This section shall be the answer to question (II.2). After giving some basic notions and results we will define anabstract nonsingular curve and prove that every nonsingular projective curve is isomorphic to such an abstract nonsingular curve for a given function field of dimension 1 overk. The answer we are looking for is an easy consequence of this.
We will use the following convention:
II.7. ABSTRACT NONSINGULAR CURVES 35 Convention. By afunction field F/kwe always mean a function field F of transcen-dence degree one overk such that kis the full constant field of F. From the previous section, we know that this is no restriction for our purposes.
By acurve we always mean an absolutely irreducible curve.
At first, we give a summary of results from commutative algebra.
Definition II.7.1. Let (A,mA) and (B,mB) be local rings contained in a field F. If A⊆B and mB∩A=mA, we say thatB dominates A.
Theorem II.7.2. Let R be a local ring contained in a fieldF. Then:
1. R is a valuation ring ofF if and only if R is a maximal element of the set of all local rings contained inF with respect to the relation of domination.
2. R is dominated by some valuation ring of F. Proof. See [Bou59, Ch. VI, §1, p. 3].
Definition II.7.3. Let A be a Noetherian domain of dimension one. A is called a Dedekind domain, if it is integrally closed.
Theorem II.7.4. LetAbe a Noetherian domain of dimension one. Then the following are equivalent:
1. A is a Dedekind domain.
2. Ap is a discrete valuation ring for any nonzero prime ideal p.
Proof. See [AM69, Ch. 9, theorem 9.3, p. 95].
Theorem II.7.5. The integral closure of a Dedekind domain in a finite extension field of its quotient field is again a Dedekind domain.
Proof. This is [ZS75, Ch. V, theorem 19, p. 281].
Theorem II.7.6. LetAbe an integral domain, which is a finitely generatedk-algebra and putL:= Quot(A). IfF/L is a finite field extension, then the integral closure ofA inF is finitely generated as an A-module and as a k-algebra.
Proof. See [ZS75, Ch. V, theorem 9, p. 267].
Now, we are able to prove the first step towards our final goal, namely that every discrete valuation ring of a function fieldF/kis isomorphic to the local ring of a point on some nonsingular affine curve.
Lemma II.7.7. IfF/k is a function field andx∈F, then the set {R∈CF |x /∈R}
is finite, whereCF denotes the set of all discrete valuation rings of F/k.
Proof. For (R,mR)∈CF we know that k⊆R, i.e. we may assumex /∈k. Certainly, x /∈R ⇐⇒ 06=y:= 1
x ∈mR,
which implies that {R ∈CF |x /∈R} ={R ∈CF |y ∈mR}. But k is assumed to be algebraically closed inF, i.e. y is transcendental overkasy /∈k. So by definition of a function field,F is a finite field extension ofk(y). Clearly, k[y] is a Dedekind domain (e.g. by [AM69, Ch. 9, exm. 1, p. 96]) and so the integral closure ofk[y] inF, denoted byB, is a Dedekind domain by theorem II.7.5. Also, by theorem II.7.6, B is a finitely generated k-algebra. This means by lemma II.1.13 together with lemma II.1.11 that B is the affine coordinate ring of an affine curve C ⊆ An for some n ∈ N (see also proposition II.6.6). Moreover, C is nonsingular by theorem II.7.4, since OP ∼=BpP for every point P ∈C.
On the other hand, y ∈ mR, so k[y] ⊆ R, which implies that B ⊆ R, as R is integrally closed inF by theorem II.5.6. Clearly, n:=mR∩B 6= 0 (as 06=y ∈n) is a maximal ideal ofB, as it is the contraction of a maximal ideal, so we can formBn, the localization ofB at the nonzero primen.
Claim. Bn=R. In particular, Quot(B) =F.
Proof. First of all, we havek⊆B ⊆Bn⊆Quot(B)⊆F by definition and the fact that Bis an integral domain. We denote the valuation corresponding toRbyvR. If ab ∈Bn, thenvR(ab) =vR(a)−vR(b)≥0 asa∈B ⊆R and b /∈n(i.e. vR(b) = 0), showing that Bn⊆R. SincenBn=nn (see [AM69, Ch. 3, p. 41]) is the unique maximal ideal ofBn, a similiar calculation yields thatmR∩Bn=nBn, i.e. R dominatesBn.
On the other hand, Bn is a discrete valuation ring of F/k by theorem II.7.4. But this in turn means by theorem II.7.2(1) thatBn=R.
Now, by lemma II.5.9 we can choose the unique pointP on C corresponding to n, which means thaty, regarded as an element ofk[C], vanishes at P.
For differentR∈CF we get different pointsP on C(recall thatC depends only on y) since R =BpP for some point P ∈ C by the above. But y 6= 0, so it vanishes only at a finite set of points, which means that there can only be finitely manyR∈CF. Corollary II.7.8. Any discrete valuation ring of F/k is isomorphic to the local ring at a point on some nonsingular affine curve.
Proof. Pick R ∈ CF and y ∈ R\k. As in the proof of the previous lemma, we can construct a smooth affine curve C such that k[C]∼= B, where B denotes the integral closure of k[y] in F. Furthermore, we have seen that it exists a point P on C with OP ∼=k[C]pP ∼=R.
This corollary is the reason why we sometimes call the elements ofCF points, and write P ∈ CF, where P actually means a discrete valuation ring RP ∈ CF. The following lemma tells us something about the size ofCF:
II.7. ABSTRACT NONSINGULAR CURVES 37 Lemma II.7.9. LetV be a nonempty affine variety. Then:
dimV = 0 ⇐⇒ V consists of only finitely many points.
Proof. This is [Kun85, Ch. II, proposition 3.11, p. 56].
This means that an affine curve always consists of infinitely many points. Together with lemma II.5.16, the corollary implies thatCF is infinite. We can define a topology onCF by taking∅, finite sets and CF asclosed sets.
Definition II.7.10. LetU ⊆CF be an open subset. We define O(U) := \
P∈U
RP
asthe ring of regular functions onU. Its elements are called regular functions.
Indeed,f ∈ O(U) defines a function fromU toA1 by takingf(P) asf modulomP, the maximal ideal ofRP. This is an element in K since
RP/mP ∼=k[C]pP/(pPk[C]pP)∼=k[C]/pP ∼=k(ξ1, . . . , ξn),
where eachξi is the image of Xi in k[C]pP/(pPk[C]pP) and is therefore algebraic over k. So we may always think of RP/mP as embedded in K, and we have the following tower of fields:
k⊆RP/mP ⊆K.
Some easy but important consequences can be drawn from the definitions:
Lemma II.7.11. Let F/kbe a function field over k. Then:
1. CF is an irreducible topological space.
2. CF is Noetherian and therefore compact.
3. In the sense of section II.3, the fieldR(CF) of rational functions ofCF is simply F itself.
Proof. 1. This is example II.1.4.
2. Let V1 ⊇ V2 ⊇ · · · be a descending chain of closed subsets of CF. If Vi = CF
for all i, we are done. Otherwise, it exists anisuch thatVi $CF, which implies that Vj is finite for all j ≥i. But then it exists a k≥j such that either Vk =∅ or Vk = Vk+l for all l ∈ N. So CF is Noetherian and compact by proposition II.1.14(2).
3. Letf, g ∈ O(U) for some open subset U ⊆CF with f(P) =g(P) for all P ∈U. Then, f−g∈mP for all P ∈U. But U has infinitely many elements and so, by lemma II.7.7, f =g.
The rest follows easily from the fact that every f ∈F/k is a regular function on some open setU ⊆CF. Iff /∈kthenf ∈mP for finitely manyP ∈CF by lemma II.7.7 and so f is regular on the complement of the set of those points. Iff ∈k, it is regular on CF.
We can now define what we mean by anabstract nonsingular curve and how it fits into our world of varieties.
Definition II.7.12. Let F/k be a function field. An open subset U ⊆CF, with the induced topology and the induced notion of regular functions on its open subsets, is called anabstract nonsingular curve.
By the remark at the end of section II.4 we are now able to include abstract non-singular curves to our notion of morphisms between varieties as follows:
Definition II.7.13. A morphism ψ : V → W between abstract nonsingular curves or varieties is a continuous mapping such that for every open set U ⊆ W, and every regular functionf :U →A1,f◦ψ is a regular function onψ−1(U).
Our next aim is to prove that every nonsingular projective curve is isomorphic to an abstract nonsingular curve.
Proposition II.7.14. Every nonsingular projective curve C is isomorphic to an ab-stract nonsingular curve.
Proof. LetF =k(C) be the function field of C. We already know that for each point P ∈C, the local ring OP is a discrete valuation ring ofF/k by proposition II.5.6. Let U ⊆CF be the set of all local rings of C. We define a map ψ :C →U by P 7→ OP. This map is surjective by definition and injective by lemma II.5.16. In order to show that ψ is an isomorphism, we need to show that U is an open subset of CF (i.e. an abstract nonsingular curve).
The local ring of a projective variety is isomorphic to the local ring of one of its nonempty affine parts (cf. section II.5), so we may assume that C is affine. The coordinate ringA:=k[C] ofC is a finitely generatedk-algebra (sayA=k[x1, . . . , xn]) by lemma II.1.13, with Quot(A) = F. By what we have done in section II.5, every local ring inU is a localization ofAat its maximal ideal and therefore containsA. We have the following equivalence:
A⊆RP ⇐⇒ x1, . . . , xn∈RP. This impliesU =Tn
i=1Ui whereUi := {P ∈CF |xi ∈RP}. But {P ∈CF |xi ∈/ RP} is a closed set by lemma II.7.7 and so allUi are open, hence U is open.
ψis certainly continuous as finite sets are closed in C. Now, letV ⊆C be an open subset, then we already know thatOC(V) =T
P∈V OC,P. For an open subset W ⊆U
II.7. ABSTRACT NONSINGULAR CURVES 39 we have
OU(W) = \
P∈W
RP = \
Q∈ψ−1(W)
ψ(Q) = \
Q∈ψ−1(W)
OC,Q=OC(ψ−1(W)).
So regular functions on any open set are the same and we have shown thatψ is indeed an isomorphism.
We want to prove the converse, namely that every abstract nonsingular curve is isomorphic to a nonsingular projective curve. To be able to do so, we need the following result about the unique extension of morphisms from curves to projective varieties.
Proposition II.7.15. LetC be an abstract nonsingular curve, letP ∈C, let V be a projective variety, and let ψ:C\P → V be a morphism. Then there exists a unique morphismψ:C→V extendingψ.
Proof. We have proved every result necessary to understand the proof given in [Har77, Ch. I, proposition 6.8, p. 43]. There, the ground field is assumed to be algebraically closed. This restriction, however, is not needed in the proof by using our vocabulary and results instead of the ones given there.
This allows us state the main result of this section:
Theorem II.7.16. Let F/kbe a function field. Then, the abstract nonsingular curve CF is isomorphic to a nonsingular projective curve.
Proof. This is [Har77, Ch. I, theorem 6.9, p. 44]. Again, in the proof given there, it is not needed thatkis algebraically closed.
Corollary II.7.17. Every curve is birationally equivalent to a nonsingular projective curve.
Proof. LetC be a curve with function fieldF =k(C). By the previous result, we know thatCF is a nonsingular projective curve with function fieldF and so,Cis birationally equivalent toCF by theorem II.3.10.
Finally, we are able to answer our question (II.2): Proposition II.1.12 becomes a one-to-one correspondence (up to isomorphism) if we restrict ourselves to function fields of transcendence degree 1 overk(kis the full constant field ofF) and absolutely irreducible nonsingular projective curves. This follows from the next result:
Corollary II.7.18. There is a one-to-one correspondence (up to isomorphism) be-tween absolutely irreducible nonsingular projective curvesC and function fieldsF/kof transcendence degree 1 overksuch that kis the full constant field ofF.
Proof. This is [Har77, Ch. I, corollary 6.12, p. 45]. Note that one only needs k to be algebraically closed inF for the proof to work.
This allows us to work in the theory of algebraic function fields instead of the geometric theory of algebraic curves.