Holomorphy rings of function fields

Transcription

1 Holomorphy rings of function fields Bruce Olberding Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico Dedicated to Robert Gilmer 1 Introduction In his 1974 text, Commutative Ring Theory, Kaplansky states that among the examples of non-dedekind Prüfer domains, the main ones are valuation domains, the ring of entire functions and the integral closure of a Prüfer domain in an algebraic extension of its quotient field [Kap74, p.72]. A similar list today would likely include Kronecker function rings, the ring of integervalued polynomials and real holomorphy rings. All of these examples of Prüfer domains have been fundamental to the development of multiplicative ideal theory, as is evidenced in the work of Robert Gilmer over the past 40 years. These rings have been intensely studied from various points of views and with different motivations and tools. In this article we make some observations regarding the ideal theory of holomorphy rings of function fields. A holomorphy ring is an intersection of valuation rings having a common quotient field F. The terminology arises from viewing elements of F as functions on collections of valuation rings having quotient field F. To formulate this more precisely, let F be a field and D be a subring of F. The Zariski-Riemann space of F is the collection Σ(F D) of all valuation rings V containing D and having quotient field F. If D is the prime subring of F, then we write Σ(F ) for Σ(F D). One can introduce a topology on Σ(F D) in a natural way [ZS75, p. 110]. In Section 2 we will consider the Zariski patch topology on Σ(F D). If S Σ(F D), then x F is holomorphic on S if x has no pole on S. More precisely, for each V S, let φ V : F F V { } be the place corresponding to V, where F V is the residue field of V. Then x assigns to V the value φ V (x). Thus x is holomorphic on S if and only if x is finite on each The author was supported in part by a grant from the National Security Agency.

2 2 Bruce Olberding V S; if and only if x V for all V S. The holomorphic functions form a subring H of F called the holomorphy ring of S. Evidently, H = V S V. The holomorphy rings we consider originate with a 1965 theorem of Dress that states that if F is a field in which 1 is not a square, then the subring of F generated by {1/(1 + t 2 ) : t F } is a Prüfer domain such that the set of valuation overrings is the set of valuation rings in Σ(F ) for which 1 is not a square in the residue field [Dre65]. In Section 2 of the 1969 paper [Gil69] Gilmer provides a very general foundation for such examples by showing that if F is a field and f(x) is a monic nonconstant polynomial in F [X] having no root in F, then the integral closure H of the subring of F generated by {1/f(t) : t F } is a Prüfer domain having quotient field F. Moreover, the valuation overrings of H are precisely the valuation rings in Σ(F ) such that f does not have a root in the residue field. Thus by considering the polynomial f(x) = 1 + X 2, the result of Dress is an immediate consequence of Gilmer s generalization. Unaware of Gilmer s work in [Gil69], P. Roquette in 1973 proved a similar result in Theorem 1 of [Roq73]. Later, in Corollary 2.6 of his 1994 article [Lop94] K. A. Loper also rediscovered this result. More recently, D. Rush has extended this theorem in several interesting ways [Rus01]. We give here Roquette s formulation, since it will be the most convenient for our purpose. Theorem 1.1. (Gilmer [Gil69]; Roquette [Roq73]; Loper [Lop94]) Let F be a field, let S Σ(F ), and let H = V S V. Suppose that there exists a nonconstant monic polynomial f(x) H[X] such that for every V S, there is no root of f(x) in the residue field of V. Then H is a Prüfer domain with quotient field F and for every finitely generated ideal I of H, I k is a principal ideal, where k is a power of deg (f(x)). The following special case is of interest. Let K be a field that is not algebraically closed, and let F be a field extension of K. Since K is not algebraically closed, there exists a monic nonconstant polynomial f(x) K[X] such that f(x) does not have a root in K. For any valuation ring V having quotient field F, we may identify K with its image in the residue field F V of V. If we take the intersection H of all valuation rings V with quotient field F such that f does not have a root in the residue field F V (assuming such a valuation ring exists!), then by the theorem H is a Prüfer domain. Taking this a step farther, we may restrict our intersection to valuation rings V in Σ(F K) such that the residue field F V has the property that a polynomial in K[X] has a root in K if and only if it has a root in F V. We in fact assume a stronger property: a field K is existentially closed in an extension field L if for any m, k > 0 and every choice of polynomials f 1,..., f k, g in K[X 1,..., X m ] such that f 1,..., f k have a common zero in L m that is not a zero of g, then they also have a common zero in K m that is not a zero of g; equivalently, every finitely generated K-subalgebra A of L admits a K-homomorphism φ : A K [BJ85, Theorem 1.1]. We define the absolute K-holomorphy ring H of F K to be the intersection of all valuation rings V in

3 Holomorphy rings of function fields 3 Σ(F K) such that K is existentially closed in the residue field of V. If there does not exist a valuation ring V in Σ(F K) such that K is existentially closed in F, then we set H = F. We will consider only holomorphy rings defined in this way via the notion of existential closure. We refer to [BS86] and [KP84] for some closely related constructions of holomorphy rings using the notion of existential closure. These approaches all encompass the important example of the real holomorphy ring of F K; that is, the case where K is real closed, F is formally real and the holomorphy ring of interest is the intersection of all the formally real valuation rings in Σ(F K). That this case falls into our setting follows from the fact that if K is a real closed field and F is an extension field of K, then F is formally real if and only if K is existentially closed in F [BCR98, Proposition 4.1.1]. Because of its importance in real algebraic geometry this case motivates much of the research on holomorphy rings. See for example [Sch82b],[Bec82], [BK89] and [Ber95]. In addition real holomorphy rings have provided the only known examples of Prüfer domains having finitely generated ideals that cannot be generated by two elements; see [OR], this volume, and its references for more on this aspect of real holomorphy rings. Another interesting example, and one that is central to our approach later, is the case where K is a field of characteristic 0 that is not algebraically closed and F is a finitely generated subfield of a purely transcendental extension of K. Then by the following basic proposition K is existentially closed in F, and we may consider the absolute K-holomorphy ring H of F K. By Proposition 3.1 H is a Prüfer domain having Krull dimension the same as the transcendence degree of F K. Proposition 1.2. (Ribenboim [Rib84, Proposition 1]) Let K L be infinite fields such that K is existentially closed in L. If F is a subfield of a purely transcendental extension of L, then K is existentially closed in F. We restrict throughout this article to the case that F K is a function field. For in this case if K is existentially closed in F, then there are infinitely many valuation rings in Σ(F K) having residue field K (Theorem 2.2). Hence the absolute K-holomorphy ring of F K is distinct from F. Moreover, as we show in Corollary 3.5 H is simply the intersection of the DVRs in Σ(F K) with residue field K. As we see in Section 4, the ideal theory of the K-holomorphy ring is quite complicated. This is due to the richness of the valuation theory of function fields, as is discussed in Section 2, where we rely heavily on recent work of F. Kuhlmann. 2 Discrete valuations of function fields In this section we survey several strong results regarding the existence of good valuations on function fields F K. All the main results of the two subsequent sections are consequences of these existence theorems. The basic

4 4 Bruce Olberding idea, and one which we will exploit in the next section, is to trade arbitrary valuations for discrete valuations with nice residue fields that behave the same way with respect to a given finite set of data. There are many other interesting existence and local uniformization results for such valuations given in [Kuh04a], but we cite here only those needed for our purposes in the next sections. For more background on valuations, see [ZS75]. Let K be a field, and let F be an extension field of K. Then F is a function field in n variables over K (or, F K is a function field in n variables) if F is a finitely generated field extension of K of transcendence degree n. A valuation on F K is a valuation on the field F such that v(x) = 0 for all x K. Let V be the valuation subring of F corresponding to v. We will use the following notation. F v = residue field of v. We view K as a subfield of F v. G v = value group of v. m(v) = maximal ideal of V. φ v : F F v { } is the place corresponding to v defined by φ v (x) = x + m(v) if x V and φ v (x) = otherwise. dim K (v) = transcendence degree of F v over K. rr(v) = dimension of Q Z G as a vector space over Q. Thus rr(v) is the rational rank of the abelian group G v. rank(v) = the rank of the totally ordered abelian group G v. This number coincides with the Krull dimension of V. Moreover, rank(v) rr(v) When we wish to emphasize the valuation ring V rather than the valuation v we write F V, G V, m(v ), φ V, rr(v ), dim K (V ) and rank(v ) analogously. The rational rank and dimension of a valuation on F K are governed by Abhyankar s inequality [Abh56, Theorem 1]: dim K (v) + rr(v) n, (1) where n is (as above) the transcendence degree of F over K. The valuation v on F K is discrete if G v is isomorphic as a totally ordered abelian group to the lexicographic sum k i=1 Z for some k > 0. The valuation ring V corresponding to v is discrete if v is discrete. We follow the usual convention and reserve the abbreviation DVR for a discrete valuation ring V of rank 1; equivalently, G V is isomorphic to Z. Thus a discrete valuation ring need not by a DVR. In this way our terminology follows [Gil92] but differs from [Kuh04a]. We define a valuation v of F K to be good if v is discrete, F v K is a function field and K is existentially closed in F v. A valuation ring V in Σ(F K) is good if its corresponding valuation is good. We denote by Good r,d (F K) the set of good valuation rings V in Σ(F K) such that rr(v ) = r and dim K (V ) = d. Since V is discrete the rational rank is also the Krull dimension of V. If F K

5 Holomorphy rings of function fields 5 has more than 1 variable, then there exist bad (i. e. not good) valuations on F K; for example, see [MLS39] or [Kuh04b]. A valuation v on F K is K-rational if K = F v. Similarly, a valuation ring V in Σ(F K) is K-rational if K = F V. If V is in Good r,0 (F K), then V is K-rational since K is existentially closed in F V. We state now a powerful theorem regarding the existence of valuations on F K with residue field finitely generated over K. The version we give was proved in 1984 by Kuhlmann and Prestel using the Ax-Kochen-Ershov Theorem. Bröcker and Schülting proved a similar result in characteristic 0 using Hironka s resolution of singularities [BS86, Theorem 2.12]. A more direct proof using a curve selection lemma is given in the case K = R by Andras in [And85, Theorem 5.1]. Recently Kuhlmann has obtained the most general version, which is valid in arbitrary characteristic [Kuh04a, Theorem 1]. If G is a totally ordered abelian group and Z is the group of integers, then any direct sum of the form Z Z G Z Z is a discrete lexicographic r-extension of G with respect to the lexicographic ordering on the sum, where r is the number of copies of Z. Theorem 2.1. (Kuhlmann Prestel [KP84, Main Theorem]) Let F K be a function field in n variables such that K has characteristic 0. Let v be a valuation on F K, and let x 1,..., x m F. Let r 1 and d 1 be integers such that dim K (v) d 1 n 1 and rr(v) r 1 n d 1. Then there exists a valuation w on F K such that: (i) dim K (w) = d 1 and F w is a subfield, finitely generated over K, of a purely transcendental extension of F v ; (ii) rr(w) = r 1 and G w is a finitely generated subgroup of an arbitrarily chosen discrete lexicographic (r 1 rr(v))-extension of G v ; and (iii) φ v (x i ) = φ w (x i ) and v(x i ) = w(x i ) for all i = 1, 2,..., m. We will need in Lemma 3.3 to replace a valuation v that is positive on a given finite set of elements by a K-rational rank one discrete valuation that is also positive on these same elements. The first step in doing so is to replace the valuation v with a K-rational discrete valuation that is nonnegative on these same elements. This we do via Theorem 2.2: Theorem 2.2. (Kuhlmann [Kuh04a, Theorem 23]) Let F K be a function field in n variables such that K is existentially closed in F. Let D be a finitely generated K-subalgebra of F. Then there are infinitely many discrete K-rational valuation rings in Σ(F D) of Krull dimension n. The second step is to replace this K-rational discrete valuation with a K-rational rank one discrete valuation that is nonnegative on the initially given set of finitely many elements. This step is enabled by the following theorem, which is formulated topologically. The Zariski patch topology on Σ(F K)

6 6 Bruce Olberding is defined by basic open sets of the form U(x 1,..., x k ; y 1,..., y m ) = {V Σ(F K) : x 1,..., x k V ; y 1,..., y m m(v )}, where x 1,..., x k, y 1,..., y m F. Theorem 2.3. (Kuhlmann [Kuh04a, Corollary 5]) If K is a perfect field, then the K-rational DVRs lie dense with respect to the Zariski patch topology in the space of all K-rational valuation rings in Σ(F K). 3 Intersection representations of holomorphy rings Let F K be a function field, and let D be K-subalgebra of F. We define the relative K-holomorphy ring of F D to be the intersection of all valuation rings V in Σ(F D) such that K is existentially closed in F V. The absolute K- holomorphy ring of F is the relative K-holomorphy ring of F K. Throughout the rest of this article we work under the following standing hypotheses: K is a field of characteristic 0 that is not algebraically closed, F K is a function field in n > 0 variables, K is existentially closed in F, D is a finitely generated K-subalgebra of F, and H is the relative K-holomorphy ring of F D. Recall from the introduction that these standing hypotheses are satisfied in the case (a) where F K is a formally real function field with K a real closed field, or (b) K is a non-algebraically closed field of characteristic 0 and F is a finitely generated subfield of a purely transcendental extension of K. Proposition 3.1. H is a Prüfer domain having Krull dimension n and quotient field F. Moreover, there exists k > 0 such that for all finitely generated ideals I of H, I k is a principal ideal of H. Proof. By Theorem 1.1 H is a Prüfer domain. Let M be a maximal ideal of H. Then H M is a valuation domain, so by Abhyankar s inequality (1), M has height at most n. Moreover, by Theorem 2.2 there is a valuation overring of H of Krull dimension n. Thus H has Krull dimension n. Finally, since K is not algebraically closed, there exists a monic nonconstant polynomial f(x) K[X] such that f does not have a root in K. Thus the final assertion follows from Theorem 1.1. Remark 3.2. The integer k in Proposition 3.1 can be chosen to be a power of the greatest common divisor of the degrees of all monic nonconstant polynomials f K[X] such that f has no root in K [Gil69, Theorem 2.2]. Hence if this gcd is 1, then H is a Bézout domain. We prove below a representation theorem for H in terms of good valuation rings, as defined in Section 2, in Σ(F D). It is a consequence of the following key lemma, which is a basic application of the existence results in Section 2.

7 Holomorphy rings of function fields 7 Lemma 3.3. Let d and r be integers such that 0 d < n and 1 r n d. The set Good r,d (F K) is dense with respect to the patch topology in the subspace S of valuation rings V in Σ(F K) such that K is existentially closed in F V. Proof. Let x 1,..., x k, y 1,..., y m F, let U = U(x 1,..., x k ; y 1,..., y m ), and suppose that U S is not empty. We prove first that there exists a K-rational DVR in U S. To do this, it is enough by Theorem 2.3 to show that there exists a K-rational valuation ring in U S. Let V be a valuation ring in S U with corresponding valuation v. Then v(x i ) 0 for all i = 1,..., k and v(y j ) > 0 for all j = 1,..., m. Thus by Theorem 2.1 there exists a valuation w on F K such that w(x i ) = v(x i ) for all i = 1, 2,..., k; w(y j ) = w(y j ) for all j = 1, 2,..., m; and F w is a finitely generated subfield of a purely transcendental extension of F v. Since K is by assumption existentially closed in F v we have by Proposition 1.2 that K is existentially closed in F w. Let W be the valuation ring corresponding to w. Since K is existentially closed in F w, we have by Theorem 2.2 that there exists a K-rational valuation ring A in Σ(F w K) such that φ w (x i ) A for all i = 1, 2,..., k. Now there is a valuation ring B W such that B/m(W ) = A; in particular, F B = K. For each i = 1, 2,..., k we have φ w (x i ) A, so that x i + m(w ) B/m(W ). Hence x 1,..., x k B. Moreover, y 1,..., y m m(w ) m(b). Consequently, B is a K-rational valuation ring in U, so by Theorem 2.3 there is a K-rational DVR W in U S. We use W now to show that U Good r,d (F D) is nonempty. Let w be the valuation ring corresponding to W. Since w is K-rational, dim K (w) = 0, and since w is discrete of rank one, rr(w) = 1. Hence by Theorem 2.1 there exists a valuation w on F K such that dim K (w ) = d; rr(w ) = r; F w is a finitely generated subfield of a purely transcendental extension of K; w (x i ) = w(x i ) for i = 1,..., k; and w(y j ) = w(y j) for j = 1,..., k. By Proposition 1.2 K is existentially closed in F w, so W Good r,d (F D). Schülting, using resolution of singularities, proves in Lemma 2.5 of [Sch82a] that if K is a real closed field, then H is the intersection of the K-rational valuation rings in Σ(F D). Becker in Theorem 1.14 of [Bec82] gives a different proof for the same result using a trace formula for quadratic forms. In Theorem 3.13 of [BS86] Bröcker and Schülting characterize in terms of direct limits of rings arising from blow-ups which collections of valuation overrings of H can be intersected to obtain H. However they assume as part of their standing hypotheses that K is existentially closed in K((t)). The fields with this latter property are shown in Theorem 15 of [Kuh04a] to be precisely the large fields, that is, the fields K such that for every smooth curve over K, the set of K- rational points is infinite if it is nonempty. On the other hand, in Theorem 2 of [KP84], Kuhlmann and Prestel prove in a context similar to the present one that the absolute K-holomorphy ring is an intersection of valuation rings in Good(F K) 1,d, where d is some fixed number < n. Thus the generality of

8 8 Bruce Olberding the following theorem appears to be new, even for real closed fields, but is not surprising in light of these other results. Theorem 3.4. Let d and r be integers such that 0 d < n and 1 r n d. Then H is the intersection of the valuation rings in Good r,d (F D). Proof. The inclusion H V Good r,d (F K) V is clear. To prove the reverse inclusion, let z F \ H. We show that there exists W Good r,d (F K) such that z W. Since K is not algebraically closed, there exists a monic nonconstant polynomial f K[X] such that f(x) 0 for all x F. Define y = (f(z)) 1. We claim that y H. Indeed, let V Σ(F D) such that K is existentially closed in F V. Suppose that y V. Then f(z) = y 1 m(v ) since V is a valuation ring. Write f(x) = t i=1 a ix i. Then since f(z) m(v ), we have 0 = φ V (f(z)) = t i=1 a iφ V (z) i. Thus φ V (z) is a root of f(x) in F V. Now K is existentially closed in F V, so this forces f to have a root in K, contrary to the choice of f. Hence y V. Therefore, y H. Next observe that y is not a unit in H. For if y is a unit in H, then f(z) H, and since H is integrally closed, z H, a contradiction. Write D = K[x 1,..., x k ], where x 1,..., x k F. Since y is not a unit in H the set U(x 1,..., x k ; y) contains a valuation ring such that K is existentially closed in its residue field. Therefore, by Lemma 3.3 there is a valuation ring W in U(x 1,..., x k ; y) Good r,d (F K). Hence y m(w ), so f(z) = y 1 W. Consequently, z W. Since W Good r,d (F K) this shows that V Good r,d (F D) V H. Corollary 3.5. H is the intersection of the K-rational DVRs in Σ(F D). Proof. By Theorem 3.4 H is the intersection of all the valuation rings in the set Good 1,0 (F K). Each valuation ring V in Good 1,0 (F K) is a DVR such that F V is algebraic over K and K is existentially closed in F V. It follows that F V = K. Corollary 3.6. H is a completely integrally closed Prüfer domain. Proof. By Corollary 3.5 H is an intersection of completely integrally closed domains. Hence H is completely integrally closed. Corollary 3.7. A valuation ring V Σ(F D) is an overring of H if and only if K is existentially closed in F V. Proof. If V Σ(F D) and K is existentially closed in F V, then clearly H V. Conversely, suppose that V Σ(F D) and H V. We claim that K is existentially closed in F V. It is enough to show that for every m, k > 0 and every choice of polynomials f 1,..., f k K[X 1,..., X m ], whenever f 1,..., f k have a common zero in (F V ) m, they have a common zero in K m [BJ85, Theorem 1.1]. Let m, k > 0, and let f 1,..., f k K[X 1,..., X m ]. Suppose that for some a = (a 1,..., a m ) V m, f 1 (a),..., f k (a) m(v ). Let

9 Holomorphy rings of function fields 9 I = (f 1 (a),..., f k (a))h. By Proposition 3.1 some power of I is a principal ideal of H, say xh. We claim that x is contained in the maximal ideal of some K-rational DVR in Σ(F D). Indeed, by Corollary 3.5 H is the intersection of the K-rational DVRs in Σ(F D). Hence if x is not contained in the maximal ideal of any of these DVRs, then x 1 H V, a contradiction to the fact that x m(v ). Therefore there is a K-rational DVR W in Σ(F D) such that x m(w ). Thus xh m(w ) since H W, and since xh is a power of I, I m(w ). Hence f 1 (a),..., f k (a) m(w ). Since K = W/m(W ), we may write for each i = 1, 2,..., m, a i = u i + x i for some u i K and x i m(w ). Let u = (u 1,..., u m ). Then for each i = 1,..., k, f i (u) m(w ) K = {0}. Hence f 1,..., f k have a common zero in K. Let R be a commutative ring. For x 1,..., x k, y R, let U(x 1,..., x k, y) = {P Spec(R) : x 1,..., x k P and y P }. The patch topology on Spec(R) is given by the basic open sets U(x 1,..., x k, y), where x 1,..., x k, y R. We say that a prime ideal P of H is good if H P is a good valuation ring in Σ(F D). As usual, the dimension of a prime ideal P of a commutative ring R is the Krull dimension of the ring R/P. The height of P is the Krull dimension of R P. Corollary 3.8. Let d and h be integers such that 0 d < n, 0 < h n and d + h n. Then the set of good prime ideals of H of height h and dimension d is dense in Spec(H) with respect to the patch topology. Proof. By Corollary 3.7 Σ(F H) is precisely the set of valuation rings V in Σ(F D) such that K is existentially closed in F V. Thus by Lemma 3.3 Good h,d (F D) is dense in Σ(F H). Moreover, since H is a Prüfer domain, the mapping Σ(F H) Spec(H) that sends a valuation to its center on H is bijective. It is easily checked that this mapping is in fact a homeomorphism with respect to patch topologies. Hence the corollary follows. The next corollary was proved for the case where K is a real closed field in Lemma 2.5 of [Sch82a]. Corollary 3.9. Let H 0 be the absolute K-holomorphy ring of F. Then H = H 0 [D]. Proof. Clearly H 0 [D] H. Conversely, since H 0 [D] is an overring of the Prüfer domain H 0, then H 0 [D] is a Prüfer domain, and hence an intersection of its valuation overrings. By Corollary 3.7 each valuation overring V of H 0 [D] has the property that V Σ(F D) and K is existentially closed in F V. Hence H V. It follows that H H 0 [D]. To conclude this section we show that if F K has 2 variables, then H is a Hilbert ring. This is noted for real closed fields on p of [Sch82a]. Lemma If P is a prime ideal of H, then H/P is the relative K- holomorphy ring of its quotient field with respect to the image of D in H/P.

10 10 Bruce Olberding Proof. Let L denote the quotient field of H/P, and let α : H P L denote the canonical map. Let B = α(d). A subring W of L is a valuation ring in Σ(L B) if and only if α 1 (W ) is a valuation ring in Σ(F B) [FHP97, Proposition 1.1.8]. Moreover, if W is a valuation ring, then F α(w ) = L W. Thus the valuation rings W in Σ(L B) such that K is existentially closed in L W are precisely the images of the valuation rings V in Σ(F D) such that V H P and K is existentially closed in F V. The claim now follows. Proposition If the function field F K has 2 variables, then H is a Hilbert ring. Proof. If P is a nonmaximal prime ideal of H, then from Proposition 3.1 it follows that H/P is a Prüfer domain of Krull dimension 1. The quotient field of H/P is isomorphic to H P /P P. Since F K has 2 variables, H P /P P is a finitely generated field extension of K [ZS75, Corollary ]. Thus since H/P is an integrally closed domain of Krull dimension 1, H/P is a Dedekind domain. Moreover, by Corollary 3.7 K is existentially closed in H P /P P. Thus by Lemma 3.10 and Theorem 2.2 H/P is the intersection of infinitely many valuation rings. Hence H/P has infinitely many maximal ideals, and the intersection of these maximal ideals is the zero ideal of H/P. Hence P is the intersection of the maximal ideals in H containing it. Remark H need not be a Hilbert ring if F K has > 2 variables. Schülting in [Sch82a, p. 1257] constructs such an example in the case where K is a real closed field. The obstacle in degree > 2 is that in this case the residue field of a valuation need not be finitely generated over K. 4 Radical ideals of holomorphy rings We maintain the standing hypotheses of Section 3. We examine in this section the representation of radical ideals in holomorphy rings as an intersection of prime ideals. With the exception of a version of Corollary 4.9, which we discuss below, the results in this section are evidently new, even for real holomorphy rings. We prove first some preliminary lemmas for Prüfer domains. Lemma 4.1. Let I be a nonzero ideal of a Prüfer domain R. Then I 1 = R if and only if I is not contained in any proper finitely generated ideal of R. Proof. Observe that I 1 = R if and only if every finite intersection J of principal fractional ideals of R containing I contains R. Since R is a Prüfer (hence coherent) domain, any such ideal J is finitely generated. On the other hand, every finitely generated ideal of R is invertible, hence a finite intersection of principal fractional ideals. Thus the lemma follows. If I is an ideal of a ring R and P is a prime ideal of R, then P is a Zariski-Samuel associated prime ideal of I if there exists x R such that

11 Holomorphy rings of function fields 11 I :R x = P. If R is a Noetherian ring, then every ideal has a Zariski-Samuel associated prime ideal, but for non-noetherian commutative rings, this need not be the case; see for example [FHO05] and its references. Lemma 4.2. Let R be a Prüfer domain, let C be a collection of height 1 maximal ideals of R, and let D = Max(R) \ C. Suppose that C and D are nonempty and R = M C R M = M D R M. Then the following statements hold for R. (i) No nonzero prime ideal of R is the radical of a finitely generated ideal. (ii) No nonzero finitely generated ideal of R has a Zariski-Samuel associated prime ideal. (iii) Every nonzero finitely generated proper ideal of R has infinitely many prime ideals minimal over it. Proof. (i) Let P be a nonzero nonmaximal prime ideal of R. Since R is a Prüfer domain and P is a nonmaximal prime ideal of R, then P 1 = End(P ) [FHP97, Corollary 3.1.8]. Since R is an intersection of DVRs, R is a completely integrally closed domain. Hence P 1 = End(P ) = R. Also, since R is a Prüfer domain and P is a nonmaximal prime ideal, P 1 = R P ( α R M α ), where {M α } is the collection of prime ideals of R not containing P [FHP97, Theorem 3.26]. Thus since P 1 = R, we have for any maximal ideal M containing P, R M = R P ( α R M α )R M. Since R M is a valuation domain, this forces α R M α R P. If P is the radical of a finitely generated ideal I, then since I is invertible, I = q 1 R q k R for some q 1,..., q k in the quotient field of R. Thus there exists i such that I R Rq i P. Since the radical of I is P, it follows that q 1 i α R M α \ R P, a contradiction. Therefore, P is not the radical of a finitely generated ideal of R. We consider next the case of maximal ideals. If M is a maximal ideal of R and M is the radical of a finitely generated ideal I of R, then using an argument such as the one above, we have N M R N R M, where N ranges over the maximal ideals of R distinct from M. However, if M C, then N D R N = R R M, a contradiction. On the other hand, if M C, then N C R N = R R M, which is also a contradiction. Hence M is not the radical of a finitely generated ideal of R. (ii) If I is a finitely generated ideal with a Zariski-Samuel associated prime ideal P, then P = I : R b for some b R. However, since R is a Prüfer domain, I : R b is a finitely generated ideal, so P is the radical of a finitely generated ideal. Thus by (i), P = 0. Hence I = 0. (iii) If an ideal I of a Prüfer domain has only finitely many prime ideals minimal over it, then each prime ideal minimal over I is a Zariski-Samuel associated prime of I [FHO05, Lemma 5.9]. If I is finitely generated, then by (ii) this forces I = 0. Lemma 4.3. If J is a nonzero finitely generated radical ideal of a Prüfer domain R, then every prime ideal containing J is a maximal ideal of R.

12 12 Bruce Olberding Proof. Suppose that P is a prime ideal of R that is minimal over J, and let M be a maximal ideal of R containing P. Since J is a radical ideal of R and R M is a valuation domain, J M = P M. Hence P M is a nonzero prime ideal of R M that is finitely generated, which since R M is a valuation domain is possible only if P M = M M. Hence we conclude that P is a maximal ideal of R. Lemma 4.4. Let R be a domain, and let C be a collection of prime ideals of R such that R = P C R P. If I is a nonzero divisorial ideal of R, then I = P C I P. Proof. We may assume that I is a proper ideal of R. Clearly, I P C I P R. To prove that the first inclusion reverses, let x R \ I. Then x 1 I R is a proper divisorial ideal of R, so there exists q in the quotient field of R such that x 1 I R qr R R. Now qr R = P C (qr P R P ), so there exists P C such that x 1 I R qr R P. Consequently, x I P. This proves the lemma. Lemma 4.5. Let R be a Prüfer domain, and let C be a collection of heightone maximal ideals of R. Suppose that C is nonempty and R = M C R M. If J is a nonzero radical ideal contained in a proper finitely generated ideal I of R, then I is a radical ideal and any prime ideal containing I is a maximal ideal of R. Proof. It suffices by Lemma 4.3 to show that I is a radical ideal of R. We claim in fact that I is the intersection of the ideals in C that contain it. If M C and I M, then J M I M. Since J is a nonzero radical ideal of R and R M has Krull dimension 1, it must be that J M = M M. Hence I M = M M. Thus by Lemma 4.4, I = ( M C,M I I M ) R = ( M C,M I M M ) R = M C,M I M. Recall the standing hypotheses of this section and the last. Lemma 4.6. Let d and h be nonnegative integers. Then P is a good prime ideal of height h and dimension d if and only if H P is a discrete valuation ring of Krull dimension h, (H P /P P ) K is a function field in d variables and K is existentially closed in H P /P P. Proof. Suppose that P is a good prime ideal of height h and dimension d. By assumption H/P has Krull dimension d. Also, by Lemma 3.10 H/P is the relative holomorphy ring of its quotient field with respect to the image of D in H/P. Thus by Proposition 3.1 d is the transcendence degree of the quotient field of H/P over K. Since the quotient field of H/P is isomorphic to H P /P P, the assertion follows. Conversely, it suffices to show P has dimension d. Let B denote the image of D in H P /P P. By Theorem 2.2 there exists a K-rational discrete valuation ring V in Σ(H P /P P B) such that V has Krull dimension d. Let W be the subring of H P such that W/P P = V. Then W is a K-rational discrete valuation ring of Krull dimension d + h and P P is a prime ideal of W of height h. Since P = P P H, it follows that P has dimension d.

13 Holomorphy rings of function fields 13 Theorem 4.7. If F K has transcendence degree > 1, then the following statements hold for H. (i) (ii) No nonzero prime ideal of H is the radical of a finitely generated ideal. No nonzero finitely generated ideal of H has a Zariski-Samuel associated prime ideal. (iii) If I is a proper nonzero finitely generated ideal of H and d and h are integers such that 0 d < n, 0 < h n and d + h n, then there exist infinitely many good prime ideals of H of dimension d and height h that are minimal over I. (iv) J 1 = H for all nonzero radical ideals J of H. Proof. To prove (i) and (ii) it is enough to show that H satisfies the hypotheses of Lemma 4.2. By Corollary 3.5 H is an intersection of K-rational DVRs. For each such DVR V, V = H M for some maximal ideal M of H. Thus H = M C H M, where C is the collection of height-one maximal ideals in M. We claim next that H is the intersection of all H M, where M Max(H)\C. Indeed, since F K has transcendence degree > 1, we have by Theorem 3.4 that H is the intersection of the valuation rings in Good(F D) 1,1. If V is such a valuation overring, then V = H P for some prime ideal P of H, and by Lemma 4.6 P is a nonmaximal prime ideal of H. It follows that H = M Max(H)\C H M. Thus H satisfies the hypotheses of Lemma 4.2, and (i) and (ii) follow. (iii) For each proper nonzero finitely generated ideal I of H, let G d,h (I) denote the collection of dimension d, height h good prime ideals of H that are minimal over I. To prove (iii) it is enough to show that for each nonzero proper finitely generated ideal I of H, there are at least 2 prime ideals in G d,h (I) that are minimal over I. For suppose there exists a proper nonzero finitely generated ideal I of H such that G d,h (I) is finite. Then by prime avoidance there exists x H such that G d,h (I + xh) has one element, a contradiction. Let I be a proper nonzero finitely generated ideal of H. By Proposition 3.1 there exists e > 0 such that I e = xh for some x I. Since I = xh we may assume without loss of generality that I = xh. By Theorem 3.4 H is the intersection of the K-rational DVRs of H. Thus since x is a non-unit in H, there exists a K-rational rank one discrete valuation v on F D such that v(x) > 0. By Theorem 2.1 there exists a discrete valuation ring W in Σ(F D) such that W has Krull dimension h; F W K is a function field in d variables; K is existentially closed in F W (apply Proposition 1.2); and IW = m(w ) e, where e = v(i) (for by the theorem we may select the position of G v arbitrarily in the value group of w). Therefore, by Lemma 4.6 P := m(w ) H is a member of G d,h (I) since H m(w ) H = W. We claim next that P is not the only member of G d,h (I). Let y H \ P such that I + yh H. (If no such y exists, then P is necessarily maximal

14 14 Bruce Olberding with P = I, contrary to (i).) As above we may find a K-rational discrete rank one valuation v such that v (I + yh) > 0. Thus v (I) > 0 and v (y) > 0. Now, using again Theorem 2.1, there exists a valuation overring W of H such that W has Krull dimension h; F W K is a function field in d variables; K is existentially closed in F W ; and IW and yw are m(w )-primary ideals of W. We have used in this last assertion that v (I) and v (y) are linearly dependent elements of G v and the position of G v can be chosen arbitrarily in the value group of W. Thus by Lemma 4.6 m(w ) H is a member of G d,h (I). Since y (m(w ) H) \ m(w ), it follows that G d,h (I) contains at least 2 elements. (iv) Let J be a nonzero radical ideal of R, and suppose that J 1 H. Then by Lemma 4.1 J is contained in a proper finitely generated ideal I of H. Now by Theorem 3.4 H is the intersection of the valuation rings in Good 1,1 (F D), so by Lemma 4.6 H is an intersection of the localizations H P, where P is a dimension 1 prime ideal. Thus by Lemma 4.4 I is contained in a nonmaximal prime ideal of H. This implies that J is contained in a nonmaximal prime ideal of H, which contradicts Lemma 4.5. Remark 4.8. In [Nak53] Nakano gives an example of a Prüfer domain of Krull dimension 1 such that no nonzero finitely generated ideal has a Zariski-Samuel associated prime ideal. By Theorem 4.7 there exists for any n > 1 a Prüfer domain of Krull dimension n also having this property. In [Sch82a] Schülting proves a Nullstellensatz for finitely generated ideals of real holomorphy rings. For an ideal J of H, let V(J) to be the collection of all K-rational places φ in F K such that φ(x) = 0 for all x I. For a collection V of K-rational places φ in F K, define I(V) = {x F : φ(x) = 0 for all φ V}. Schülting proves in Theorem 2.6 of [Sch82a] that if I is a proper finitely generated ideal of H, where K is real closed, then I = I(V(I)). In terms of ideals, this is equivalent to the assertion that I is the intersection of the maximal ideals M containing I such that H M is a K-rational valuation ring. Thus I is an intersection of prime ideals of H of dimension 0. In the following corollary we extend Schülting s theorem in several ways by generalizing it from invertible ideals of absolute real holomorphy rings to divisorial ideals of relative K-holomorphy rings, where K need not be real closed. We also may restrict to good prime ideals and allow constraints on the dimension and height of the prime ideals involved in the intersection representation. Corollary 4.9. Assume that n > 1, and let I be a proper nonzero divisorial ideal of H. Let d and h be integers such that 0 d < n and 1 h n d. Then I is the intersection of infinitely many good prime ideals P of dimension d and height h. Proof. We prove that I is the intersection of good prime ideals P of dimension d and height h, for then by Theorem 4.7(iii) this intersection is infinite.

15 Holomorphy rings of function fields 15 Since every nonzero divisorial ideal of a Prüfer domain is an intersection of finitely generated ideals, it is enough to prove the theorem in the case where I is a proper finitely generated ideal. Let J be the intersection of all good prime ideals P of dimension d and height h that contain I, and let a J. Since by Theorem 3.1 some power of I is a principal ideal, we may assume without loss of generality that I = bh for some b I. We claim that 1/b H[1/a]. By Theorem 3.4 and Corollary 3.9, H[1/a] is the intersection of all V Good h,d (F K) such that a 1 V. Let V Good h,d (F K) and suppose that a 1 V. If 1/b V, then b m(v ). Hence I = bh m(v ). Since V Good h,d (F K) we have by the choice of J that a J m(v ), a contradiction to the assumption that a 1 m(v ). Therefore, 1/b V, and we conclude that 1/b H[1/a]. Hence there exists m > 0 such that a m bh I. This proves that J = I. Corollary Assume that n > 1. Let I be a proper nonzero divisorial ideal of H, and let d be an integer such that 0 d < n. Let X be the set of good prime ideals of height 1 and dimension d containing I. There is a sequence {e P } P X of positive integers such that for any k 0, if Y k = {P X : e P k}, then I = P Y k P e P. Proof. Let X denote the set of good prime ideals of height 1 and dimension d that contain I. By Theorem 3.4 and Lemma 4.6 H = P X H P. Thus since I is divisorial, we have by Lemma 4.4 that I = P X I P H. Since H P is a DVR, we have for each P X, that there exist e P > 0 such that I P = P e P P. Since H is a Prüfer domain, P e P P H = P e P, and we conclude that I = P X P e P. Now let k > 0, and let Y k be as above. Define Z k = X \Y k. We may assume that Z k is not empty. Write J = P Z k P e P and B = P Y k P e P. We claim that J 1 = H. Indeed, if A = P Z k P, then A k J, so J 1 (A k ) 1. Since by Theorem 4.7 A 1 = H, an easy induction shows that (A k ) 1 = H. Hence J 1 = H, as claimed. Now JB J B = I = (I 1 ) 1, so BJI 1 H. Hence BI 1 J 1 = H, so B (I 1 ) 1 = I. Thus B = I. The previous corollaries are framed in terms of divisorial ideals. However, I do not know of an example in the present context of a non-invertible divisorial ideal: Question For which (if any) choices of K, F and D does there exist a nonzero divisorial ideal I of H such that I is not invertible? An integral domain for which every nonzero divisorial ideal is invertible is termed a generalized Dedekind domain in [Zaf86] and a pseudo-dedekind domain in [AK89]. The ring of entire functions is such a Prüfer domain [Zaf86], while the ring of integer-valued polynomials on Z is not [Lop97], though like the holomorphy ring H and the ring of entire functions it is completely integrally closed.

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