and let $g:X\to {\mathbb{P}}_k^r$ be a morphism. Suppose $X$ irreducible and normal and $\dim g(X)\ge 2$. Let $H$ be a hyperplane of ${\mathbb{P}}_k^r$ and let $Y=X{\times }_{{\mathbb{P}}^r}H$. Then $Y$ is connected, and the homomorphism ${\pi }_1(Y)\to {\pi }_1(X)$ is surjective.

These assertions follow from:

Corollary.

Under the preceding conditions, let $X^{\prime }$ be a connected étale covering of $X$, and let $Y^{\prime }=X^{\prime }{\times }_{X}Y=X^{\prime }{\times }_{{\mathbb{P}}^r}H$ be the induced covering on $Y$. Then $Y^{\prime }$ is connected.

Since $X$ is normal, $X^{\prime }$ is normal; being connected, it is irreducible, and its image in ${\mathbb{P}}_k^r$ has dimension $\ge 2$. A well-known lemma due to Zariski, called the Bertini theorem, implies that if $H_1^{\prime }$ is the generic hyperplane in ${\mathbb{P}}_k^r$, defined over an extension $K$ of $k$, then $X^{\prime }{\times }_{{\mathbb{P}}^r}{H}_1$ is universally irreducible, hence universally connected over $K$. Zariski’s connectedness theorem (EGA III 4) then implies that for every hyperplane $H$, defined over any extension of $k$, $X^{\prime }{\times }_{{\mathbb{P}}^r}H$ is geometrically connected. This proves X.2.11, hence X.2.9.

Corollary (Lang-Serre).

Under the conditions of X.2.9, for every finite group $G$, the set of isomorphism classes of principal coverings of $X$ with group $G$ is finite.

Remark.

Under the conditions of X.2.10, when $\dim g(X)\ge 3$ we shall prove, at least when $g$ is an immersion and $X$ regular, a sharper result known in algebraic geometry as the Lefschetz theorem: ${\pi }_1(Y)\to {\pi }_1(X)$ is an isomorphism. [Translator note: the corrected source refers to the subsequent seminar SGA 2, theorem X 3.10.] In the classical cases there are analogous statements for homology groups and higher homotopy groups, which sooner or later must be incorporated into abstract algebraic geometry. Even for Hodge cohomology $H^p(X,{\Omega }^q)$, the question does not seem to have been studied; moreover, it is hardly likely that for the latter the Lefschetz theorems remain valid as stated in characteristic $p>0$.

Remark (M. Raynaud).

Let $R$ be a complete discrete valuation ring, with algebraically closed residue field $k$ of characteristic $p>0$, fraction field $K$, and let $Y$ be a proper, smooth, connected curve of genus $g$ over $R$. There is a surjective specialization morphism $sp:{\pi }_1(Y_{\bar{K}})\to {\pi }_1(Y_k)$. We have already observed that if $K$ has characteristic 0, sp is not an isomorphism as soon as $g\ge 1$. Suppose $K$ has characteristic $p$, so that $R$ is isomorphic to the power series ring k[[T]].

In equal characteristic $p>0$, the fundamental group is not determined by the genus $g$, as one already sees for elliptic curves, which may be ordinary or supersingular. We quote the recent result of A. Tamagawa, not yet published. If $G$ is a profinite group, write $G^{res}$ for the profinite quotient of $G$ that is the inverse limit of the finite solvable topological quotients of $G$.

Theorem (A. Tamagawa). Suppose $g\ge 2$, that the special fiber $Y_k$ is definable over a finite field, and that the morphism sp^res: π₁(Y_K̄)^res → π₁(Y_k)^res deduced from sp by passage to the quotient is bijective. Then the curve $Y$ is constant over $R$.

Notice that the Galois group of $\bar{K}/K$ is solvable. The preceding statement can also be translated as follows: suppose that every finite étale covering $X_K\to Y_K$ of the generic fiber Y_K, Galois with solvable Galois group, has potentially good reduction, that is, extends to a finite étale covering of $Y$ after possibly replacing $R$ by its normalization in a finite extension of $K$. Then $Y$ is constant over $R$.

3. Application of the Purity Theorem: Continuity Theorem for Fundamental Groups of Fibers of a Proper and Smooth Morphism

Recall without proof the following theorem. [Translator note: the source refers for a proof to SGA 2 X.3.4.]

Purity Theorem (Zariski-Nagata).

Let $f:X\to Y$ be a quasi-finite and dominant morphism of integral preschemes, with $X$ normal and $Y$ regular locally noetherian, and let $Z$ be the set of points of $X$ at which $f$ is not étale, that is, where $f$ is ramified (equivalently, I.9.5(ii)). If $Z\ne X$, then $Z$ has codimension 1 in $X$ at all its points; that is, for every irreducible component $Z^{\prime }$ of $Z$ with generic point $z$, the Krull dimension of ${\mathcal{O}}_{X,z}$ is equal to 1.

Recall that a prescheme is called normal, respectively regular, if its local rings are normal, respectively regular, and that the relation $Z\ne X$ also means that the finite extension $R(Z)/R(X)$, where $R$ denotes the field of rational functions, is separable. Placing ourselves at the generic point $z$ of a component $Z^{\prime }$ of $Z$, and localizing at the point $y$ of $Y$ below $z$, one obtains

the equivalent statement:

Corollary.

Let $A$ be a regular noetherian local ring, and let $A\to B$ be an injective local homomorphism such that $B$ is normal, a localization of a finite-type $A$-algebra, and quasi-finite over $A$. Suppose moreover that $\dim A(=\dim B)\ge 2$, and that for every prime ideal $\mathfrak{p}$ of $B$ distinct from the maximal ideal, $B$ is étale over $A$ at $\mathfrak{p}$, that is, $B_{\mathfrak{p}}$ is étale over $A_{\mathfrak{q}}$, where $\mathfrak{q}=A\cap \mathfrak{p}$. Then $B$ is étale over $A$.

It is not difficult to reduce this last statement to the case where $A$ is a complete local ring, hence where $B$ is finite over $A$. Zariski [X.5] gives a simple proof of this result, valid in the equal-characteristic case. The general case is due to Nagata [X.3], who relies on a delicate result of Chow; the latter was not verified by any of the participants in the seminar, and should be the subject of a later exposé.

We record here only the very simple proof in the special case $\dim A=2$, which is enough for the most important application we shall make of it in the present number. Since $B$ is normal, it is a $B$-module of depth (old terminology: cohomological codimension) $\ge 2$; hence it is an $A$-module of depth $\ge 2$. Since $A$ is regular of dimension 2, it follows that $B$ is a free module over $A$. [Translator note: the source refers to EGA 0_IV 17.3.4.] It then follows from I.4.10 that the set of prime ideals $\mathfrak{q}$ of $A$ at which $B$ is ramified over $A$ is the subset of $\mathrm{Spec}(A)$ defined by a principal ideal (generated by the discriminant of a basis of $B$ over $A$). Thus it is empty if it is contained in the closed point of $\mathrm{Spec}(A)$, proving X.3.2 when $\dim A=2$.

We shall mainly use X.3.1 in the following equivalent form:

Corollary.

Let $X$ be a locally noetherian regular prescheme, and let $U$ be an open subset of $X$ whose complement is a closed subset $Z$ of $X$ of codimension $\ge 2$. Then the functor $X^{\prime }\mapsto X^{\prime }{\times }_{X}U$ from the category of étale coverings of $X$ to the category of étale coverings of $U$ is an equivalence

of categories. In particular, if $a$ is a geometric point of $U$, the canonical homomorphism ${\pi }_1(U,a)\to {\pi }_1(X,a)$ is an isomorphism.

The last assertion is plainly a consequence of the first; for the first, one may plainly suppose $X$ connected, hence irreducible. The normality of $X$ already implies that the functor $X^{\prime }\mapsto X^{\prime }{\times }_{X}U$ from the category of locally free coverings (not necessarily étale) of $X$ to the category of coverings of $U$ is fully faithful, because the functor $\mathcal{E}\mapsto \mathcal{E}|U$ from locally free Modules on $X$ to locally free Modules on $U$ is fully faithful.

It remains to prove that for every étale covering $U^{\prime }$ of $U$, there exists an étale covering $X^{\prime }$ of $X$, necessarily unique by what precedes, such that $U^{\prime }$ is isomorphic to $X^{\prime }{\times }_{X}U$. One may plainly suppose $U^{\prime }$ connected, hence irreducible since $U^{\prime }$ is normal ($U$ being normal). Let $K$ be the field of rational functions on $X$, or on $U$, which is the same, and let $K^{\prime }$ be that of $U^{\prime }$. Then $U^{\prime }$ identifies with the normalization of $U$ in $K^{\prime }$ (I.10.3). Let $X^{\prime }$ be the normalization of $X$ in $K^{\prime }$ (EGA II 6.3); then $X^{\prime }{\times }_{X}U\simeq U^{\prime }$. Moreover $X^{\prime }$ is normal and integral, and the structural morphism $f:X^{\prime }\to X$ is finite and dominant, since $X$ is normal and $K^{\prime }/K$ is a finite separable extension. It is étale over $U^{\prime }=f^{-1}(U)$, and since $Z$ has codimension $\ge 2$ in $X$, $f^{-1}(Z)$ has codimension $\ge 2$ in $X^{\prime }$. We conclude from X.3.1 that $X^{\prime }$ is étale over $X$, completing the proof.

Now let $f:X\to Y$ be a rational map from a locally noetherian regular prescheme $X$ to a prescheme $Y$, and suppose $f$ is defined on an open subset $U$ whose complement is a closed subset of codimension $\ge 2$. Then X.3.3 gives a functor, defined up to isomorphism, from the category of étale coverings of $Y$ to the category of étale coverings of $X$; hence for every geometric point $a$ of $U$, with image $b$ in $Y$, a canonical homomorphism

$${\pi }_1(X,a)\to {\pi }_1(Y,b),$$

deduced from the canonical homomorphism ${\pi }_1(U,a)\to {\pi }_1(Y,b)$ by means of the isomorphism ${\pi }_1(U,a)\simeq {\pi }_1(X,a)$. When $f$ is a dominant morphism, with $X$ and $Y$ integral of function fields $K$ and $L$, so that $K$ is an extension of $L$, and with $Y$ normal, these correspondences become more precise in terms of field extensions: for every finite extension $L^{\prime }$ of $L$ unramified over $Y$, the $K$-algebra $K^{\prime }=L^{\prime }{\otimes }_{L}K$ is unramified over $X$.

In particular, these reflections show that the fundamental group of connected locally noetherian regular preschemes, pointed by geometric points localized in codimension $\le 1$, is a functor when as morphisms in this category one takes dominant rational maps defined on complements of closed subsets of codimension $\ge 2$. Recalling, for example, that a rational map from a normal scheme over a field $k$ to a proper scheme over $k$ is defined on the complement of a set of codimension $\ge 2$, one obtains:

Corollary. Birational Invariance of the Fundamental Group.

Let $k$ be a field, let $X$ and $Y$ be two proper regular schemes over $k$, let $f:X\to Y$ be a birational map from $X$ to $Y$, and let $\Omega$ be an algebraically closed extension of the function field $K$ of $X$ allowing one to define the fundamental group of $X$ and the fundamental group of $Y$. These groups are then canonically isomorphic.

This also means that for a finite extension $K^{\prime }$ of $K$, if it is unramified over one nonsingular proper “model” $X$ of $K$, it is unramified over every other nonsingular proper model.

Remark.

For other applications of the purity theorem, see the work of Abhyankar presented in [X.4], inspired by the results of Zariski [X.6, Chapter VIII], proved by topological methods. These latter results are far from having been assimilated by “abstract” algebraic geometry and deserve renewed effort.

We shall need a few elementary facts from ramification theory. Let $V$ be a discrete valuation ring with fraction field $K$ and residue field $k$; let $L$ be a Galois extension of $K$ with group $G$; let $V^{\prime }$ be the normalization of $V$ in $L$, which is a free $V$-module of rank $n=[L:K]$; let ${\mathfrak{m}}^{\prime }$ be a maximal ideal of $V^{\prime }$; let $G_d$ be the subgroup of $G$ consisting of the elements that leave ${\mathfrak{m}}^{\prime }$ invariant, so that $G_d$ acts on the residue extension $k^{\prime }=V^{\prime }/{\mathfrak{m}}^{\prime }$ of $k$; and let $G_i$ be the subgroup of elements of $G_d$ acting trivially. Recall that $G_d$ and $G_i$ are called respectively the decomposition and inertia subgroups of $G$.

One says that $L$ is tamely ramified over $V$ if $n_i=[G_i:e]$ is of order prime to the characteristic $p$ of $k$, a condition always satisfied if $k$ has characteristic 0. It is well known that $G_i$ then embeds canonically in the group $k^{\prime }\ast$, and is therefore isomorphic to the group of $n_i$-th roots of unity in $k^{\prime }\ast$. In particular, $G_i$ is cyclic. The typical case is $L=K[t]/(t^n-u)$, where $u$ is a uniformizer of $V$ and $n$ is prime to $p$: if $K$ contains the $n$-th roots of unity, $L$ is a totally ramified Galois extension of $K$, with Galois group $G=G_i$ isomorphic to $\mathbb{Z}/n\mathbb{Z}$.

Lemma (Abhyankar’s Lemma).

Let $V$ be a discrete valuation ring with fraction field $K$. Let $L$ and $K^{\prime }$ be two Galois extensions of $K$ tamely ramified over $V$, and let $n$ and $m$ be the orders of the corresponding inertia groups. Let $L^{\prime }$ be a composite extension of $L$ and $K^{\prime }$ over $K$. If $m$ is a multiple of $n$, then $L^{\prime }$ is unramified over the localizations of the normal closure $V^{\prime }$ of $V$ in $K^{\prime }$.

Indeed, let $W^{\prime }$ be the normalization of $V^{\prime }$ in $L^{\prime }$, let ${\mathfrak{m}}^{\prime }$ be a maximal ideal of $V^{\prime }$, let ${\mathfrak{n}}^{\prime }$ be a maximal ideal of $W^{\prime }$ above ${\mathfrak{m}}^{\prime }$, and let $\mathfrak{n}$ be the maximal ideal that it induces on the normalization $W$ of $V$ in $L$. Let $G$, $H$, $M$ be the Galois groups of $L$, $K^{\prime }$, $L^{\prime }$ over $K$, and let $G_i$, $H_i$, $M_i$ be the inertia groups corresponding to the chosen maximal ideals. Then $M$ embeds in $G\times H$ and $M_i$ in $G_i\times H_i$, in such a way that the projections $M\to G$ and $M\to H$, and $M_i\to G_i$

and $M_i\to H_i$, are surjective (the standard intermediate-field sorites). It already follows, since $G_i$ and $H_i$ are by hypothesis cyclic of orders $m$ and $n$ prime to $p$, that $M_i$ has order prime to $p$, hence is cyclic. Since $m$ is a multiple of $n$, all elements of $G_i\times H_i$ have $m$-th power equal to the identity; hence $M_i$ has order dividing $m$, and therefore order exactly $m$ because $M_i\to H_i$ is surjective. This last homomorphism is therefore also injective. But its kernel is the inertia group of ${\mathfrak{n}}^{\prime }$ over ${\mathfrak{m}}^{\prime }$, which proves that $L^{\prime }$ is unramified over $K^{\prime }$ at ${\mathfrak{n}}^{\prime }$. This proves the lemma.

Place ourselves now under the conditions of X.2.4, where one has a surjective specialization homomorphism

$${\pi }_1({\bar{X}}_1,{\bar{a}}_1)\to {\pi }_1({\bar{X}}_0,{\bar{a}}_0)$$

relative to a proper and separable morphism $f:X\to Y$. We want to make its kernel more precise. Proceeding as in the proof of X.2.4, one sees that for this question one may always suppose that $Y$ is the spectrum of a complete discrete valuation ring $V$ with algebraically closed residue field, since one can always find such a ring and a morphism from its spectrum $Y^{\prime }$ into $Y$ whose image is $y_0,y_1$. Then $X_0={\bar{X}}_0$, $\kappa (y_0)=k$ is the residue field of $V$, and $\kappa (y_1)=K$ is the fraction field of $V$. Let $K_s$ be the separable closure of $K$, $\bar{K}$ its algebraic closure, and for every subring $W$ of $\bar{K}$ containing $V$ put $X_W=X{\otimes }_{V}W$. In particular,

X_V = X,    X_K = X₁,    X_K̄ = X̄₁.

Moreover the canonical morphism ${\bar{X}}_1=X_{\bar{K}}\to X_{K_s}$ induces an isomorphism on fundamental groups (IX.4.11). Thus, taking into account the isomorphism X.2.1, ${\pi }_1(X_0)\to {\pi }_1(X)$, we are reduced to studying the surjective homomorphism

π₁(X_K_s) → π₁(X)

associated with the canonical morphism $X_{K_s}\to X$.

Determining the kernel of this latter homomorphism is equivalent to solving the following problem: given a connected principal covering $Z_{K_s}$ of $X_{K_s}$ with group $G$ (hence associated with a homomorphism from ${\pi }_1(X_{K_s})$ to $G$), determine under what conditions it is isomorphic to the inverse image of a principal covering $Z$ of $X$ with group $G$.

First note that $K_s$ is the filtered increasing union of its finite subextensions $K^{\prime }$ over $K$, and therefore $Z_{K_s}$ is isomorphic to the inverse image of a principal covering $Z_{K^{\prime }}$ of $X_{K^{\prime }}$ for a suitable $K^{\prime }$. Be careful, however, that for fixed $K^{\prime }$, $Z_{K^{\prime }}$ is not uniquely determined. To say that $Z_{K_s}$ is isomorphic to the inverse image of a principal covering $Z$ of $X$ means that there exists a finite subextension $K^{\prime \prime }\supset K^{\prime }$ of $K_s$ such that $Z_{K^{\prime \prime }}=Z_{K^{\prime }}{\otimes }_{K^{\prime }}{K}^{\prime \prime }$ is isomorphic to $Z{\otimes }_V{K}^{\prime \prime }$.

Now, for a finite subextension $K^{\prime }$ of $K_s$, denote by $V^{\prime }$ the normalization of $V$ in $K^{\prime }$. This is a complete discrete valuation ring with residue field $k$. The canonical morphism $X_{V^{\prime }}\to X_V$ therefore induces an isomorphism on the fibers above the closed points of $Y=\mathrm{Spec}(V)$ and $Y^{\prime }=\mathrm{Spec}(V^{\prime })$; applying X.2.1 to X_V and $X_{V^{\prime }}$, it follows that the induced homomorphism on fundamental groups ${\pi }_1(X_{V^{\prime }})\to {\pi }_1(X_V)$ is an isomorphism. Equivalently, every principal covering of $X_{V^{\prime }}$ is the inverse image of a principal covering of X_V, determined up to isomorphism. This implies:

Lemma.

Let $Z_{K^{\prime }}$ be a connected principal covering of $X_{K^{\prime }}$ with group $G$, and let $Z_{K_s}$ be its inverse image on $X_{K_s}$. Then $Z_{K_s}$ is isomorphic to the inverse image of a principal covering $Z$ of $X$ if and only if there exists a finite extension $K^{\prime \prime }\supset K^{\prime }$ of $K$ in $K_s$ such that the principal covering $Z_{K^{\prime \prime }}$ of $X_{K^{\prime \prime }}$ is induced by a principal covering of $X_{V^{\prime \prime }}$.

Suppose in particular that the $X_{V^{\prime \prime }}$ are normal. It is enough, for example, that $X_0$ be normal, and a fortiori that $X_0$ be simple; cf. I.9.1.

Since they are connected, they are then irreducible. Let $L$ be the field of rational functions of $X$ and X_K, let $L^{\prime }$ be the field for $X_{V^{\prime }}$ and $X_{K^{\prime }}$, and let $L^{\prime \prime }$ be the field for $X_{V^{\prime \prime }}$ and $X_{K^{\prime \prime }}$. Under the conditions of X.3.7, $Z_{K^{\prime }}$ defines a finite separable extension $R^{\prime }$ of $L^{\prime }$, and $Z_{K^{\prime \prime }}$ defines the extension $R^{\prime \prime }=R^{\prime }{\otimes }_{L^{\prime }}{L}^{\prime \prime }=R^{\prime }{\otimes }_{K^{\prime }}{K}^{\prime \prime }$. The condition considered in X.3.7 therefore also means that there exists a finite separable extension $K^{\prime \prime }$ of $K^{\prime }$ such that $R^{\prime \prime }=R^{\prime }{\otimes }_{K^{\prime }}{K}^{\prime \prime }$ is unramified over the normal scheme $X_{V^{\prime \prime }}$ with function field $L^{\prime \prime }=L^{\prime }{\otimes }_{K^{\prime }}{K}^{\prime \prime }$, and not only over the open part $X_{K^{\prime \prime }}$ of $X_{V^{\prime \prime }}$.

From now on suppose that $f:X\to Y$ is a smooth morphism. Then the morphisms $X_{V^{\prime }}\to \mathrm{Spec}(V^{\prime })$ are smooth, and therefore the schemes $X_{V^{\prime }}$ are regular. Notice that the fiber of the closed point of $\mathrm{Spec}(V^{\prime })$ in $X_{V^{\prime }}$ is irreducible and of codimension 1. Let ${\mathfrak{o}}^{\prime }$ be its local ring; this is a discrete valuation ring with fraction field $L^{\prime }$ and residue field isomorphic to the field of rational functions of $X_0$, hence with the same characteristic as $k$. Define similarly ${\mathfrak{o}}^{\prime \prime }$ in $L^{\prime \prime }$; it is plainly the normalization of ${\mathfrak{o}}^{\prime }$ in $L^{\prime \prime }$. It then follows from the purity theorem X.3.1, or X.3.3, that $R^{\prime \prime }$ is unramified over $X_{V^{\prime \prime }}$ if and only if $R^{\prime \prime }$ is unramified over ${\mathfrak{o}}^{\prime \prime }$, the normalization of ${\mathfrak{o}}^{\prime }$ in $L^{\prime \prime }$.

Now note that if $u^{\prime }$ is a uniformizer of $V^{\prime }$, it is also a uniformizer of ${\mathfrak{o}}^{\prime }$. If $n$ is an integer prime to the characteristic $p$ of $k$, and if one takes $K^{\prime \prime }=K^{\prime }[t]/(t^n-u^{\prime })$, then $K^{\prime \prime }$ is a finite Galois extension of $K^{\prime }$ and $L^{\prime \prime }$ is isomorphic to $L^{\prime }[t]/(t^n-u^{\prime })$, hence is tamely ramified over ${\mathfrak{o}}^{\prime }$ with inertia group of order $n$. Suppose now that $G$ has order prime to $p$. Then $R^{\prime }$ is tamely ramified over ${\mathfrak{o}}^{\prime }$. Take $n$ to be a multiple prime to $p$ of the order of the inertia group of $R^{\prime }$ over ${\mathfrak{o}}^{\prime }$, for example $n=[G:e]$. Applying Abhyankar’s lemma X.3.6, one sees that the condition considered in X.3.7 is satisfied.

This proves the following theorem:

Theorem.

Let $f:X\to Y$ be a proper and smooth morphism with geometrically connected fibers, with $Y$ locally noetherian. Let $y_0$ and $y_1$ be two points of $Y$ such that $y_0\in cl(y_1)$, let ${\bar{X}}_0$ and ${\bar{X}}_1$ be the corresponding geometric fibers, and consider the specialization homomorphism of X.2.4

$${\pi }_1({\bar{X}}_1)\to {\pi }_1({\bar{X}}_0).$$

This homomorphism is surjective, and every continuous homomorphism from ${\pi }_1({\bar{X}}_1)$ to a finite group $G$ of order prime to the characteristic $p$ of $\kappa (y_0)$ comes from a homomorphism from ${\pi }_1({\bar{X}}_0)$ to $G$.

In other words:

Corollary.

If $\kappa (y_0)$ has characteristic zero, then the specialization homomorphism is an isomorphism. If $\kappa (y_0)$ has characteristic $p>0$, then the kernel of the specialization homomorphism is contained in the intersection of the kernels of the continuous homomorphisms from ${\pi }_1({\bar{X}}_1)$ to finite groups of order prime to $p$, or equivalently in the closed normal subgroup generated by a $p$-Sylow subgroup of the group of galoisian type ${\pi }_1({\bar{X}}_1)$. Thus, if ${\pi }_1({\bar{X}}_1{)}^p$ denotes the quotient group of ${\pi }_1({\bar{X}}_1)$ by the preceding closed subgroup, and if ${\pi }_1({\bar{X}}_0{)}^p$ is defined similarly, then the specialization homomorphism induces an isomorphism

$${\pi }_1({\bar{X}}_1{)}^p\simeq {\pi }_1({\bar{X}}_0{)}^p.$$

Notice that the proof of X.3.8 is purely algebraic. Proceeding as in X.2.6, one concludes by transcendental methods:

Corollary.

Let $X_0$ be a proper, smooth, connected curve of genus $g$ over an algebraically closed field of characteristic $p$. With the notation introduced in X.3.9, the group ${\pi }_1(X_0{)}^p$ is isomorphic to ${\Gamma }^p$, where $\Gamma$ is the group of galoisian type generated by generators $s_i,t_i$, $1\le i\le g$, bound by the relation

(s₁t₁s₁⁻¹t₁⁻¹)⋯(s_gt_gs_g⁻¹t_g⁻¹) = 1.

Remarks.

When $\kappa (y_0)$ has characteristic zero, the result X.3.9 is well known by transcendental methods. Notice that the proof of X.3.10 appeals to the purity theorem in the unequal-characteristic case, but only for rings of dimension 2, where the proof of that theorem is easy and was recalled in the text.

Bibliography

[X.1] K. Kodaira, “On compact analytic surfaces,” Annals of Mathematics 71 (1960), pp. 111–152.

[X.2] S. Lang and J.-P. Serre, “Sur les revêtements non ramifiés des variétés algébriques,” American Journal of Mathematics 79 (1957), pp. 319–330.

[X.3] M. Nagata, “On the purity of branch loci in regular local rings,” Illinois Journal of Mathematics 3 (1959), pp. 328–333.

[X.4] J.-P. Serre, Revêtements ramifiés du plan projectif (d’après S. Abhyankar), Séminaire Bourbaki, May 1960.

[X.5] O. Zariski, “On the purity of the branch locus of algebraic functions,” Proceedings of the National Academy of Sciences USA 44 (1958), pp. 791–796.

[X.6] O. Zariski, Algebraic Surfaces, Ergebnisse, 1948; Chelsea, New York.