Fields Institute Communications Volume 00, 0000

Regular curves over Prüfer domains

Hagen Knaf

Fraunhofer Institut Techno- und Wirtschaftsmathematik 67663 Kaiserslautern, Germany

                          knaf@itwm.fhg.de

Abstract. In the present article integral separated schemes C of finite type over a Prüfer domain R, having one-dimensional fibres C ×_{R} k(p), p ∈ Spec(R)—R-curves for short—are investigated with respect to regularity: here a not necessarily noetherian local ring O is called regular iff every finitely generated ideal I ⊲ O has finite projective dimension. It is shown that for an R-curve C the set Reg(C) of points x ∈ C such that the local ring O_{C,x} is regular, ca n be characterized in terms of the weak homological dimension: Reg(C) = {x ∈ C : wdim(O_{C,x}) ≤ 2}. This result implies that an R-curve C having the property Reg(C) = C—a regular R-curve—has regular fibres C ×_{R} k(p) over those primes p ∈ Spec(R) which have the property that pR_{p} is not finitely generated. Moreover, the maximal ideal M_{C,x} ⊲ O_{C,x} for a point x ∈ Reg(C) closed in some fibre C ×_{R} k(p), p ≠ 0, is either not finitely generat ed or generated by a regular sequence of length two.

In analogy to the noetherian case, regularity of an R-curve C is preserved under an étale base change S|R and ascends in finite étale morphisms C′ → C.

Extending a result valid for normal curves over a valuation domain R, it is finally proved that for a proper regular R-curve C Weil divisors of the generic fibre C ×_{R} K, K = Frac(R), can be specialized to Cartier divisors of the closed fibre C ×_{R} k(M), M ⊲ R the maximal ideal of R, preserving the partial ordering, principal divisors and the degree.

1 Introduction

A valuation v of a function field F |K is called a constant reduction, if the residue field F v of F with respect to v is a function field over the residue field Kv of the same transcendence degree as F |K. This type of valuation is well-studied and plays a role for example in the investigation of diophantine problems in F |K—the so-called Local Skolem property—as well as in the Galois theory of the field F.

A geometric approach to constant reductions is given by integral normal schemes X over a valuation domain O_{v} having fibres of a common dimension n ∈ ℕ: every

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

2000 Mathematics Subject Classification. Primary: 13D05, Secondary: 13B40, 14C20.

local ring O_{X ,η} in a generic point η of the closed fibre X ×_{O_{v}} Kv of such a scheme is a valuation domain of the function field F := K(X). The corresponding valuation of F prolongs v and is a constant reduction, because its residue field is the function field of one of the irreducible components o f X ×_{O_{v}} Kv.

In the one-dimensional case this approach can be elaborated to a category equivalence if one restricts the considered classes of schemes resp. valued function fields as is shown in [6]: the category of integral proper normal O_{v}-schemes with one-dimensional fibres is equivalent to a certain subcategory of function fields of transcendence degree one over K equipped with finitely many constant reductions prolonging v. This equivalence extends the classical equivalence between integral proper norm al K-curves and function fields F |K of one variable. The proof for the equivalence (unfortunately) depends strongly on the fact, that the considered schemes are of finite type over O_{v}—from now on an integral O_{v}-scheme of finite type with one-dimensional fibres will be called an O_{v}-curve.

Browsing through the results obtained for O_{v}-curves it seems desirable to have a subclass of the class of normal O_{v}-curves at hand comparable to regular O_{v}-curves in the noetherian case. This subclass should include the O_{v}-curves with connected regular fibres (the case of so-called good reduction at v), but its members should behave smoother than normal O_{v}-curves for example with respect to the properties of their reflexive ideal sheaves.

A generalization of the concept of regularity to non-noetherian (local) rings was given by J. Bertin in [2] and was subsequently studied by various authors obtaining remarkable results for coherent regular local rings. Motivated by the results valid for normal O_{v}-curves it is natural to investigate regular O_{v}-curves in the sense of Bertin or slightly more generally, regular curves over Prüfer domains. This is what is done in the present article.

The presented material is grouped into three sections beside the introduction: section 2 contains preliminary results on curves over Prüfer domains with emphasis on normal curves.

In the main section, 3, Bertin’s generalization of regularity to non-noetherian local rings is applied to curves over a Prüfer domain R. The principal results of the section demonstrate the similarity of regular R-curves in the sense of Bertin with regular arithmetic surfaces: the regularity of a point x ∈ C on an R-curve C can be characterized by the homological condition wdim(O_{C,x}) ≤ 2 for the weak dimension of the local ring at x. Moreover regularity is stable under étale morphisms. Finally regular R-curves have a “nice divisor theory” in that reflexive fractionary ideal sheaves on these schemes are invertible.

All these results depend strongly on the properties of coherent regular local rings of small weak dimension, that also allow one to prove a surprising fact about the fibres of a regular R-curve: namely, that all fibres over primes p ∈ Spec(R) such that pR_{p} is not finitely generated are regular. In particular it follows that the regularity of a proper O_{v}-curve C over a valuation domain with non-finitely generated maximal ideal is equivalent to the regularity of its closed fibre. Consequently one has to draw the conclusion that regular proper O_{v}-curves are not too different from O_{v}-models of proper K = Frac(O_{v})-curves having good reduction at v—in contrast to what one might have expected.

In the final section, 4, the existence of an arithmetic homomorphism

Div(C) → CaDiv(C ×_{O_{v}} Kv)

from the Weil divisors of the generic fibre C of an integral proper regular O_{v}-curve C to the Cartier divisors of its closed fibre is proved. This result supplements a theorem proven for a special class of integral proper normal O_{v}-curves in [9].

2 Curves over Prüfer domains

Considering multiplicative ideal theory Prüfer domains are the non-noetherian generalization of Dedekind domains. Replacing the Dedekind base ring in the definition of an arithmetic surface by a possibly non-noetherian Prüfer domain, and requiring necessary finiteness conditions, thus yields a natural extension of the class of arithmetic surfaces. The members of this class are the central objects of the present article.

Throughout the whole article R denotes a Prüfer domain with field of fractions K ≠ R. We will frequently consider the local case only; in this case R is a valuation ring and is then denoted by O_{v}, where v is a non-archimedian valuation of K having valuation ring O_{v}. The maximal ideal of O_{v} is denoted with M_{v}. We will consider prolongations w of v to extension fields L of K—written w|v or (L, w)|(K, v). The symbols e(w|v), ϵ(w|v), f(w|v) and d(w|v) de note the ramification index, the initial index, the degree of the residue field extension and the defect of w|v respectively— see [5] and [12] for their definition.

For a O_{v}-algebra A the residue classes a + M_{v}A ∈ A/M_{v}A are briefly written as av; in particular this holds for the elements of the residue field O_{v}/M_{v} =: Kv.

Definition 2.1 An integral separated scheme ϕ : C → Spec(R) of finite type over the Prüfer domain R is called an R-curve, if the generic fibre C := C ×_{R} K has dimension one and the structure morphism ϕ is surjective.

The surjectivity of the structure morphism is only included into the definition to avoid strange cases.

Examining the other requirements in this definition two things immediately come to mind: first, what do the fibres C_{p} := C ×_{R} k(p), p ∈ Spec(R), k(p) := Frac(R/p), look like? Second, being of finite type is usually to weak when dealing with non-noetherian schemes. What can one say about this matter in the present situation?

The following result clarifies the situation with respect to these points; it holds for a larger class of schemes than just R-curves, but throughout this article we focus on the one-dimensional case:

Theorem 2.2 An R-curve C has the following properties:

1. C is of finite presentation and faithfully flat over R.

2. The structure sheaf O_{C} is coherent.

3. The fibres C_{p}, p ∈ Spec(R), are purely one-dimensional k(p)-schemes.

For a proof of these properties see [11] resp. the references given there.

It should be noted that the class of R-curves is stable under localization in the sense that if C is an R-curve, and S ⊂ R is a multiplicative set, then C ×_{R} (S^{−1}R) is a S^{−1}R-curve, S^{−1}R being a Prüfer domain. This fact frequently allows one to pass to the case of a valuation ring as a base.

Property 3 of theorem 2.2 shows, that an R-curve C can be understood to be a family of one-dimensional algebraic sets, where the fibres C_{p} for p ≠ 0 are specializations of the generic fibre C. To compare the fibres certain subschemes of C are particularly useful: for a closed point P ∈ C of the generic fibre let P be the Zariski closure of P in C; equipped with the reduced induced subscheme structure the resulting subscheme of C is called a horizontal curve on C.

Proposition 2.3 The horizontal curves on an R-curve C are of finite type over R and possess coherent structure sheaves. The horizontal curves on a proper R-curve C are finite R-schemes and thus affine.

Proof In an affine open set U ⊆ C, U = Spec(A), a horizontal curve P is defined by a prime ideal q ∈ U lying over 0 ⊲ R. Since A is integral and finitely generated over R [7], Thm. 7.4.3 yields that q is finitely generated. This proves the first assertion, because O_{C} and thus the ring A are coherent.

By [4], (8.11.1), a proper, quasi-finite, locally finitely presented morphism is finite. As a closed subscheme of the proper R-scheme C the horizontal curve P is proper over R too. Moreover we have already shown that P|R is finitely presented. Thus the quasi-finiteness remains to be verified.

The generic point of the horizontal curve P is a closed point P ∈ C of the generic fibre, therefore the function field K(P) is a finite extension of K = Frac(R). Let x ∈ P be some point lying over p ∈ Spec(R); we have to show that O_{P,x}/pO_{P,x} is a finite k(p)-module, and we can assume that R is a valuation domain and p its maximal ideal. Let U = Spec(A) be an affine open neighborhood of x on P. Since R is a valuation domain in the algebraic extension K(P)|K, every a ∈ A is a root of a polynomial f ∈ R[X] with at least one coefficient different from the constant coefficient a unit in R. It follows that A/pA is algebraic hence finite over the field R/p finishing the proof.

Next we are taking a closer look at normal R-curves, which are closely related to the valuation theory of their function fields. In the noetherian case this relationship is expressed in the theory of Weil divisors. Starting from the notion of divisorial or reflexive sheaves one can build a similar theory for certain non-noetherian schemes including R-curves—see [11] and the references given there. In this article we only need a few facts about divisorial ideal sheaves and certain valuatio ns induced by an R-curve on its function field. The reader should nevertheless keep in mind that the proper context to treat the subsequent results is divisor theory.

Let C be an R-curve with function field F; since the generic fibre C is dense in C one has F = K(C), i.e., F |K is a function field of transcendence degree one.

A fractionary O_{C}-ideal is a O_{C}-submodule I of the constant sheaf \underline{F}. Two fractionary O_{C}-ideals I, J can be multiplied to get the fractionary O_{C}-ideal IJ, furthermore the O_{C}-module (I : J) ≅ Hom_{O_{C}}(J , I) is a fractionary O_{C}-ideal. In particular the bidual Î := (O_{C} : (O_{C} : I)) is a fractionary O_{C}-ideal. A divisorial O_{C}-ideal is a fractionary O_{C}-ideal with the property

I = Î.  (2.1)

This property can be checked stalkwise, provided that I is coherent. Therefore we restrict our considerations to the set D_{coh}(C) of coherent divisorial O_{X} -ideals.

D_{coh}(C) is an abelian semigroup with the multiplication (I, J) ↦ ÎJ and with neutral element O_{C} (see theorem 2.2). If C is normal, this semigroup is in fact a group as one can prove using the multiplicative ideal theory of normal coherent rings. The group D_{coh}(C) can be described using valuations of F, and is isomorphic to the group of Weil divisors in the noetherian case.

We will not make use of these facts in the sequel, but the reader may recall them now and then when reading the forthcoming paragraphs.

Proposition 2.4 A normal R-curve C has the following properties:

1. The ideal sheaf J_{P} defining a horizontal curve P ⊂ C is divisorial.

2. The map P ↦ J_{P} on the prime divisors of C extends to an injective homomorphism h : Div(C) → D_{coh}(C), where Div(C) is the group of Weil divisors of the generic fibre C of C.

Proof For every open affine U ⊆ C intersecting with P the localization O_{C}(U)_{q}, q := J_{P} (U), is a discrete valuation ring. This implies that q satisfies

q = ⋃_{I⊆q, I fin. gen.} Î

—see for example [14], Cor. 4.2. The divisoriality of q now follows, since it is finitely generated by proposition 2.3.

Div(C) is a free abelian group, therefore it only remains to show that the

homomorphism h is injective. Let D = ∑_{i=1}^{r} n_{i}P_{i} ∈ Div(C) be some divisor; by

definition of h we have

J_{D} = (∏_{i=1}^{r} J_{P_{i}}^{n_{i}}) .

(Of course one implicitly uses here the well-known identity ÎJ = ÎĴ.)

Taking stalks commutes with taking the bidual due to the coherence of the involved sheaves; for i = 1, ..., r we therefore obtain

(J_{D})_{P_{i}} = ((J_{P_{i}})_{P_{i}}^{n_{i}}) ,

where the outermost lower index denotes localization at the point P_{i} ∈ C ⊂ C. Now the right hand side of this equation is a non-trivial principal ideal and thus J_{D} ≠ O_{C} provided D ≠ 0.

After having investigated the horizontal curves on a normal R-curve we turn to their fibres that one may also call vertical curves.

The behavior of the fibres depends on the structure of Spec(R); in particular the set

nonfgSpec(R) := {p ∈ Spec(R) \ {0} : pR_{p} is not finitely generated}

plays a crucial role. We also have to deal with the valuations corresponding to the valuation rings R_{p}—denote these valuations of K by v_{p}. We start with a useful finiteness result:

Proposition 2.5 An algebraic extension (L, w)|(K, v) of valued fields with the property that for some finitely generated O_{v}-algebra A ⊆ L, O_{w} = A_{q}, q ∈ Spec(A), satisfies d(w|v) = 1 and e(w|v) = ϵ(w|v).

The proof of this result is based on the following generalization of [5], (18.5):

Lemma 2.6 Let (L, w)|(K, v) be a finite extension of valued fields and let (O, M) be a local ring with the properties:

• L = Frac(O),

• O_{v} ⊆ O ⊆ O_{w} where all extensions are local,

• w(M) = w(M_{w}) and O/M = Lw.

Then dim_{Kv}(O/M_{v}O) = ϵ(w|v)f(w|v) holds.

Proof The proof can be copied from [5], (18.5 a), where it is given in the case O = O_{w}. We sketch it here to make clear, where the different assumptions enter in.

Let δ_{1} = 0 < δ_{2} < ... < δ_{ϵ}, ϵ := ϵ(w|v), be the different positive elements of wL smaller than vK^{>0} and define O_{i} := {a ∈ O| wa ≥ δ_{i}}, i = 1, ..., ϵ, and O_{ϵ+1} = M_{v}O. The chain O_{ϵ+1} ⊂ O_{ϵ} ⊂ ... ⊂ O_{1} = O of O_{v}-modules satisfies dim_{Kv}(O_{i}/O_{i+1}) = f(w|v): by assumption there exist elements b_{1}, ..., b_{f} ∈ O, f := f(w|v), such that b_{1}w, ..., b_{f}w form a Kv-basis of Lw. Furthermore for every i = 1, .., ϵ there exists an x_{i} ∈ M such that wx_{i} = δ_{i}. The elements x_{i}b_{1} + O_{i+1}, ..., x_{i}b_{f} + O_{i+1} then form a Kv-basis of O_{i}/O_{i+1}. The assertion now follows by a straightforward calculation.

We proceed with the proof of proposition 2.5:

Proof One considers henselizations O_{v^{h}} ⊆ O_{w^{h}} of O_{v} resp. O_{w} and their fields of fractions K^{h} resp. L^{h} = L.K^{h}. Inside L^{h} sits the tower of rings O_{v^{h}} ⊆ AO_{v^{h}} ⊆ O_{w^{h}}. Since O_{v^{h}} is henselian the ring O := AO_{v^{h}} is a local ring and the extensions O|O_{v^{h}} resp. O_{w^{h}}|O are local because O_{w^{h}}|O_{v^{h}} is integral. The other requirements of lemma 2.6 are also satisfied: the maximal ideal M of O lies over q ⊲ A, thus w^{h}(M_{w^{h}}) = w(M_{w}) = w(q) ⊆ w^{h}(M) ⊆ w^{h}(M_{w^{h}}). Moreover one has the natural embeddings Lw = A/q ↪ O/M ↪ L^{h}w^{h} = Lw.

Since A is a finitely generated O_{v}-algebra the extension AO_{v^{h}}|O_{v^{h}} is a finite thus free O_{v^{h}}-module. Its rank equals [L^{h} : K^{h}] because Frac(AO_{v^{h}}) = L^{h}.

Combining the fundamental equality

[L^{h} : K^{h}] = e(w^{h}|v^{h})f(w^{h}|v^{h})d(w^{h}|v^{h}) = e(w|v)f(w|v)d(w|v)

with lemma 2.6 finally yields:

e(w|v)f(w|v)d(w|v) = rank_{O_{v^{h}}} (O)
= dim_{Kv}(O/M_{v^{h}}O)
= ϵ(w^{h}|v^{h})f(w^{h}|v^{h})
= ϵ(w|v)f(w|v)
≤ e(w|v)f(w|v),

and thus the assertion.

As an application we next describe the valuations induced by the vertical curves on the function field F:

Proposition 2.7 Let C be a normal R-curve and let η ∈ C be a generic point of the fibre C_{p}, p ∈ Spec(R) \ 0. The local ring O_{C,η} then is a valuation ring O_{v_{p}} of a valuation v_{p} of F = K(C) prolonging v_{p} with the properties:

1. F v_{p}|Kv_{p} is a function field of transcendence degree one.

2. e(v_{p}|v_{p}) = ϵ(v_{p}|v_{p}) and d(v_{p}|v_{p}) = 1.

3. ∀p ∈ nonfgSpec(R) : e(v_{p}|v_{p}) = 1.

Proof We can assume that R is a valuation ring O_{v}. The local ring O_{C,η} has the form A_{q} for some finitely generated normal integral O_{v}-algebra A, and A/q is a one-dimensional Kv-algebra due to the choice of η. Let x ∈ A be an element such that x + q is transcendental over Kv; it is then straightforward to check that q ∩ O_{v}[x] = M_{v}[x] holds in the polynomial ring O_{v}[x] ⊆ A. It follows that the integral closure O of the local ring O_{v}[x]_{M_{v}[x]} in F is contained in A_{q}. Now since O_{v}[x]_{M_{v}[x]} is a valuation ring, namely the valuation ring of the so-called Gauß prolongation v_{x} of v to K(x), the integral closure O in the algebraic extension F |K(x) is Prüfer and thus A_{q} is a valuation domain O_{v} of some valuation v|v.

Property 1 of the valuation v is now clear by construction. Property 2 follows from proposition 2.5 applied to the extension (F, v)|(K(x), v_{x}), the fact that e(v_{x}|v) = 1 and d(v_{x}|v) = 1 hold, and the multiplicative nature of these quantities. Concerning property 3 one argues as follows: if M_{v} is not finitely generated the value group v(K) has no smallest positive element. By definition of v_{x} the same holds for v_{x}(K(x)) and therefore also for v(F). Conseque ntly ϵ(v|v) = 1, which gives the assertion using property 2.

As a consequence of proposition 2.7, that is quite surprising at the first glance, we get:

Corollary 2.8 The fibres C_{p}, p ∈ nonfgSpec(R), of a normal R-curve are reduced. If C is proper over R all fibres C_{p}, p ∈ Spec(R)\MaxSpec(R), are reduced.

Proof To verify the first assertion one shows that reducedness of some fibre C_{p} is equivalent to the unramifiedness of all valuations v_{p} obtained from the generic points of this fibre as described by proposition 2.7.

Again one can assume R to be a valuation ring O_{v} and p = M_{v}. If the fibre Cv over M_{v} is reduced, and O_{C,η} =: O_{v} is the local ring at a generic point of Cv, then M_{v}O_{v} is a radical ideal in a valuation domain, hence prime. Since this prime is lying over M_{v} it equals the maximal ideal M_{v}: by proposition 2.7 point 1 the valuation rings O_{v} and O_{v} have the same rank. Now according to proposition 2.7 point 2 ramific ation of v|v can only appear if ϵ(v|v) > 1; in this case M_{v} must be a principal ideal tO_{v} and therefore M_{v} = tO_{v} as we just proved. The latter implies e(v|v) = ϵ(v|v) = 1 as desired.

Vice versa, let V be the set of constant reductions obtained from the generic points of the fibre Cv, and assume e(v|v) = 1 for all v ∈ V. Let x ∈ C be a point on the closed fibre Cv; we have to show that M_{v}O_{C,x}⊲O_{C,x} is a radical ideal. According to [11], Thm. 2.15 the local ring of x admits an intersection representation

O_{C,x} = B ∩ ⋂_{v∈V_{x}} O_{v},

where B ⊆ F is a K-algebra and V_{x} ⊆ V is the subset of constant reductions corresponding to those irreducible components of Cv passing through x. Obviously M_{v}O_{C,x} ⊆ ⋂_{v∈V_{x}}(O_{C,x} ∩ M_{v}); we show that actually equality holds thus verifying

that M_{v}O_{C,x} is radical. Let a ∈ ⋂_{v∈V_{x}}(O_{C,x} ∩ M_{v}); by assumption there exists

c ∈ M_{v} such that v(c) ≤ v(a) for all v ∈ V_{x}. Therefore \frac{a}{c} ∈ ⋂ lim its_{v∈V_{x}}O_{v} and since moreover c ∈ M_{v} ⊂ K ⊆ B^{×} we get \frac{a}{c} ∈ O_{C,x}. This ends the verification: a = c\frac{a}{c} ∈ M_{v}O_{C,x}.

The first assertion of the corollary now follows from point 3 of proposition 2.7. To verify the second statement fix primes p ∈ Spec(R) \ MaxSpec(R) and

p′ ∈ MaxSpec(R) with p ⊂ p′. Let η ∈ C_{p} be a generic point of the fibre; the Zariski closure \overline{η} intersects the fibre C_{p′} due to the properness of C over R. Moreover at least one generic point ξ ∈ C of C_{p′} lies in this intersection: take an affine neighborhood U ⊆ C of a point x ∈ C in the intersection and consider the finitely g enerated R_{p′}-algebra A := O_{C}(U)⊗_{R} R_{p′}. Let q ⊂ q′ ∈ Spec(A) be the primes corresponding to η and x respectively. Then trdeg(A/q|R_{p′}/pR_{p′}) = 1 and by lemma 2.1 of [15] every prime \overline{q} ∈ Spec(A/q) minimal among those containing p′R_{p′}/pR_{p′} satisfies trdeg((A/q)/\overline{q}|k(p′)) = 1, hence \overline{q} gives rise to a prime q′′ ⊃ q of A lying over p′ and corresponding to a generic point of the fibre C_{p′}.

Let O_{v} = O_{C,ξ}; the valuation ring O_{C,η} then equals a localization O_{v_{q}} = (O_{v})_{q}, q ∈ Spec(O_{v}) lying over p. Let O_{v} := R_{p′}, then O_{v_{q}} is a prolongation of O_{v_{p}} = (O_{v})_{p}. In this situation by general valuation theory we have a commutative diagram

0 → Δ_{q} → vF → v_{q}F → 0
↑ ↑ ↑
0 → Δ_{p} → vK → v_{p}K → 0

with exact rows, where Δ_{p} resp. Δ_{q} are convex subgroups of vK resp. vF. From this diagram we deduce the index formula

e(v|v) = e(v_{q}|v_{p})(Δ_{q} : Δ_{p}).  (2.2)

If the fibre C_{p′} is reduced, we already proved that e(v|v) = 1 and thus (2.2) yields e(v_{q}|v_{p}) = 1, which in turn implies reducedness of the fibre C_{p}.

If C_{p′} is not reduced we know that e(v|v) > 1 and thus from proposition 2.7 point 3 that the value group vF possesses a minimal positive element γ. The set {0, γ, 2γ, ..., (e(v|v) − 1)γ} forms a set of representatives of vF/vK. On the other hand the subgroup ℤγ ⊆ vF is convex and thus ℤγ ⊆ Δ_{q}. We conclude that (Δ_{q} : Δ_{p}) ≥ e(v|v) hence e(v_{q}|v_{p}) = 1 as asserted.

A similar behavior of the fibres is present in the context of regularity and leads to an unexpected structure theorem about regular R-curves.

3 Regular curves over Prüfer domains

A noetherian local ring O is regular iff its global homological dimension is finite. This characterization resp. definition can be generalized to non-noetherian local rings by requiring the finitely generated ideals I ⊲ O to have finite projective dimension. Regular rings in this sense need not have finite global dimension. It is not even true in general that their weak dimension is finite. However the central statement of the present section is that a local ring O_{C,x} of an R-curve C is r egular iff wdim(O_{C,x}) ≤ 2. This result enables one to apply several properties of local rings of weak dimension two (or smaller) published in Vasconcelos’ classification [19] of rings of dimension two, to eventually arrive at a simple structure theorem for regular R-curves. These properties essentially are consequences of Burch’s theorem: an ideal I ⊲ R in a commutative ring R possessing a resolution

0 → R^{n−1} →^{f} R^{n} → I → 0

and containing a nonzero divisor, has the form I = dJ, where d ∈ R is a nonzero divisor and the ideal J is generated by the subdeterminants of order n − 1 of a matrix associated with f.

3.1 Regularity. The following definition of regularity stems from the work [2] of J. Bertin and is suitable for non-noetherian rings:

Definition 3.1 The local ring O is called regular if the projective dimension pdim_{O}(I) of every finitely generated ideal I ⊲ O is finite.

A point x on a scheme X is called a regular point if the local ring O_{X ,x} is regular. The regular locus of X is the set Reg(X) := {x ∈ X : O_{X ,x} is regular} and X is called regular if X = Reg(X).

This extension of the concept of regularity has been studied by various authors— see [7] for an overview—and proved to be useful in particular in connection with coherence.

The most simple non-noetherian examples of regular local rings are the valuation rings; thus Spec(R) for every Prüfer domain R is a regular scheme. To provide more interesting examples one uses tools from homological algebra: recall that the weak dimension wdim_{R}(M) of a module M over the commutative ring R is the shortest resolution

0 → F_{d} → F_{d−1} → ... → F_{0} → M → 0

of M using flat R-modules F_{i}, provided such a resolution exists. Otherwise one sets wdim_{R}(M) = ∞. As usual the weak dimension of the ring R itself is defined as wdim(R) = sup(wdim_{R}(M) : M an R-module).

Assume now that O is a coherent local ring, i.e., that all finitely generated ideals I ⊲O are finitely presented, and assume moreover that wdim(O) ≠ ∞. Then every finitely generated ideal I has a flat resolution

0 → F_{d} → F_{d−1} → ... → F_{0} → I → 0,

where the modules F_{i} can be choosen to be finitely presented since O is coherent. It follows that the F_{i} are projective since they are flat and O is local, hence pdim_{O}(I) ≤ d, that is we have proved:

Lemma 3.2 A local coherent ring of finite weak dimension is regular.

In general the implication of lemma 3.2 cannot be reversed as for noetherian local rings, but it turns out that for R-curves this is possible:

Theorem 3.3 The regular locus of an R-curve C can be characterized as

Reg(C) = {x ∈ C : wdim(O_{C,x}) ≠ ∞} = {x ∈ C : wdim(O_{C,x}) ≤ 2}.

Remark: Observe that in the case of a Dedekind domain R one has

wdim(O_{C,x}) = gldim(O_{C,x}) = dim(O_{C,x}) ≤ 2,

where the equality of the weak and the global dimension gldim(O_{C,x}) holds for every noetherian local ring ([18], Thm. 9.22).

The proof of theorem 3.3 essentially consists of a combination of two results about non-noetherian grade of ideals: for a finitely generated ideal I ⊲ A of the commutative ring A and an A-module M the non-noetherian grade of I on M is defined as

Grade_{A}(A/I, M) := sup(n ∈ ℕ : ∃ faithfully flat A-algebra B
and a M ⊗_{A} B-regular sequence (b_{1}, ..., b_{n}) ∈ (IB)^{n})

—see [1] for details in particular for the extension of this definition to non-finitely generated ideals I. For a prime ideal I this extension amounts to

Grade_{A}(A/I, M) = sup(Grade_{A}(A/I_{0}, M) : I_{0} ⊆ I a finitely generated ideal).

Note that if A is a noetherian ring Grade_{A}(A/I, M) is just the length of the longest M-regular sequence in the ideal I, i.e., coincides with the usual notion of grade.

The use of non-noetherian grade becomes visible in the important formula

wdim(O) = Grade_{O}(O/M, O)  (3.1)

for a coherent regular local ring O with maximal ideal M—see [8], lemma 3.

To calculate Grade_{O}(O/M, O) for a local ring of an R-curve we use the following special case of [1], Cor. 2.10: let A be a flat finitely presented R-algebra, q ∈ Spec(A), p := q ∩ R, then:

Grade_{A_{q}}(k(q), A_{q}) = Grade_{R_{p}}(k(p), R_{p}) + Grade_{A_{q}⊗_{R}k(p)}(k(q), A_{q} ⊗_{R} k(p)) (3.2)

holds. We are now prepared to prove theorem 3.3:

Proof Let O_{C,x} = A_{q}, q ∈ Spec(A), for some finitely generated integral R-algebra A, and set p = q ∩ R. The algebra A is finitely presented and flat over R—see theorem 2.2. We can thus apply formula (3.2). Taking into account that R_{p} is a valuation domain or a field and thus has wdim(R_{p}) ≤ 1, that A_{q} ⊗_{R} k(p) is noetherian and thus satisfies

Grade_{A_{q}⊗_{R}k(p)}(k(q), A_{q} ⊗_{R} k(p)) ≤ dim(A_{q} ⊗_{R} k(p)) ≤ 1,

the combination of the formulas (3.1) and (3.2) yields:

wdim(A_{q}) = Grade_{A_{q}}(k(q), A_{q}) ≤ 1 + 1 = 2

as asserted.

Regular points on an R-curve can now be classified:

Corollary 3.4 The regular points on an R-curve fall into the following classes:

1. wdim(O_{C,x}) = 0: in this case x is the generic point of C.

2. wdim(O_{C,x}) = 1: in this case x is either a closed point on the generic fibre C or a generic point of a fibre C_{p}, p ≠ 0. In the first case O_{C,x} is a discrete valuation ring, in the second case it is the valuation ring of a constant reduction of K(C).

3. wdim(O_{C,x}) = 2: in this case x is a closed point of a fibre C_{p}, p ≠ 0. If M_{C,x} is finitely generated a minimal set of generators consists of two elements.

Proof The first of the three cases is clear. Furthermore it is well-known that a local ring of weak dimension one is a valuation ring. Therefore the second case is verified (see proposition 2.7) once one knows that the local ring O_{C,x} in a closed point of some fibre C_{p}, p ≠ 0, is no valuation ring. This has been proved in [11], Lemma 2.8. It remains to check the statement about a minimal set of generators of M_{C,x} in the finitely generated case. This follows from [19], Thm. 5.22.

We now turn to the investigation of the structure of a regular R-curve C. In this context the dual of the Zariski tangent space at a point x ∈ C plays a role: as in the noetherian case it is defined as the k(x)-vector space M/M^{2}, where (O, M) is the local ring at the point x ∈ C.

A priori it is not clear that dim_{k(x)}(M/M^{2}) is finite. However in the present situation we know even more than that:

Proposition 3.5 The dimension dim_{k(x)}(M/M^{2}) at a closed point x of a fibre C_{p} of an R-curve C is non-zero and finite. For p ∈ nonfgSpec(R) there is a natural isomorphism M/M^{2} ≅ \overline{M}/\overline{M}^{2}, where (\overline{O}, \overline{M}) := (O/pO, M/pO).

Proof The local ring \overline{O} is noetherian, therefore finite generation of pR_{p} implies finite generation of the maximal ideal M of O. The space M/M^{2} then has positive finite dimension.

Now assume that pR_{p} is not finitely generated. In this case one has pR_{p} = (pR_{p})^{2}, hence pO ⊆ M^{2}, since R_{p} is a valuation domain. The exact sequence

0 → (pO + M^{2})/M^{2} → M/M^{2} → \overline{M}/\overline{M}^{2} → 0

therefore yields an isomorphism M/M^{2} ≅ \overline{M}/\overline{M}^{2}. In particular M/M^{2} has positive finite dimension.

It is already clear that finite generation of pR_{p} plays a particular role in our investigation; the next result clarifies the relationship between finite generation of pR_{p} and that of M:

Lemma 3.6 Let (O, M) be the local ring at a regular point x of an R-curve, and assume that x is a closed point on a fibre C_{p}, p ≠ 0. The maximal ideal M then is finitely generated if and only if pR_{p} is finitely generated.

Proof It only remains to verify the implication ⇒. Let a ∈ M be a member of a minimal set of generators of M, in particular a ∉ M^{2} by Nakayama’s Lemma. The regularity of O implies that finitely many elements of O always have a greatest common divisor ([7], Cor. 6.2.10), and thus that a is a prime element since it is irreducible. One can now apply results from [19]: the arguments on page 54 yield pdim_{O/aO}(M/aO) ≠ ∞ and thus wdim(O/a cO) ≠ ∞ by theorem 5.22. Applying the dimension formula [7], Cor. 3.1.4 combined with theorem 3.3 of the present article gives

wdim(O/aO) = wdim(O) − 1 = 1.

It follows that O/aO is a valuation domain dominating the valuation domain R_{p}/p_{a}R_{p}, p_{a} := aO ∩ R ⊆ p.

The prime aO is either a generic or a closed point of the fibre C_{p_{a}}. In the first case p_{a} ≠ 0 and therefore the ring O/aO is the local ring at a regular closed point on an R/p_{a}-curve. Due to corollary 3.4 point 3 the maximal ideal M/aO must be minimally generated by two elements unless p_{a} = p. Since on the other hand the finitely generated ideal M/aO in the valuation domain O/aO is principal, we indeed have p_{a} = p. In this case the valuation ring O_{aO} dominates R_{p} and has principal maximal ideal; the latter is inherited by the valuation ring R_{p}, because O_{aO} belongs to a constant reduction v_{p} of F (proposition 2.7) and therefore (v_{p}(F) : v_{p}(K)) ≠ ∞.

The case that aO is a closed point of C_{p_{a}} can only appear if p_{a} = 0: otherwise, the local ring O_{aO} in this closed point would have principal maximal ideal in contradiction to corollary 3.4 point 3.

Now if p_{a} = 0 the valuation domain O/aO dominates R_{p} in a finite extension of K = Frac(R). Again the property of having a principal maximal ideal is then inherited by R_{p}, completing the proof.

We are finally in the position to clarify the structure of regular R-curves to a certain extent:

Theorem 3.7 The fibres C_{p}, p ∈ nonfgSpec(R), of a regular R-curve C are regular k(p)-schemes.

Proof Let x ∈ C_{p} be a closed point on some fibre C_{p}, p ≠ 0, and assume p ∈ nonfgSpec(R). Lemma 3.6 then shows that M_{C,x} is not finitely generated. Now we have to use Vasconcelos’ results on the space M_{C,x}/M_{C,x}^{2} for local rings of weak dimension two: proposition 5.24 of [19] states that

dim_{k(x)} (M_{C,x}/M_{C,x}^{2}) ≤ 2,

and by corollary 5.25 to this proposition equality can only occur if M_{C,x} is finitely generated. Using proposition 3.5 we conclude that

1 = dim_{k(x)}(M_{C,x}/M_{C,x}^{2}) = dim_{k(x)}(\overline{MC,x}/\overline{MC,x}^{2})

holds, which implies the regularity of the noetherian ring O_{C,x}/pO_{C,x}.

In the case of proper O_{v}-curves the preceeding result can be sharpened:

Corollary 3.8 A proper O_{v}-curve C over a valuation ring O_{v} with non-finitely generated maximal ideal M_{v} is regular if and only if the closed fibre Cv is a regular Kv-scheme.

Proof Every proper R-curve C having regular closed fibres is regular: the properness implies that every point of C specializes into a point on a closed fibre of C, hence by the stability of regularity under localization ([7], Thm. 6.2.3) it suffices to show that the points on the closed fibres are regular points of C.

Let x ∈ C_{p} be a regular point on the closed fibre C_{p}; then by [7], Thm. 3.1.3 one gets:

wdim(O_{C,x}) ≤ wdim(O_{C,x}/pO_{C,x}) + wdim_{O_{C,x}}(O_{C,x}/pO_{C,x}).

By assumption wdim(O_{C,x}/pO_{C,x}) = 1; furthermore since O_{C,x}|R is flat one has pO_{C,x} = p ⊗_{R} O_{C,x}, which is thus a flat O_{C,x}-module. This in turn implies wdim_{O_{C,x}}(O_{C,x}/pO_{C,x}) ≤ 1. The assertion x ∈ Reg(C) now follows from lemma 3.2.

Vice versa, if C is a regular O_{v}-curve and M_{v} is non-finitely generated the assertion follows from theorem 3.7.

One reason for introducing and investigating regular schemes in the noetherian case is their nice divisor theory: Weil and Cartier divisors coincide due to the theorem of Auslander-Buchsbaum on the factoriality of local regular rings. Bertin’s generalization of regularity leads to an entirely analogues result with which we finish this subsection:

Theorem 3.9 On a regular R-curve C every coherent divisorial O_{C}-ideal is invertible, thus the group of Cartier divisors CaDiv(C) coincides with the group D_{coh}(C).

Proof It suffices to verify that a coherent divisorial O_{C}-ideal I is stalkwise principal. The stalk I := I_{x} in some point x ∈ C is a finitely generated divisorial fractionary ideal of the local ring O_{C,x} hence satisfies

I = ⋂_{I⊆fO_{C,x}} fO_{C,x}.  (3.3)

By assumption O_{C,x} is a regular ring, which by [7], Cor. 6.2.10 implies that finitely many elements of O_{C,x} always have a greatest common divisor. Consequently the set of principal fractionary ideals containing a given finitely generated fractionary ideal possesses a minimal element. Applied to the fractionary ideal I and taking (3.3) into account yields I = gO_{C,x} for some g ∈ F as desired.

3.2 Étale morphisms. The purpose of this subsection is to verify the permanence of regularity of an R-curve under étale covering and base change. As in the preceeding subsection the results strongly depend on the properties of rings of weak dimension two.

Theorem 3.10 Let (O, M) be a local ring of an R-curve. If O is regular then every étale O-algebra B is regular too.

Proof We distinguish according to the weak dimension of O.

wdim(O) = 0: the local ring O is then a field; thus B is a finite product of finite separable extensions of O.

In the remaining cases it suffices to verify the regularity of the local rings B_{N} for prime ideals N ∈ Spec(B) lying over M: for an arbitrary prime ideal N the tensor product B ⊗_{O} O_{N ∩O} is an étale extension of the regular local ring O_{N ∩O} by the permanence of étale extensions under base change ([17], Ch. II, Prop. 2) and of regularity under localization ([7], Thm. 6.2.3).

Observe also that B is a coherent ring: by definition B is a finitely presented O-algebra, thus it suffices to verify the coherence of the polynomial rings O[X_{1}, ..., X_{m}]. Since O is a localization of a finitely presented integral (thus flat) R-algebra, the same holds for the polynomial rings O[X_{1}, ..., X_{m}], which are therefore coherent by [7], Cor. 7.3.4.

wdim(O) = 1: this is the case of a valuation domain O. Since by [17], Ch. VII, Prop. 2 the local ring B_{N} is normal, it contains the integral closure O′ of O in L := Frac(B_{N}). The field extension L|Frac(O) is algebraic, hence O′ is a Prüfer domain and it follows that B_{N} is a valuation domain.

wdim(O) = 2: this is the most complicated case; we apply several results from [19] to treat it. First by proposition 3.5 and [19], Prop. 5.24 dim_{O/M}(M/M^{2}) ∈ {1, 2}; these two cases are treated separately.

dim_{O/M}(M/M^{2}) = 2: in this case [19], Cor. 5.25 shows that M is generated by two elements; since by [17], page 38 we have NB_{N} = MB_{N} the maximal ideal of B_{N} is also generated by two elements.

Next by theorem 5.22 of [19] and theorem 3.3 the equality pdim_{O}(M) = 1 holds; tensoring a projective resolution of M with B_{N} hence yields

pdim_{BN} (NB_{N}) ≤ 1.

A further application of theorem 5.22 gives wdim(B_{N}) ≤ 2 and thus the desired regularity by lemma 3.2.

dim_{O/M}(M/M^{2}) = 1: every a ∈ M \ M^{2} is a prime element such that O/aO is a valuation domain ([19], Prop. 5.27). The ring B/aB is an étale O/aO-algebra, the same reasoning as in the case wdim(O) = 1 now shows that B_{N} /aB_{N} is a valuation domain. Finally we apply [7], Cor. 3.1.4 to obtain

wdim(B_{N}) = wdim(B_{N} /aB_{N}) + 1 = 2,

which completes the proof.

Finite étale morphisms between R-curves can now be treated easily:

Corollary 3.11 Let X → C be a finite étale R-morphism of the integral separated scheme X to the regular R-curve C, then X is a regular R-curve.

Proof The assertion that X be regular follows directly from theorem 3.10. X is of finite type over R and has non-empty fibres since this holds for C|R and the morphism X → C. The generic fibre of X is a finite covering of the generic fibre of C.

Remark: In the situation of the corollary let η ∈ C be a generic point of some fibre C_{p}, p ≠ 0, and let v_{p} be the constant reduction of F = K(C) associated to the local ring O_{C,η} as described in proposition 2.7. The points ξ ∈ X lying above η are then generic points of the fibre X_{p} and the corresponding set of valuations consists of the prolongations w_{p} of v_{p} to E = K(X). By assumption O_{w_{p}} is a localization of a finitely generated O_{v_{p}}-algebra. Application of proposition 2.5 yields d(w_{p}|v_{p}) = 1 and e(w_{p}|v_{p}) = ϵ(w_{p}|v_{p}). Since the morphism X → C is étale we have M_{v_{p}}O_{w_{p}} = M_{w_{p}} and thus e(w_{p}|v_{p}) = 1. We see that the extension (E, w_{p})|(F, v_{p}) is an unramified defectless extension of valued fields. Next we deal with (local-ind-)étale base change of R-curves. Here the behavior of normality is as interesting as the behavior of regularity:

Corollary 3.12 Let C be a regular resp. normal R-curve with absolutely irreducible generic fibre, and let S|R be an étale extension of Prüfer domains, then C ×_{R} S is a regular resp. normal S-curve.

Let C be a regular resp. normal O_{v}-curve with absolutely irreducible generic fibre, and let (L, w)|(K, v) be an algebraic extension such that (L, w) is contained in the absolute inertia field of (K, v), then C ×_{O_{v}} O_{w} is a regular resp. normal O_{w}-curve.

Proof The scheme C ×_{R} S is covered by spectra of algebras B := A ⊗_{R} S, where A is an R-algebra. Thus the assumption about S|R implies that B is an étale A-algebra, and theorem 3.10 shows the regularity of C ×_{R} S provided C is regular. Normality of C ascends to C ×_{R} S by [17], Ch. VII, Prop. 2.

Faithful flatness is stable under base change, thus the fibres of C ×_{R} S over S are not empty.

By assumption the generic fibre of C ×_{R} S is irreducible, together with the normality of the local rings this yields integrality.

Concerning the second assertion [17], Ch. X, Thm. 1 shows that O_{w}|O_{v} is local-étale in the case of a finite extension L|K, thus in this case the assertions are already proved. In the infinite case the limit O_{w} = lim_{M^{→}}(O_{w} ∩ M), M running through the finite subextensions of L|K, for every point x ∈ C yields a limit

O_{C,x} ⊗_{O_{v}} O_{w} = lim_{M^{→}}(O_{C,x} ⊗_{O_{v}} (O_{w} ∩ M)).  (3.4)

Since the extensions (O_{w} ∩M)|O_{v} are local-étale we conclude that the tensor products O_{C,x} ⊗_{O_{v}} (O_{w} ∩ M) are regular. They are also coherent and for M_{1} ⊆ M_{2} the extension O_{C,x} ⊗_{O_{v}} (O_{w} ∩ M_{2})|O_{C,x} ⊗_{O_{v}} (O_{w} ∩ M_{1}) is flat. In this situation [7], Thm. 6.2.2 can be applied and gives the regularity of the left hand side of (3.4).

It remains to verify the normality of C ×_{O_{v}} O_{w} provided C is normal: for a point y ∈ C ×_{O_{v}} O_{w} lying over x ∈ C we can localize the equation (3.4) at the primes induced by M_{C×_{Ov} O_{w},y} in the rings O_{C,x} ⊗_{O_{v}} (O_{w} ∩ M) thus showing that O_{C×_{Ov} O_{w},y} is a local-ind-étale O_{C,x}-algebra. The assertion now follows from [17], Ch. VIII, Thm. 3.

4 Divisor reduction

In the sequel we provide an addendum to the results on the reduction of Weil divisors of a function field of transcendence degree one with respect to a finite set of constant reductions published in [9]: let (F |K, V|v) be a function field of one variable equipped with a finite set V of constant reductions prolonging v. To compare the arithmetic of F |K with that of the residue fields F v|Kv one seeks to construct a homomorphism

r_{n} : Div(F |K) → ∏_{v∈V} Div(F v|Kv)

preserving the partial ordering of the divisor groups, the degree of divisors, and mapping principal divisors to such. A homomorphism with these properties is called arithmetic. The meaning of the index in the symbol r_{n} becomes clear below.

In [9], section 2 a map r_{n} is defined for valued function fields subject to rather strong conditions and it is shown, that r_{n} is an arithmetic homomorphism on a certain subgroup of Div(F |K). The difficulties when trying to define a (natural) homomorphism r_{n} become visible if one takes a geometric point of view using normal O_{v}-curves. Doing this one should nevertheless be aware of the fact, that in general there is no complete correspondence between valued function fields and proper normal O_{v}-curves.

Let C be a proper normal O_{v}-curve and (F |K, V|v) the corresponding valued function field as described by proposition 2.7. The group of Weil divisors of F |K is then equal to the group Div(C) of Weil divisors of the generic fibre C of C. By proposition 2.7 the irreducible components of the closed fibre Cv are in bijection with the elements of V. We use the symbol Cv for the irreducible component of Cv corresponding to v ∈ V equipped with the reduced subscheme structure. The residue field F v|Kv then is its function field and Div(F v|Kv) equals Div(Cv′), where Cv′ is the normalization of Cv in F v. The relation between Cv and the components Cv′ is described by the maps:

⋃_{v∈V} Cv′ = (Cv_{red})′ ⟶^{η} ⋃_{v∈V} Cv = Cv_{red} ⟶^{ρ} Cv,

where ρ resp. η denote removing nilpotents resp. normalization in the total ring of quotients K(Cv_{red}) = ∏_{v∈V} F v.

For invertible sheaves L on Cv a degree function deg_{Cv}(L) defined via the lengths ℓ_{O_{Cv,x}}(O_{Cv,x}/L_{x}) (in the case L ⊲ O_{Cv}) exists, and it is well-known that this function is invariant under normalization

deg_{Cv}(L) = deg_{Cv′}(η^{∗}L)  (4.1)

provided that Cv is reduced.

The effect of ρ to the degree of a invertible sheaf is given by the formula

deg_{Cv}(L) = ∑_{v∈V} e(Cv, Cv) deg_{Cv}((ρ^{∗}L)|_{Cv}),  (4.2)

where e(Cv, Cv) := ℓ_{O_{Cv,η}}(O_{Cv,η}), η the generic point of Cv, and ℓ denoting the length of a module.

In the sequel we identify the invertible subsheaves of the sheaf of rational functions K(Cv) with the group of Cartier divisors CaDiv(Cv).

Now let j : Cv = C ×_{O_{v}} Kv → C be the natural morphism, then we have the following result concerning the reduction (or specialization) of Weil divisors:

Theorem 4.1 For a proper regular O_{v}-curve C the divisor reduction map

r : Div(C) → CaDiv(Cv), D ↦ j^{∗}(J_{D})

is an arithmetic homomorphism—see proposition 2.4 for the definition of J_{D}. The homomorphism

r_{n} : Div(C) → ∏_{v∈V} Div(Cv′), D ↦ η^{∗}ρ^{∗}r(D)

preserves the partial ordering and maps principal divisors to such. Moreover the equality

deg_{C}(D) = ∑_{v∈V} e(v|v) deg_{Cv′}(r_{n}(D)|_{Cv′})

holds. If the fibre Cv is reduced r_{n} is degree preserving.

Remarks: 1. The homomorphism r_{n} coincides with the map r defined in [9], section 2, where reducedness of the closed fibre Cv is assumed. In [9], theorem 2.2 it is shown that r is an arithmetic homomorphism on a subgroup of Div(C) generated by prime divisors, which are not mapped to divisors intersecting the singular locus of Cv. Comparing with theorem 4.1 this shows, that the obstructions for defining a natural arithm etic homomorphism

Div(F |K) → ∏_{v∈V} Div(F v|Kv)

are in fact the non-reducedness of the closed fibre of a proper normal O_{v}-model of F |K and the singularities of this model, provided such a model exists.

2. In the noetherian case theorem 4.1 can be proved using the Riemann-Roch-theorem and the invariance of the arithmetic genus in flat families—see [3], Ch. 9, Prop. 2. This approach can also be generalized to the non-noetherian case.

Proof The map r is well-defined: by theorem 3.9 the sheaf J_{D} is invertible due to the regularity of C. This also implies that r is a homomorphism since the assignment (D ↦ J_{D}) is a homomorphism. Furthermore it is clear that r preserves principal divisors and the partial ordering. Thus it remains to show that r is degree preserving.

It suffices to check the preservation of degrees for prime divisors. Let P ∈ Div(C) be a prime divisor and P ⊂ C the horizontal curve defined by P. By proposition 2.3 the scheme P is finite over O_{v}: P ≅ Spec(A), where A is a finite O_{v}-algebra and therefore a free O_{v}-module. The field of fractions Frac(A) equals the residue field K(P) of the closed point P ∈ C. It follows that rank_{O_{v}}(A) = deg_{C}(P) and thus by Nakayama’s lemma that dim_{Kv}(Av) = deg_{C}(P) (Av = A/M_{v}A) holds.

On the other hand we have j^{∗}(J_{P})_{x} = ((J_{P})_{jx} + M_{v}O_{C,x})/M_{v}O_{C,x}, which shows that

ℓ_{O_{Cv,x}}(O_{Cv,x}/j^{∗}(J_{P})_{x}) = ℓ_{O_{Pv,x}}(O_{Pv,x}),

where Pv := P ×_{O_{v}} Kv, thus giving

dim_{Kv}(Av) = ∑_{q∈MaxSpec(Av)}[k(q) : Kv] ℓ_{Av_{q}}(Av_{q})
= ∑_{x∈Pv}[k(x) : Kv] ℓ_{O_{Pv,x}}(O_{Pv,x})
= deg_{Cv}(j^{∗}J_{P}).

The assertions about the homomorphism r_{n} are straightforward to verify except the degree equation: we apply the formulas (4.2) and (4.1) in the case L = j^{∗}(J_{P}) = r(P) and obtain

deg_{Cv}(r(P)) = ∑_{v∈V} e(Cv, Cv) deg_{Cv}((ρ^{∗}r(P))|_{Cv})
= ∑_{v∈V} e(Cv, Cv) deg_{Cv′}(η_{v}^{∗}[(ρ^{∗}r(P))|_{Cv}])
= ∑_{v∈V} e(Cv, Cv) deg_{Cv′}((η^{∗}ρ^{∗}r(P))|_{Cv′}),

where η_{v} : Cv′ → Cv is the normalization morphism of the component Cv. Using the degree preservation of r it now suffices to verify the formula e(Cv, Cv) = e(v, v).

The sequence of O_{v}-modules

M_{v}O_{v} ⊂ O_{1} ⊂ ... ⊂ O_{e} = O_{v},

where O_{i} := {a ∈ O_{v} : v(a) ≥ γ_{i}} and 0 = γ_{e} < γ_{e−1} < ... < γ_{1} are the different positive elements of vF smaller than vK^{>0}, gives rise to a composition series for O_{v}/M_{v}O_{v} of length e(v|v) due to proposition 2.7 point 2. This proves the formula e(Cv, Cv) = e(v, v).

The degree preservation of r_{n} in the reduced case follows directly from formula (4.1).

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