Elementary GAGAKay Werndli

University of Basel26th July, 2011

Master’s Thesisunder the supervision of

Prof. Dr. Hanspeter Kraft

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2 Paragraph 1. Introduction

[. . . ] to myself I seem to have been only like a boy playing on thesea-shore, and diverting myself in now and then finding a smootherpebble or a prettier shell than ordinary, whilst the great ocean oftruth lay all undiscovered before me.

Sir Isaac Newton

1. Introduction

The study of algebraic geometry may be traced back in history almost arbitrarily far. Forexample, the old Greek, Arab and Persian mathematicians already used geometric methods tosolve cubic equations, although one could argue about whether this is considered real algebraicgeometry (no pun intended). The next vital step towards algebraic geometry was certainlythe introduction of algebra into geometry, i.e. the introduction of Cartesian coordinates in the17th century. Again one can argue whether this should be considered algebraic geometry ornot. However, one of the first definite guises of algebraic geometry can be found in Newton’sclassification of cubic curves (as published in John Harris’s Lexicon Technicum, vol. 2 of 1710).This list goes on via the advent of projective and Kleinian geometry to Bernhard Riemann’sdevelopment of Riemann surfaces and the Italian school of algebraic geometry.

Obviously neither Descartes, nor Newton, nor Riemann knew what a “Zariski topology”was (or a “topology” for that matter) let alone a (maximal) spectrum of a ring or a structuresheaf. If there was some form of topology lurking in the background then it was certainly theanalytic one induced by the standard Euclidean metric. It was not until the middle of the 20th

century that the Zariski topology and later on the étale topology (although not a topology in theusual sense) were introduced and became the standard way of studying varieties and schemes.

The interplay of the Zariski topology on an (affine) algebraic variety and the classicalone induced by the Euclidean norm of an ambient complex affine space has always been anintensely studied question. However, the theory of (complex analytic) manifolds turns out to beinsufficient for this because an algebraic variety may have singularities. This obstacle was quicklyremoved in 1956 only shortly after the introduction of the Zariski topology when Jean-PierreSerre published his influential paper [GAGA], introducing the notion of analytic spaces. To thisdate, Serre’s original paper remains a primary source of information about the above mentionedinterplay between algebraic and analytic geometry and its importance is perhaps best reflectedin the fact that the study of this is nowadays colloquially referred to simply as “GAGA”.

However, Serre’s paper is mainly concerned with his newly developed sheaf cohomologyand the relations between coherent algebraic and analytic sheaves, while only addressing the nec-essary topological questions. The standard reference containing a lot more comparison theoremsbetween the topological structures is [SGA 1, Exposé XII] which has the obvious drawback ofbeing highly technical because everything is considered in the more general context of schemesof finite type over C, which is usually not necessary in classical algebraic geometry. There arehowever more elementary accounts like [Mumford, 1999] and [Neeman, 2007] but the former isonly concerned with the analytification of complete varieties and the latter uses GAGA as amere motivational device to develop scheme theory from scratch. For this reasons it would bedesirable to have an elementary account at least for some of the more important topologicalaspects of GAGA and this is exactly the goal of this thesis. The topological aspects that werepicked out are the following three theorems:(1.1) Theorem. If ϕ : X → Y is a finite morphism between algebraic varieties, then theanalytification ϕan : Xan → Yan is a closed map.(1.2) Theorem. If ϕ : X → Y is a flat morphism between algebraic varieties, then theanalytification ϕan : Xan → Yan is an open map.

Paragraph 2. Topological Preliminaries 3

(1.3) Theorem. If X is an irreducible algebraic variety, then Xan is connected.

The first theorem was settled in a highly satisfactory manner and relies on a short andelementary proof of Chow’s lemma for quasi-projective varieties, whose presence in the literaturethe author is not aware of. Unfortunately, we were not able to address the second question intime and it turns out to be very complicated. However, we included a proof for the étale case(rather than the flat one), where the analytification is even a local homeomorphism although theproof can hardly be considered elementary because it relies on the fact that any étale morphismlooks locally like a standard étale one, which in turn relies on Zariski’s main theorem. As forthe last theorem, we were able to give two proofs. The first one has the advantage of beingpurely algebraic but with the drawback of needing a weak form of the famous Riemann-Rochtheorem and admittedly proving the theorem in a roundabout way. It establishes the theoremfor affine algebraic curves, enabling us to generalise it to arbitrary dimensions because on anirreducible affine variety, any two points are always contained in an irreducible one-dimensionalaffine closed subvariety, as shown in [De Capitani, 2006]. The second proof doesn’t take a detourvia algebraic curves and is more elementary but uses tools from complex analysis.

A quick word on the notations and names used in the text: We are always workingover the ground field C, the words “algebraic variety” and “variety” are used interchangeablyand the word “curve” is used as a synonym for a one-dimensional variety. To distinguish thetwo topologies on the complex numbers that we are concerned with, we use A1 for the Zariskitopology and C for the one induced by the Euclidean norm (or any other norm for that matter).Unfortunately, there is no such notation for projective space. By a “morphism” we always meana morphism of algebraic varieties and a morphism to A1 is also called a “regular function”. Thestructure sheaf of an algebraic variety X is denoted by OX and its stalk at a point x ∈ Xby OX,x. However, we will usually only consider affine varieties, so that only its global sections(a.k.a. its coordinate ring) OX(X) are important. Finally, if X is an irreducible affine varietyits “rational function field” is the field of fractions Frac OX(X) and is denoted by C(X).

I’d like to finish this introduction with the most important thing, namely to thank allthe people who supported me, the above mentioned little boy on the sea-shore. I especially wantto emphasise the importance of the role my adviser Hanspeter Kraft played in the creation ofthis work, without whose excellent supervision I would have failed desperately. Not a bit lessimportant was the help of his PhD-student Immanuel Stampfli, who provided me with all kindsof mathematical insights during our very frequent and lively discussions. Last but not least,all the mathematical insight in the world won’t suffice if there isn’t a single person providingmoral support. For that reason I’d like to thank all my friends who accompanied me during thisimportant and hard but at the same time very exciting time of my life.

2. Topological Preliminaries

For readability’s sake, we gather some topological results in this section that we are going to useat some point(s) in the text. The reader can skip this section without further ado and returnto it later, when a result stated here is needed.

(2.1) Proposition. Let X, Y be topological spaces with Y Hausdorff, D ⊆ X a dense subsetand f , g : X → Y continuous maps that agree on D (i.e. f |D = g|D). Then even f = g.

Proof. Because Y is Hausdorff, the diagonal ∆Y := {(y, y) | y ∈ Y } ⊆ Y × Y is closed and

(f, g)−1∆Y = {x ∈ X | fx = gx} ⊇ D.

But then the continuity of f and g implies that D = X ⊆ (f, g)−1∆Y and the claim follows. �

4 Paragraph 2. Topological Preliminaries

(2.2) Definition. Recall that a continuous map ϕ : X → Y is called proper iff for everyother topological space Z, the pullback

ϕ× 1Z : X × Z → Y × Z

along the projection Y ×Z → Y is closed. Equivalently, ϕ is proper iff it is closed and all its fibresare quasi-compact. This implies that for ϕ proper, the preimages of quasi-compact subsets arequasi-compact. The converse of this implication is true for X Hausdorff and Y locally compactHausdorff. Recall furthermore, that a continuous map ϕ : X → Y is called finite iff it is closedand all its fibres are finite. In particular, a finite map is proper.

(2.3) Proposition. Let ϕ : X → Y a continuous map such that Y has an open cover (Yi)i∈I ,satisfying that for each i ∈ I the restriction ϕ−1Yi → Yi of ϕ is closed/proper/finite. Then ϕ isclosed/proper/finite.

Proof. We put Xi := ϕ−1Yi and first check the claim for closedness. If A ⊆ X is closed andB := ϕA ⊆ Y , we write

Bi := B ∩ Yi = ϕA ∩ Yi = ϕ(A ∩Xi),

where the last equality holds because Xi = ϕ−1Yi. Therefore, by hypothesis, all the Bi ⊆ Yiare closed. Now if y ∈ Y \ B there is some i ∈ I, such that y ∈ Yi and because Bi ⊆ Yi isclosed, there is some open neighborhood V ⊆ Yi of y such that Vi ∩ Bi = ∅. But V ⊆ Yi andBi = Yi∩B, thence V ∩B = V ∩Yi∩B = V ∩Bi = ∅, so that B = ϕA ⊆ Y is indeed closed. Asfor properness (resp. finiteness), we only need to check that all fibres are quasi-compact (resp.finite), which is obvious by hypothesis. �

(2.4) Definition. A continuous map p : E → X between two topological spaces is called alocal homeomorphism (or étale) iff for every e ∈ E, there is some open neighborhood U ⊆ Eof e, such that pU ⊆ X is open and p restricts to a homeomorphism U ∼= pU . Furthermore,we say that a continuous map p : E → X has the (unique) path-lifting property iff for everypath γ : [0, 1] → X with starting point x0 := γ0 and every e0 ∈ p−1x0 there is a (unique) pathγ : [0, 1]→ E that starts at e0 and satisfies p ◦ γ = γ.

E

p

��

e0_

��

[0, 1]

∃γ==

γ// X x0

(2.5) Remark. It follows immediately from the definition, that local homeomorphisms havediscrete fibres and are open.

(2.6) Example. It is well-known and probably part of every introduction to the homotopytheory of topological spaces, that covering spaces (which are instances of local homeomorphisms)have the unique path-lifting property (for a proof, see e.g. or [Hatcher, 2002, p. 60]). However,proving that a continuous map is actually a covering is sometimes a bit of a hassle. But we dohave the following useful criterion from [Forster, 1981, p. 29].

(2.7) Proposition. Any étale and proper continuous map p : E → X with E Hausdorff iseven a covering map.

Paragraph 3. Algebraic Preliminaries 5

Proof. First, we note that p has finite fibres (they are discrete because p is étale and quasi-compact by properness), so that for x ∈ X, we have p−1x = {e1, . . . , en} ⊆ E for some pairwisedistinct e1, . . . , en ∈ E. As p is étale, we find for every i ∈ {1, . . . , n} some open neighborhoodUi ⊆ E of ei, which is mapped homeomorphically onto the open set pUi ⊆ X via p. Moreover,since E is Hausdorff, we can assume without loss of generality, that U1, . . . , Un are pairwisedisjoint. By the lemma below, there is some open neighborhood V ⊆ pU1 ∩ . . . ∩ pUn of x, suchthat p−1V ⊆ U1 ∪ . . . ∪ Un. Putting Vi := Ui ∩ p−1V for i ∈ {1, . . . , n}, one readily checks that

p−1V = V1 ∪ . . . ∪ Vnis a disjoint union of open subsets of E, with p restricting to a homeomorphism Vi ∼= V for eachi ∈ {1, . . . , n}. As x ∈ X was an arbitrary point, p is indeed a covering map. �

(2.8) Lemma. Let p : E → X be a closed continuous map, x ∈ X and U ⊆ E open withp−1x ⊆ U . Then there is some open neighborhood V ⊆ X of x such that p−1V ⊆ U .

Proof. A := p(E \ U) ⊆ X is closed and V := X \A is the sought for open neighborhood. �

A direct consequence of this lemma, which we shall find useful to relate the closures ofa constructible set in the Zariski- and the strong topology, is the following.(2.9) Lemma. Let ϕ : X → Y be a closed continuous surjection, x ∈ X with ϕ−1ϕx = {x}and S ⊆ X. If ϕx is not an interior point of ϕS, then x is not an interior point of S either.Proof. Assume there is some U ⊆ X open with x ∈ U and U ⊆ S. Then by the previouslemma, there is some open neighborhood V ⊆ Y of ϕx such that ϕ−1V ⊆ U ⊆ S, implying thatV = ϕϕ−1V ⊆ ϕS, which is a contradiction. �

Finally, we need a result concerning the interplay between a covering and the connectedcomponents of its total space, following from the above mentioned path-lifting property.(2.10) Proposition. Let p : E → X be a surjective covering map withX pathwise connectedand let (Ei)i∈I be the connected components (or alternatively the path-components) of E. Thenevery restriction p|Ei : Ei → X with i ∈ I is again surjective. �

3. Algebraic Preliminaries

As mentioned in the introduction, we didn’t fully succeed in giving elementary proofs for alldesired facts. Obviously, the notion of being elementary is highly subjective and depends heavilyon the mathematical history and talents of the “definer” (that is the person defining “elemen-tary”). However, while our proof that étale morphisms are local homeomorphisms in the strongtopology is a lost cause for being classified as elementary by someone not at ease with the theoryof schemes, the strong connectedness of algebraic curves might be less so.

Still, we will need some very weak form of the Riemann-Roch theorem in section 7 andthis is usually not considered to be elementary although most people with some background inalgebraic geometry will probably have encountered it at some point during their studies. We shallstate the theorem needed in this preliminary section rather than the section it is actually used.The reason for this is that we want the structure of the text to reflect the mental disjunctionbetween this non-elementary part of the proof and the more elementary part found in the abovementioned section.(3.1) Theorem. Let K be a field containing C with trdegCK = 1. Then there is somef ∈ K \ C such that for every affine algebraic curve C together with a chosen isomorphism ofC-algebras C(C) ∼= K (thus enabling us to view f as a rational function on C), f has at mostone zero on C. �

6 Paragraph 4. Analytification

4. Analytification

By definition, any affine algebraic variety (over the complex numbers) can be embedded asa closed subvariety into some affine space An. Moreover, any algebraic variety is “patchedtogether” by (finitely many) affine varieties. More precisely, any algebraic variety X has a finiteopen cover (X1, . . . , Xn) by affine varieties, meaning that for each i ∈ {1, . . . , n} the locallyringed space (Xi,OXi) is an affine variety (where OXi is the restriction of OX to the opensubspace Xi).

(4.1) Theorem. (Analytification) There is a functor called analytification

−an : AlgVarC → Top

from the category of algebraic varieties over the complex numbers to the category of topologicalspaces, which is characterised by the following properties:

(a) if Sp: AlgVarC → Top is the functor, which sends each algebraic variety to its underlyingtopological space and U : Top → Sets the forgetful functor, then U ◦ −an = U ◦ Sp andwe have a natural transformation −an ⇒ Sp, whose components are the identity maps;

(b) −an sends open (resp. closed) immersions to open (resp. closed) embeddings;

(c) −an preserves products;

(d) A1an = C with the usual topology induced by any norm on C.

(4.2) Remark. Property (a) in the theorem just says that X = Xan as sets and the topologyof Xan is stronger (i.e. finer) than the Zariski topology on X. Because of this, we speak of the“strong” (or “analytic”) topology on X. Moreover, a morphism ϕ : X → Y is sent to the samemap between sets and turns out to be continuous for the strong topologies on X and Y .

Proof. It is immediate that for n ∈ N, we must have Anan = Cn and moreover, if ϕ : Am → An isa morphism of affine spaces, that the analytification ϕan : Cm → Cn (which by definition is thesame as ϕ if viewed as a mere map between sets) is continuous.

More generally, for X any affine algebraic variety, we have X = Xan as a set and asthere is some closed immersion X ↪→ An into an affine space, Xan must carry the subspacetopology induced by Xan ↪→ Cn. This is independent of the closed immersion chosen, forif ϕ : X ↪→ Am and ψ : X ↪→ An are two closed immersions, and X ′an, X ′′an carry the initialtopology with respect to the inclusions ϕan : X ′an ↪→ Cm and ψan : X ′′an ↪→ Cn respectively, wecan lift the identity morphism idX : X → X to a morphism χ : Am → An, which then gives us acommutative square

X ′an_�

ϕan

��

1Xan // X ′′an_�

ψan��

Cm χan// Cn ,

where the upper morphism must be continuous because because it is a restriction of χan. Byreversing the roles of X ′an and X ′′an we also see that 1Xan : X ′′an → X ′an is continuous, so thatindeed X ′an = X ′′an as topological spaces.

For a general algebraic variety X and U ⊆ X an affine open subvariety, the inclusionUan ↪→ Xan must be an open embedding. Hence, the topology on Xan is coarser than the final

Paragraph 4. Analytification 7

topology with respect to all inclusions Uan ↪→ Xan with U ⊆ X any affine open subvariety of X.But it must be finer, too, because the set

U := {U ⊆ X | U affine and open}

provides an open cover of X and if we write iUan : Uan → Xan for the standard inclusions, eachof these must be open and continuous. Thence, if V ⊆ Xan is such that Uan ∩ V is open in Uanfor every U ∈ U , then we have

V =⋃U∈U

iUan(Uan ∩ V ),

which is must be open in Xan by openness of the iUan . �

One should observe that for proving the fact that for a general variety X, the topologyon the analytification Xan must be the final topology with respect to all inclusions Uan ↪→ Xanfor U ⊆ X affine and open, we only needed that these subvarieties cover all of X and in fact, bythe same argument, any cover by affine open subvarieties will do.

(4.3) Proposition. If X is an algebraic variety and X = X1∪ . . .∪Xn a cover by affine opensubvarieties, then Xan carries the final topology with respect to all inclusions (Xi)an ↪→ Xan fori ∈ {1, . . . , n}. �

Our approach to equip the set Xan = X with a topology is the one taken in probablymost of the more elementary expository texts, e.g. [Mumford, 1999] and [Neeman, 2007]. Analternative and quicker definition of the analytification that is mentioned in the original paper[GAGA] by Serre is the following: For an affine variety X, we put Xan := X as a set and equipit with the initial topology with respect to all regular functions X → C in OX(X) but withthe codomain C carrying the strong topology. Another alternative in a more abstract settingis provided by [SGA 1], which defines the analytification of a scheme of finite type over C as arepresenting object for a certain functor. Even just formulating this would require a vast amountof definitions (e.g. sheaves, locally ringed spaces and complex analytic spaces) but in principle,it boils down to the same characterisation as in the above theorem.

(4.4) Remark. As already implicitly mentioned in the last paragraph, analytification is notjust a mere functor to Top butXan carries the structure of a so-called analytic space (whence thename “analytification”), which can be viewed as a generalisation of a complex analytic manifoldto include the possibility of it having singularities. A special case of this is the following well-known theorem.

(4.5) Theorem. If we write SmAlgVarC for the full subcategory of AlgVarC with objectsthe smooth algebraic varieties and AnManC for the the category of complex analytic manifoldsthen we have a commutative square of functors

SmAlgVarC−an //

� _

��

AnManC

U��

AlgVarC −an// Top ,

where U is the forgetful functor. The functor −an : SmAlgVarC → AnManC is characterisedby conditions analogous to (a) – (d) in (4.1) with (d) in this context reading as “A1

an = C withthe usual complex analytic structure”.

8 Paragraph 5. Finite and Proper Morphisms

Proof. If X is a d-dimensional irreducible smooth affine variety then we choose a closed immer-sion X ↪→ An and furthermore, upon identifying X with its image in An, we choose generatorsf1, . . . , fs ∈ C[x1, . . . , xn] of IAn(X). By [Kraft, 2005, p.36, Proposition 4.3] we know that forevery x ∈ X the rank of the Jacobian matrix (∂fi/∂xj |x)i,j is n− d and w.l.o.g. we assume that(∂fi/∂xj |x)i,j∈{d+1,...,n} is invertible. Hence, by the implicit function theorem, there are U ⊆ Cd,V ⊆ Cn−d open and g : U → V holomorphic such that x ∈ U × V and

X ∩ (U × V ) = Γg = {(a, ga) | a ∈ U} .

Put differently, the standard projection Cn = Cd×Cn−d → Cd restricts to a biholomorphic mapX ∩ (U × V )→ U with inverse a 7→ (a, ga). This proves that Xan ⊆ Cn is a closed submanifoldand just as in the proof of (4.1), one checks that the complex analytic structure of Xan does notdepend on the chosen closed immersion of X into affine space.

If X is a general smooth variety then we can assume that it is irreducible because itsirreducible components are disjoint (i.e. they are even the connected components). Choosing anycover (X1, . . . , Xn) of X by irreducible affine open subvarieties yields an atlas for Xan becausethe transition functions are all polynomial. By a similar argument, one easily checks that ifϕ : X → Y is a morphism of smooth varieties and we equip Xan and Yan with the complexanalytic structures just defined, then ϕan is holomorphic. �

(4.6) Nomenclature. If P is any property then strongly P is the new property that we getby replacing each algebraic variety and each morphism of algebraic varieties occurring in P bytheir analytification. For instance, a subset U ⊆ X of an algebraic variety X is called stronglydense if U lies dense in Xan.

5. Finite and Proper Morphisms

It is proved in [SGA 1] (in the more general context of schemes of finite type over C) that theanalytification of proper morphisms (cf. the definition below) is again proper (in the topologicalsense). For the proof of this, a very general version of Chow’s lemma is needed. We will notinvoke this but instead prove a stronger version of it for morphisms with a quasi-projectivedomain. The trade-off is that we will only be able to prove the strong properness of propermorphisms for the case of a quasi-projective domain. However, our version of Chow’s lemma isenough to show that the analytification of a finite morphism is again finite (that is, closed withfinite fibres).

(5.1) Definition. Recall that a morphism ϕ : X → Y between affine varieties gives OX(X)the structure of an OY (Y )-algebra, via its comorphism ϕ∗ : OY (Y )→ OX(X). Now, ϕ is calledfinite iff OX(X) is a finite OY (Y )-algebra (i.e. it is finitely generated as an OY (Y )-module). Ageneral morphism ϕ : X → Y between algebraic varieties (not necessarily affine) is called finiteiff Y has a cover Y = Y1 ∪ . . . ∪ Yn by affine open subvarieties such that Xi := ϕ−1Yi ⊆ X isaffine and the restriction ϕ|Xi

Yi: Xi → Yi is finite for all i ∈ {1, . . . , n}.

(5.2) Definition. A morphism ϕ : X → Y of algebraic varieties is called proper iff for anyalgebraic variety Z, the product morphism

ϕ× 1Z : X × Z → Y × Z

is closed.

Paragraph 5. Finite and Proper Morphisms 9

(5.3) Example. Finite morphisms (between affine varieties) are proper and in particular,closed immersions are proper. To wit, if ϕ : X → Y is finite, then

OX(X) = OY (Y )f1 + . . .+ OY (Y )fmfor suitable f1, . . . , fm ∈ OX(X). If now, Z is any other affine variety, then OX×Z(X × Z), asan Abelian group, is generated by all regular functions of the form

f · h : X × Z → C, (x, z) 7→ fx · hz

with f ∈ OX(X) and h ∈ OZ(Z). Similarly for OY×Z(Y × Z). Hence, f1 · 1, . . . , fm · 1 generateOX×Z(X ×Z) as an OY×Z(Y ×Z)-module, meaning that ϕ× 1Z : X ×Z → Y ×Z is finite andin particular closed.

(5.4) Example. If ϕ : X → Y is a proper morphism and A ⊆ X a closed subvariety, thenthe restriction ϕ|A : A → Y is proper, too, which is an instance of the more general fact thatthe composite of two proper morphisms is proper.

(5.5) Proposition. A morphism ϕ : X → Y is proper iff it is universally closed. That is,the base-change X ×Y Y ′ → Y ′ of ϕ along any ψ : Y ′ → Y is closed.

Proof. The direction “⇐” is obvious, for if Z is any variety, the product map ϕ× 1Z is nothingbut the base-change of ϕ along the standard projection Y ×Z → Y . As for “⇒”, we look at thefollowing commutative diagram (where the pullback is taken in the category AlgVarC)

X × Y ′

ϕ×1Y ′��

X ×Y Y ′_�? _oo

��

// X

ϕ

��

Y × Y ′ Y ′? _oo // Y

and observe that the left-hand horizontal maps are both closed immersions (by separatedness).But the map on the left is closed (by hypothesis) and thence, so is the middle one. �

(5.6) Corollary. If ϕ : X → Y is a proper morphism and Y ′ ⊆ Y a subvariety, then therestriction ϕ−1Y ′ → Y ′ of ϕ is proper, too. �

(5.7) Theorem. Let ϕ : X → Y be a proper morphism and ψ : X → Z any other morphism.Then (ϕ,ψ) : X → Y × Z is again proper.

Proof. For W any variety, we need to check that

X ×W (ϕ,ψ)×1W−−−−−−→ Y × Z ×W

is a closed morphism. Let us write X ×W for the closure of the image in Y × Z × W andconsider the diagonal X × W

∆−→ X × W × X ×W , which has the standard projection toX × W as an obvious retraction and is thus a closed immersion. But observe that becauseX ×W ↪→ Y × Z ×W is a closed immersion, its product with 1X×W

X ×W ×X ×W ↪→ X ×W × Y × Z ×W ∼= X ×W × Y ×W × Z

is a closed immersion, too. Now, we consider the commutative diagram

X ×W � � ∆ //

(ϕ,ψ)×1W

��

X ×W ×X ×W � � // X ×W × Y ×W × Zϕ×1W×Y×W×Z

��

Y × Z ×W ∼=// Y ×W × Z � �

∆′×1Z

// Y ×W × Y ×W × Z ,

10 Paragraph 5. Finite and Proper Morphisms

where ∆′ : Y ×W → (Y ×W )2 is the diagonal and ϕ × 1W×Y×W×Z is closed by propernessof ϕ. It follows that the image of a closed A ⊆ X ×W in Y ×W × Y ×W × Z, which we callA, is closed and the injectivity of (∆′ × 1Z) implies that

((ϕ,ψ) × 1W

)A = (∆′ × 1Z)−1A. In

particular, the image of A in Y × Z ×W is closed. �

(5.8) Corollary. (Chow’s Lemma, Quasi-Projective Case) Let X ⊆ Pn be locallyclosed (i.e. X quasi-projective) and ϕ : X → Y a proper morphism of algebraic varieties. Then(ϕ, i) : X ↪→ Y × Pn is a closed immersion, where i : X ↪→ Pn denotes the inclusion.

Proof. By the above theorem, the image X of X in Y × Pn is closed and the the standardprojection Y × Pn → Pn restricts to an inverse of the morphism X → X, induced by (ϕ, i). �

(5.9) Corollary. In the same situation as in Chow’s Lemma, the analytification of ϕ isproper (in the topological sense). In particular, ϕ is strongly closed.

Proof. By Chow’s Lemma, ϕ factors as

X� � //

ϕ##HHHHHHHHHH Y × Pn

prY����

Y ,

where the horizontal morphism is a closed immersion. By the fact that analytification preservesclosed immersions and products, as well as the compactness of Pn in the strong topology, itfollows that ϕan is proper. �

(5.10) Corollary. The analytification of a finite morphism is finite (in the topological sense).In particular, finite morphisms are strongly closed.

Proof. Let ϕ : X → Y be finite and Y = Y1∪. . .∪Yn a cover of Y by affine open subvarieties, suchthat Xi := ϕ−1Yi ⊆ X is affine and the restriction ϕ|Xi

Yi: Xi → Yi is finite for all i ∈ {1, . . . , n}.

Now, by the last corollary, all these restrictions are strongly closed and by (2.3) from thetopological preliminaries, ϕ is itself strongly closed. �

This corollary can now be used to prove that the analytic topology on a variety, whilebeing stronger than the Zariski topology, is not “too strong” in the sense that the situationfrom Cn where Zariski-open subsets lie dense, generalises to the analytification of an arbitraryvariety. This result can also be found in [Mumford, 1999, p. 58, Theorem 1] and in fact, theproof given here is essentially the same as the one in loc. cit.. However, we can use the resultsobtained above to our advantage, whereas Mumford has to give a streamlined proof involvinga minor index skirmish. For readability’s sake, we shall split off a small part of the proof as alemma.

(5.11) Lemma. Let A ⊂ An be a proper closed subset. Then Aan ⊂ Cn is nowhere dense(i.e. it has no interior points).

Proof. Let a ∈ A arbitrary, b ∈ An \A and consider the straight line L through a and b. BecauseL ∼= A1 as a closed subvariety of An and L ∩ A ( L is a proper closed subset, L ∩ A must befinite. In particular every strongly open neighborhood of a intersects L \A. �

(5.12) Proposition. For X an irreducible algebraic variety and U ⊆ X a non-empty opensubset, U is strongly dense (i.e. dense in Xan).

Paragraph 5. Finite and Proper Morphisms 11

Proof. It suffices to check this for X affine, where by Noether’s normalisation lemma, we finda finite surjective morphism ϕ : X → An for n = dimX. Let us now define A := X \ U , whichis a proper closed subset of X and we need to check that Aan ⊂ Xan is nowhere dense, forwhich we consider a ∈ Aan arbitrary and b := ϕa ∈ (ϕA)an. Because A ⊂ X is a proper closedsubset dimA < dimX = n, so that also dimϕA < n, meaning that ϕA ⊂ An is a proper closedsubset again. By the last lemma, there is some sequence (bi)i∈N ∈

(Cn \ (ϕA)an

)N with limitlimi→∞ bi = b ∈ (ϕA)an.

As a next step, let us choose some h ∈ OX(X) such that the only zero of h in the fibreϕ−1b = ϕ−1ϕa is a itself and let H ∈ C[x1, . . . , xn][y] be the minimal polynomial of h. That isto say, H is the irreducible monic polynomial

H(x1, . . . , xn, y) = yd + c1(x1, . . . , xn)yd−1 + . . .+ cd(x1, . . . , xn) ∈ C[x1, . . . , xn][y]

such that H(x1, . . . , xn, h) = 0 ∈ C[x1, . . . , xn], which exists because ϕ is finite. This gives ustwo surjective and finite morphisms (by the transitivity of integral dependence)

X(ϕ,h)−−−→ VAn+1(H) pr1−−→ An ↔ OX(X) ⊇ C[x1, . . . , xn, h] ⊇ C[x1, . . . , xn],

whose composite is ϕ. If we abbreviate x := (x1, . . . , xn), H(x, h) = 0 implies that in particular0 = H(ϕa, ha) = cd(ϕa) = cd(b) and hence limi→∞ cd(bi) = 0 by continuity of cd. On the otherhand, cd(bi) is the product of all roots of H(bi, t) ∈ C[t], whence we can find a sequence (zi)i∈N ∈CN such that H(bi, zi) = 0 for all i ∈ N and limi zi = 0. Therefore, (bi, zi)i∈N is a sequence inV (H)an \

((ϕ, h)A

)an with limit (ϕa, 0) and hence (ϕa, 0) is no interior point of

((ϕ, h)A

)an. But

now, (2.9) implies that a is no interior point of Aan either because (ϕ, h)−1(ϕa, 0) = {a}. Sincea ∈ Aan was arbitrary, Aan ⊂ Xan is indeed nowhere dense as claimed. �

(5.13) Corollary. Let X be an irreducible algebraic variety and U ⊆ Xan a non-empty opensubset. Then U is dense in the Zariski topology.

Proof. Assume U ⊆ A for some proper closed A ⊂ X. Then X \ A ⊂ X is a non-empty opensubset but its closure in the strong topology is contained in X \ U 6= X. �

(5.14) Corollary. Let X be a algebraic variety and C ⊆ X a constructible subset. Thenthe closure of C in Zariski topology and that in the strong topology are equal.

Proof. As taking the closure is distributive over taking finite unions, it suffices to check this forC locally closed, meaning that C = A∩U with A ⊆ X closed and U ⊆ X open and by the sameargument, we can even assume that A is irreducible. But now, C = A ∩ U is an open subset ofthe irreducible subvariety A ⊆ X, so that C ⊆ A is dense in the Zariski topology, as well as inthe strong topology by the above corollary. This implies that C = A in the Zariski topology, aswell as in the strong topology. �

Using this result, we immediately deduce that the converse of (5.9) also holds. Thatis to say, given a morphism ϕ : X → Y , whose analytification is proper, ϕ is itself proper. Forthis, one only needs to observe that analytification preserves products, that the Zariski-closureof a constructible set is the same as the closure in the strong topology (as shown in the lastcorollary) and finally, that images of morphisms are constructible (Chevalley’s theorem).

(5.15) Remark. One could now use what we just discussed to prove the converse of (5.10)but one needs the following (non-trivial) characterisation of finite morphisms: A morphism isfinite iff it is proper and quasi-finite (i.e. has finite fibres).

12 Paragraph 6. Étale Morphisms

The fact that a morphism ϕ : X → Y with X quasi-projective is proper iff its analytifi-cation is so enables us to use the machinery of point-set topology to prove a fundamental resultconcerning complete varieties, which would otherwise be proved by algebraic reasoning, fiddlingaround with the construction of homogeneous ideals.

(5.16) Definition. A variety X is called complete iff the unique morphism X → {∗} isproper. That is to say that for each variety Y , the standard projection prY : X × Y → Y isclosed.

(5.17) Corollary. A quasi-projective variety X is complete iff Xan is compact. �

(5.18) Corollary. Any projective variety (in particular, Pn for any n ∈ N) is complete. �

(5.19) Corollary. Projective morphisms are proper.

Proof. Let ϕ : X → Y be projective, so that it factors as

Xϕ−→ Y = X

i−→ Y × PnprY−−� Y

for some n ∈ N and some closed immersion i : X ↪→ Y × Pn. But now, if Z is another variety,we get a factorisation

X × Z � � i×1Z //

ϕ,,XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Y × Pn × Z

∼= // Y × Z × Pn

prY×Z

��

Y × Z ,

where i× 1Z is obviously closed and prY×Z is also closed, by completeness of Pn. �

6. Étale Morphisms

The usual description of étale morphisms is that they are the algebraic analogue of local home-omorphisms. In most texts, the only justification for this claim is that it is obvious in the caseof smooth varieties. However, in the next section, we shall justify this statement for arbitraryvarieties (and it can even be justified for schemes of finite type over C).

(6.1) Definition. For X an algebraic variety (over C) and x ∈ X, we write CxX for thecotangent cone at x. It is defined as the graded C-algebra

CxX := grmxOX,x =

⊕n∈N

mnx/m

n+1x

(where m0x = OX,x) with the multiplication of two homogeneous elements [x] ∈ mm

x /mm+1x and

[y] ∈ mnx/m

n+1x defined as [x][y] := [xy] ∈ mm+n

x /mm+n+1x .

(6.2) Definition. A morphism ϕ : X → Y between two varieties (over C) is called étale atx ∈ X iff it induces an isomorphism of C-algebras ϕ∗ : CϕxY ∼= CxX. ϕ is called étale iff it isétale at every point x ∈ X. Because of the short 5-Lemma, this is equivalent to requiring thatϕ∗ : OY,ϕx/m

nY,ϕx → OX,x/m

nX,x is an isomorphism for all n ∈ N.

(6.3) Remark. Another equivalent definition (although we won’t need it) is the following:Let ϕ : X → Y be a morphism between two varieties and x ∈ X. Then ϕ is étale at x iff itinduces an isomorphism ϕ∗ : OY,ϕx

∼= OX,x between the completed stalks of ϕx and x.

Paragraph 6. Étale Morphisms 13

Proof. The direction “⇒” can be found in [Atiyah and Macdonald, 1969, p. 112, Lemma 10.23].As for the other direction, we have a natural isomorphism CxX = grmx

OX,x∼= grmx

OX,x by[ibid., p. 111, Proposition 10.22] and if ϕ∗ is an isomorphism, so is grmx

ϕ∗. �

(6.4) Observation. For X a variety and TxX the tangent space at x in the guise of

TxX := {δ : OX,x → C | δ(fg) = fx · δg + δf · gx}

the set of all derivations OX,x → C at x, we have a natural isomorphism

TxX ∼= HomC(mx/m2x,C), δ 7→

(δ|mx : mx/m

2x → C, [f ] 7→ δf

),

from which we immediately conclude that if a morphism ϕ : X → Y is étale at x ∈ X, thenthe differential Dϕx : TxX → TϕxY is an isomorphism. Conversely, if x and ϕx are smooth,then CxX and CϕxY are polynomial rings (in dimxX and dimϕx Y variables respectively) withthe standard grading. In particular, an isomorphism Dϕx : TxX ∼= TϕxY can be lifted to anisomorphism CxX ∼= CϕxY and hence, ϕ is étale at x.

We can use this observation to prove that the analytification of an étale morphismϕ : X → Y between smooth varieties is a local homeomorphism. This follows readily from thefact that for X and Y smooth, these carry the structure of a (complex analytic) manifold andeven better, for x ∈ X, the algebraic tangent spaces of x and ϕx are the same as the analyticones and we can use the inverse function theorem to prove our claim.

We now wish to prove this even for non-smooth varieties. The idea is to embedX and Yinto affine spaces Am and An, so that the morphism ϕ : X → Y extends to a morphism Am → An.But these are now smooth varieties and we can use the above considerations. However, fromthis we usually cannot derive that the original ϕ must be a local homeomorphism in the strongtopology except if we can control the embeddings. As it turns out, X and Y can locallybe embedded nicely, such that the morphism ϕ between the embedded varieties is so-called“standard étale”. But we have to work a little bit to arrive at this point.

(6.5) Proposition. Étale morphisms have finite fibres.

Proof. Let ϕ : X → Y be étale and y ∈ Y . Taking the fibre above y, we get a commutativesquare as shown on the left below and taking differentials at an arbitrary point x ∈ ϕ−1y, weget a commutative square as shown on the right.

ϕ−1y� � //

��

X

ϕ

��

Tx(ϕ−1y) � � //

��

TxX

∼= Dϕx

��

{y} � � // Y 0 � � // TyY

Because Dϕx : TxX ∼= TyY is an isomorphism, we conclude that Tx(ϕ−1y) = 0 and hencex ∈ ϕ−1y is an isolated point. But as ϕ−1y ⊆ X is a closed subvariety, it is in particularquasi-compact and the claim follows. �

(6.6) Remark. Recall that if Y is an affine variety and X ⊆ Y a closed subvariety then theinclusion X ↪→ Y induces for each x ∈ X an inclusion of tangent spaces

res∗ : TxX ↪→ TxY, δ 7→ δ ◦ res,

where res : OY (Y ) � OX(X) is the restriction map. If IY (X) = 〈g1, . . . , gn〉 P OY (Y ) forsuitable g1, . . . , gn ∈ OY (Y ), then the subspace TxX 6 TxY has a more computable description:Considering (g1, . . . , gn) : Y → An and taking the differential at x, it turns out that

TxX = Ker D(g1, . . . , gn)x.

14 Paragraph 6. Étale Morphisms

Proof. If we write g := (g1, . . . , gn) : Y → An then X = g−10 is the fibre of g over 0. This fibreis obviously reduced because g∗ : C[y1, . . . , yn]→ OY (Y ), yi 7→ gi and hence

OX(X) ∼= OY (Y )/〈g∗mCn,0〉OY (Y ).

As localising commutes with taking quotients and the localisation of an ideal is the extendedideal in the localised ring, we conclude that OX,x

∼= OY,x/〈g∗mCn,0〉OY,xand hence 〈g∗mCn,0〉OY,x

is radical, i.e. the fibre X = g−10 is reduced. �

(6.7) Proposition. Let X, Y be affine varieties together with an isomorphism of C-alge-bras OX(X) ∼= OY (Y )[t1, . . . , tn]/〈p1, . . . , pn〉 for some n ∈ N, p1, . . . , pn ∈ OY (Y )[t1, . . . , tn].Moreover, let ϕ : X → Y be a morphism such that ϕ∗ : OY (Y )→ OX(X) is the canonical map

OY (Y ) ↪→ OY (Y )[t1, . . . , tn]� OY (Y )[t1, . . . , tn]/〈p1, . . . , pn〉 ∼= OX(X).

For x ∈ X, the following are equivalent:

(a) the differential Dϕx : TxX → TϕxY is an isomorphism;

(b) the Jacobi matrix

Jx := Jac(p1, . . . , pn)|x =[∂pi∂tj

(x)]i,j

is invertible.

Moreover, if these two conditions are satisfied then ϕan is a local homeomorphism at x.

Proof. By hypothesis, we can identify X with the subspace VY×An(p1, . . . , pn) ⊆ Y × An andunder this identification, ϕ is nothing but the restriction of the standard projection. That is,we have a commutative triangle

X

ϕ$$HHHHHHHHHH

� � // Y × An

prY����

Y .

By the chain rule and the fact that the differential of the inclusion X ↪→ Y × An is just theinclusion of tangent spaces, the differential Dϕx at x = (y, a) (with y ∈ Y and a = (a1, . . . , an) ∈An) is given by restricting (D prY )x to TxX. Observe that we have natural isomorphisms

Tx(Y × An) ∼= TyY ⊕ TaAn, δ 7→ (δ1, δ2), ε←[ (ε1, ε2),

where δ1g := δ(g · 1), δ2h := δ(1 · h) and ε(g · h) := ε1g · h(a) + g(y) · ε2h; as well as

TaAn ∼= An, δ 7→ (δt1, . . . , δtn), ∂v|a ←[ v.

Putting these together, p = (p1, . . . , pn) : Y × An → An gives us a commutative square

Tx(Y × An)∼=

��

Dpx// TpxAn

∼=

��

TyY ⊕ TaAn∼=

��

TyY ⊕ An // An

Paragraph 6. Étale Morphisms 15

and chasing an element (ε, v) ∈ TyY ⊕ An around the diagram, its image is(εpi(−, a) + ∂vpi(y,−)|a

)i∈{1,...,n}.

In particular for j ∈ {1, . . . , n}, the image of (0, ej) is(∂pi(y,−)∂tj

(a))i

=(∂pi∂tj

(x))i

.

To sum this up, under all identifications we made above, the differential of p at x is

Dpx =[∗[∂pi∂tj

(x)]i,j

]=[∗ Jx

]: TyY ⊕ An → An.

Now, as IY×An(X) = 〈p1, . . . , pn〉, by the above remark

TxX = KerDpx = Ker[∗ Jx

]and again upon identifying Tx(Y ×An) with TyY ⊕TaAn, (D prY )x is nothing but the standardprojection π : TyY ⊕ TaAn � TyY . Recalling that Dϕx = (D prY )x|TxX = π|TxX , we immedi-ately see that Dϕx is invertible iff Jx is so. To wit, if Dϕx is invertible then there’s for everyδ ∈ TyY exactly one v ∈ An such that (δ, v) ∈ TxX, meaning that ∗δ + Jxv = 0. In particular,putting δ = 0, there is exactly one v ∈ An such that Jxv = 0, which must be v = 0. HenceKer Jx = 0 and by the rank-nullity theorem, Jx is invertible. Conversely, if Jx is invertible thenDϕx is an isomorphism, for if δ ∈ TyY and (δ, v) ∈ TxX for some v ∈ An then ∗δ + Jxv = 0and consequentially v = −J−1

x ∗ δ, so that Dϕx is indeed bijective.For the second claim, let’s choose a closed immersion Y ⊆ Am for some m ∈ N and

construct from this the commutative diagram

Y × An

prY����

� � // Am × An

prAm

����

X, �

;;vvvvvvvvv

ϕ// Y

� � // Am .

By choosing preimages P1, . . . , Pn of p1, . . . , pn in C[y1, . . . , ym, t1, . . . , tn] we get a regular map

f : (P1, . . . , Pn) : Am × An → An

of which we know X = (Y × An) ∩ f−10 and whose differential at x = (y, a) ∈ X ⊆ Y × An is

Dfx =[∗ Jx

].

But x is a smooth point in Am ×An, so that by the implicit function theorem, we find U ⊆ Cmand V ⊆ Cn open with x = (y, a) ∈ U × V as well as a unique smooth (even holomorphic) mapg : U → V such that for all (u, v) ∈ U × V , we have f(u, v) = 0 iff v = gu. But this means thatthe restriction of (1U , g) : U → U × V to

ψ : Yan ∩ U → Xan ∩ (U × V )

(which makes sense because f(y′, gy′) = 0 for all y′ ∈ U and in particular, if y′ ∈ Y then(y′, gy′) ∈ (Y × Cn) ∩ f−10 = X) provides a local inverse of ϕan. �

16 Paragraph 7. Irreducible Algebraic Curves

(6.8) Example. Let ϕ : X → Y be a morphism between affine algebraic varieties and sup-pose that there is an isomorphism of C-algebras OX(X) ∼= (OY (Y )[t]/〈p〉)q, where p ∈ OY (Y )[t]is monic and q ∈ OY (Y )[t]/〈p〉 with [p′] invertible in (OY (Y )[t]/〈p〉)q (e.g. q = [p′]). Moreover,this isomorphism is required to be compatible with ϕ in the sense that the following trianglecommutes.

OY (Y ) ϕ∗//

''OOOOOOOOOOOOX(X)

∼=��

(OY (Y )[t]/〈p〉)q ,

where the diagonal arrow is the canonical morphism

OY (Y ) ↪→ OY (Y )[t]� OY (Y )[t]/〈p〉 → (OY (Y )[t]/〈p〉)q , g 7→ [g]/1.

In this case, the differential Dϕx is an isomorphism at every x ∈ X and hence, by the propositionjust proved, ϕan is a local homeomorphism.

Proof. We identify OX(X) ∼=(OY (Y )[t]/〈p〉

)q∼= OY (Y )[t, s]/〈p, qs− 1〉 and calculate for x ∈ X

det

∂p

∂t(x) ∂p

∂s(x)

∂(qs− 1)∂t

(x) ∂(qs− 1)∂s

(x)

= det[p′(x) 0∗ q(x)

]= p′(x)q(x) = (p′q)(x) 6= 0

because p′ and q are invertible in OX(X). As ϕ∗ is the standard projection upon the aboveidentification, the last lemma proves our claim. �

(6.9) Remark. One can even show that ϕ from this example is étale and we call a morphismof this form standard étale. In fact, the conditions for ϕ : X → Y and x ∈ X as in the propositionare equivalent to ϕ being étale at x ∈ X.

(6.10) Theorem. Étale morphisms are locally standard étale. I.e. a morphism ϕ : X → Ybetween two algebraic varieties is étale iff for every x ∈ X there are affine open neighborhoodsU 3 x and V 3 ϕx with ϕU ⊆ V and ϕ|UV : U → V standard étale.

Proof. [Milne, 1980, p. 26, Theorem 3.14] �

(6.11) Corollary. For ϕ : X → Y an étale morphism, the analytification ϕan : Xan → Yan isa local homeomorphism. �

7. Irreducible Algebraic Curves

In the analytic proof that irreducible plane algebraic curves are connected in the strong topology,we investigated some restriction of the projection onto the x-axis, which turned out to be acovering map. Afterwards, we used analytic tools to study the monodromy action of this coveringmap and were able to show that it is transitive.

Now, our algebraic proof relies on the observation that only considering the projectiononto some coordinate axis is a vast constraint and not really necessary. The intuition is thatby allowing more general morphisms (namely finite or even more general, proper ones) that arestill covering maps on a dense open subvariety, one should be able to collapse the covering’ssheets in one point, thus rendering all arguments for transitivity of the monodromy action inthat point needless.

Paragraph 7. Irreducible Algebraic Curves 17

(7.1) Lemma. Let ϕ : X → Y be a surjective proper morphism between two irreducibleaffine varieties. Then there is a dense open subset V ⊆ Y such that Ysing∩V = Xsing∩ϕ−1V = ∅and the restriction ϕ′an : (ϕ−1V )an → Van of ϕan is a covering map (in the strong topology thatis). Moreover, if ϕ is of finite degree d = [C(X) : C(Y )] then V be chosen in such a way that ϕ′anis a d-sheeted covering.

Proof. By [Kraft, 2005, p.46, Theorem 4.2] there is a dense open U ⊆ X such that Dϕu issurjective for all u ∈ U and we write S := ϕ(Xsing)∪ϕ(X\U). This is a proper closed subset of Ybecause dimXsing and dim(X \U) are strictly smaller than dimX = dimY . Putting V := Y \S,one easily checks ϕ−1V ⊆ X \Xsing ∩ U , and thus V ⊆ Y \ Ysing by definition of U . Moreover,ϕ restricts to a surjective proper and smooth (i.e. all differentials are isomorphisms) morphism

ϕ′ : ϕ−1V = X \ ϕ−1S → Y \ S = V.

By the inverse function theorem, ϕ′an is a local homeomorphism and by (5.9) it is also proper.But then (2.7) gives us that ϕ′an is a covering map. As for the last claim, if ϕ is of finitedegree d then by [Kraft, 2005, p.34, Proposition 3.8] there exists a dense open V ′ ⊆ Y suchthat #ϕ−1y = d for all y ∈ V ′ and restricting ϕan even further to

(ϕ−1(V ∩V ′)

)an → (V ∩V ′)an

yields the desired result. �

(7.2) Theorem. Let C be an irreducible affine curve, together with a point a ∈ C and aproper surjective morphism ϕ : C → A1, whose fibre above ϕa consists of a itself only. Then Cis connected in the strong topology.

Proof. By the lemma just proven, there is a dense open V ⊆ A1 such that Csing ∩ ϕ−1V = ∅and the restriction ϕ′ : ϕ−1V → V of ϕ is a covering map in the strong topology. Becausethis is a finite covering and Van is path-connected, (ϕ−1V )an has only finitely many connectedcomponents. As (ϕ−1V )an lies dense in Can, this implies that for every b ∈ Can there is asequence that is contained in a single connected component of (ϕ−1V )an and converges to b.

Hence it suffices to prove that if M ⊆ (ϕ−1V )an is a connected component, the closureof M in Can contains a. To do so, we choose a sequence (yn)n∈N in V converging to ϕa, whichwe can do because Van lies dense in C. But by (2.10) the restriction ϕ′|M : M → V is surjective,so that (yn)n∈N lifts to a sequence (xn)n∈N in M . Observing that ϕan is proper, we concludethat ϕ−1

an {yn}n∈N ⊆ Can is compact and therefore (xn)n∈N has a convergent subsequence, whoselimit we call x. But ϕan is continuous, (ϕanxn)n∈N converges to ϕana and the fibre above ϕanaconsists only of a itself, so that x = a. �

Rather than now constructing for every irreducible affine curve C a proper surjectivemorphism ϕ : C → A1 that has some one-point fibre, we start with a rational function f ∈ C(C)as in (3.1), and then manipulate the model curve C of C(C) in such a way that f becomes afinite regular function. Because f was chosen such that it behaves well under manipulation ofthe domain, this will then allow us to use the theorem just proved, and if our manipulation iscareful enough we can still deduce the strong connectedness of C from that of the manipulateddomain. But first, we shall prove the following elementary observation.

(7.3) Lemma. Let k be a finitely generated field extension of C and f1, . . . , fn a tran-scendence basis. If A is the integral closure of C[f1, . . . , fn] in k, then its field of fractions isFracA = k.

Proof. Let S := C[f1, . . . , fn] \ {0}, which is a multiplicative system because k is an integraldomain. By [Atiyah and Macdonald, 1969, p. 62, Proposition 5.12] S−1A is the integral closureof S−1C[f1, . . . , fn] = C(f1, . . . , fn) in S−1k = k. But k is integral over C(f1, . . . , fn) becausef1, . . . , fn is a transcendence basis of k over C and hence k ⊆ S−1A ⊆ FracA. �

18 Paragraph 8. Irreducible Algebraic Varieties

(7.4) Theorem. Every irreducible affine curve C is connected in the strong topology.

Proof. Without loss of generality, we may assume that C is normal, for otherwise we have asurjective finite morphism C → C with C the normalisation of C and by surjectivity, if C isstrongly connected, so is C.

Now we choose some f ∈ C(C) \ C as in (3.1) (i.e. such that for every affine algebraiccurve C ′ together with a chosen isomorphism C(C ′) ∼= C(C), f has at most one zero on C ′) andconsider the integral closure A of C[f ] in C(C), yielding a surjective finite morphism

A ⊇ C[f ] ∼= C[x] ↔ ϕ : MaxA� MaxC[f ] ∼= A1,

where MaxA is the maximal spectrum of A, carrying the Zariski topology (which is an affinealgebraic variety by [Kraft, 2005, p.51, Proposition 5.1]) and the isomorphism C[x] ∼= C[f ] isgiven by x 7→ f . By surjectivity, ϕ−10 6= ∅ and by definition of f , it is even a one-point fibre.

It therefore follows from the above theorem, that MaxA is strongly connected and bythe last lemma (7.3) C(MaxA) = C(C), so that there are (non-empty) isomorphic special opensubsets U ⊆ MaxA and V ⊆ C. But MaxA \ U is only a finite number of points, which aresmooth because A is integrally closed (i.e. MaxA is a normal curve) and therefore U ⊆ MaxAis still connected; hence so is V ⊆ C. Finally, this implies that V = C is connected, too. �

8. Irreducible Algebraic Varieties

(8.1) Lemma. Let f : Cn → C be a continuous function that is holomorphic on a non-empty Zariski-open (hence dense) subset U ⊆ Cn. If f satisfies an integral dependence rela-tion over C[x1, . . . , xn] (i.e. there is some monic p(t) = td −

∑d−1i=0 pit

i ∈ C[x1, . . . , xn][t] suchthat p(f) : Cn → C is the zero-function) then f is a polynomial function.

Proof. We observe that we find some M ∈ N, R ∈ R>1 such that |pi(x)| 6 ‖x‖M for all i ∈{0, . . . , d− 1} and all x ∈ Cn with ‖x‖ > R. If x ∈ Cn with ‖x‖ > R is such that |fx| > 1 then

|fx|d =∣∣∣∣∣d−1∑i=0

pi(x)f(x)i∣∣∣∣∣ 6

d−1∑i=0‖x‖M |fx|i 6

d−1∑i=0‖x‖M |fx|d−1 = d ‖x‖M |fx|d−1

and therefore |fx| 6 d ‖x‖M . On the other hand, if x ∈ Cn with ‖x‖ > R is such that |fx| < 1then trivially again |fx| 6 d ‖x‖M as before.

Now we proceed by induction on n. If n = 1 then C \ U is finite and by the Riemannextension theorem f is entire (i.e. holomorphic on all of C). But by the above considerations,its Taylor series around 0 has zero-coefficients in degree > M , whence f is polynomial. For theinductive step, consider f : Cn+1 → C, U ⊆ Cn+1 as in the claim and let

V := pr1 U = {x ∈ Cn | U ∩ ({x} × C) 6= ∅} ,

where pr1 : Cn × C → Cn, (x, y) 7→ x, which is open in the Zariski topology. By the inductivehypothesis f |{x}×C : {x}×C ∼= C→ C is a polynomial function for every x ∈ V , so that we havea family of functions a0, . . . , aM : V → C (with M ∈ N as above) satisfying

f(x, y) =M∑i=0

ai(x)yi for all x ∈ V, y ∈ C

(the degree is 6 M for all x ∈ V by the upper bound established above). Furthermore, if weconsider pr2 : Cn × C→ C, (x, y) 7→ y, which is again open in the Zariski topology,

W := pr2 U = {y ∈ C | U ∩ (Cn × {y}) 6= ∅}

Paragraph 8. Irreducible Algebraic Varieties 19

is a non-empty Zariski-open (i.e. cofinite) subset of C and we can choose pairwise distinctc0, c1, . . . , cM ∈ W . This then gives us a system of linear equations f(x, cj) =

∑Mi=0 ai(x)cij

for j ∈ {0, . . . ,M} and x ∈ V or in matrix notation

[f(x, cj)

]j∈{0,...,M} = C

[ai(x)

]i∈{0,...,M} where C =

1 c0 c2

0 . . . cM01 c1 c2

1 . . . cM1...

...... . . . ...

1 cM c2M . . . cMM

is the Vandermonde matrix defined by the cj . As C is invertible, we find for every i ∈ {0, . . . ,M}suitable λi,0, . . . , λi,M ∈ C (namely λi,j = (C−1)i,j with matrix indices starting at 0) such thatai(x) =

∑Mj=0 λi,jf(x, cj) for all x ∈ V and can define

g : Cn+1 = Cn × C→ C, (x, y) 7→M∑i=0

M∑j=0

λi,jf(x, cj)yi,

which is a polynomial function by the inductive hypothesis. But now f(x, y) = g(x, y) forall (x, y) ∈ V × C and this is a dense open subset of Cn+1, implying that f = g by (2.1). �

(8.2) Remark. In fact, our reasoning from the induction basis works in arbitrary dimension.A continuous function f : Cn → C that is holomorphic on a non-empty Zariski-open U ⊆ Cnis entire because Cn \ U is a thin subset of Cn in the sense of [Gunning and Rossi, 1965, p.19]and we can use the Riemann extension theorem as found in loc. cit.. Our lemma then just saysthat C[x1, . . . , xn] is integrally closed in the ring HCn(Cn) of holomorphic functions on Cn.

(8.3) Theorem. Let X be an irreducible algebraic variety. Then Xan is connected.

Proof. It suffices to consider the case where X is affine. By Noether normalisation, we choosea finite surjective morphism p : X → An, yielding a finite field extension C(x1, . . . , xn) ⊆ C(X).We take the Galois closure

C(x1, . . . , xn) ⊆ C(X) ⊆ K with Galois group G := AutC(x1,...,xn)K,

which is finite because C(x1, . . . , xn) ⊆ C(X) is finite, and consider the integral closure Aof C[x1, . . . , xn] in K. By [Kraft, 2005, p.51, Proposition 5.1] this defines an irreducible normalaffine variety Z := MaxA and because OX(X) is integral over C[x1, . . . , xn], it comes equippedwith a finite surjective morphism ϕ : Z → X. Now the Galois group G acts faithfully on Zbecause FracA = K by (7.3) and hence if f ∈ G with fa = a for all a ∈ A then A ⊆ Fix〈f〉 butthen also FracA = K = Fix{idK} ⊆ Fix〈f〉, so that f = idK . Moreover, the composition

Zq−→ An := Z

ϕ−→ Xp−→ An

is the geometric quotient Z/G, i.e. AG = C[x1, . . . , xn]. As this is surjective, it suffices to checkthat Zan is connected. For this notice that each fibre of q contains exactly one closed G-orbit(cf. [Kraft, 1984, p.96, Bemerkung 1]) and because q is finite, each fibre of q is a single G-orbit(or put differently, G acts transitively on the fibres of q) and the surjectivity of q implies thatit is in fact a quotient of Z as a G-set. By (7.1) there is some dense open U ⊆ An such thatZsing∩q−1U = ∅ and the restriction q′ : q−1U → U of q is a covering map in the strong topology.

Now let Zan = Z1 ∪ . . . ∪ Zd be the decomposition of Zan into its connected compo-nents. These are only finitely many because q′an is a finite covering and (q−1U)an ⊆ Zan isdense. If we write qi := qan|Zi : Zi → Cn for i ∈ {1, . . . , d} then the qi are surjective by (2.10)

20 References

because Zi ∩ (q−1U)an contains a connected component of (q−1U)an and Uan ⊆ Cn is dense andpath-connected. Furthermore, the qi are proper because the Zi are are closed in Z.

As seen above, G acts transitively on the fibres of q′, implying that for every two i, j ∈{1, . . . , d} there is some g ∈ G such that gZi = Zj (in particular Zi ∼= Zj). For i ∈ {1, . . . , d}let us define

Ni := NormG(Zi) = {g ∈ G | gZi = Zi} / G,

which has #Ni = n/d and acts transitively on the fibres of qi. In particular, qi induces acontinuous bijection Zi/Ni → Cn that is also closed (because qi is so), hence a homeomorphism.

Let us write C(Zi) for the C-algebra of continuous functions Zi → C (with all operationsdefined pointwise). If i ∈ {1, . . . , d} then Zi ⊆ Zan is open and therefore Zi ⊆ Z lies dense (inthe Zariski topology) by (5.13). Hence the restriction map

ϕi : OZ(Z)→ C(Zi), f 7→ f |Zi

is an injective morphism of C-algebras, so that this equips C(Zi) with the structure of a faithfulOZ(Z)-algebra. By restriction of scalars, C(Zi) becomes a C[x1, . . . , xn]-algebra and each ϕiis a morphism of C[x1, . . . , xn]-algebras. Moreover, each ϕi is Ni-equivariant and in particularϕi(OZ(Z)Ni

)⊆ C(Zi)Ni , implying that each f ∈ OZ(Z)Ni gives rise to a continuous function

f : Cn ∼= Zi/Ni → C

induced by ϕif . Furthermore, (q−1U)an is naturally a complex analytic manifold, (f |(q−1U))anis holomorphic, q′an a local homeomorphism and we have a commutative diagram

(q−1U)anq′an //

_�

��

Uan_�

��

Zanqan

//

fan%%KKKKKKKKKK Cn

f��

C ,

so that f is holomorphic on Uan. Now f is integral over C[x1, . . . , xn] (because q is finite) butas ϕi is a morphism of C[x1, . . . , xn]-algebras, so is f . By the last lemma then, f is a polynomialfunction, so that we conclude

(ϕiOZ(Z)

)Ni = C[x1, . . . , xn].But now we are done because each ϕi induces an isomorphism OZ(Z) ∼= ϕiOZ(Z)

that is Ni-equivariant, hence restricting to OZ(Z)Ni ∼=(ϕiOZ(Z)

)Ni = C[x1, . . . , xn] for everyi ∈ {1, . . . , d}. Therefore also OZ(Z)Ni = C[x1, . . . , xn], implying Ni = G for every i ∈ {1, . . . , d}and thus d = 1 as claimed. �

References

Atiyah, M. F.; Macdonald, I. G.; Introduction to Commutative Algebra; Addison-Wesley Pub-lishing Co., Reading, MA; Menlo Park, CA; London; DonMills, ON; 1969

De Capitani, O.; Can a Finite Number of Points on an Irreducible Affine Variety be Connectedby an Irreducible Curve?; Master’s thesis; University of Basel; 2006

Forster, O.; Lectures on Riemann Surfaces; Graduate Texts in Mathematics, vol. 81; Springer-Verlag, New York; 1981

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Grothendieck, A.; Raynaud, M.; Revêtements étales et groupe fondamental (SGA 1); SociétéMathématique de France; 2003

Gunning, R. C.; Rossi, H.; Analytic Functions of Several Complex Variables; Prentice-Hall Inc.,Englewood Cliffs, N.J.; 1965

Hatcher, A.; Algebraic Topology; Cambridge University Press, Cambridge; 2002

Kraft, H.; Geometrische Methoden in der Invariantentheorie; Aspects of Mathematics, D1;Friedr. Vieweg & Sohn, Braunschweig; 1984

Kraft, H.; Algebraic Transformation Groups - Part 6: Basic Algebraic Geometry; 2005; URLhttp://www.math.unibas.ch/~kraft/Notizen/AlgTG.pdf

Milne, J. S.; Étale Cohomology; Princeton Mathematical Series, vol. 33; Princeton UniversityPress, Princeton, N.J.; 1980

Mumford, D.; The Red Book of Varieties and Schemes; Lecture Notes in Mathematics, vol. 1358;Springer-Verlag, Berlin; 1999

Neeman, A.; Algebraic and Analytic Geometry; Cambridge University Press, Cambridge; 2007

Serre, J.-P.; Géométrie Algébrique et Géométrie Analytique; Ann. Inst. Fourier, Grenoble; vol. 6pp. 1–42; 1956