Essential finite generation of extensions of valuation rings
aa r X i v : . [ m a t h . A C ] J a n ESSENTIAL FINITE GENERATION OF EXTENSIONS OF VALUATION RINGS
RANKEYA DATTA
Abstract.
Given a generically finite local extension of valuation rings V ⊂ W , the question of whether W is thelocalization of a finitely generated V -algebra is significant for approaches to the problem of local uniformizationof valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially offinite type over V in terms of classical invariants of the extension of associated valuations. Knaf’s conjecturehas been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankarvaluations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky forvaluation rings of function fields of characteristic using embedded resolution of singularities. In this paper weprove Knaf’s conjecture in full generality. Introduction
Let
L/K be a finite field extension. Given a domain R with fraction field K , results that characterize whenthe integral closure of R in L is a finite type R -algebra have fundamental applications in algebraic geometry,commutative algebra and number theory. We investigate a local valuative analogue of the finite generation ofintegral closures in this paper. Suppose ω is a valuation of L with valuation ring ( O ω , m ω , κ ω ) and value group Γ ω . Let ν be the restriction of ω to K with valuation ring ( O ν , m ν , κ ν ) and value group Γ ν . Inclusion inducesa local homomorphism ( O ν , m ν , κ ν ) ֒ → ( O ω , m ω , κ ω ) , and the valuation ring O ω is a local ring of the integral closure of O ν in L [Bou98, Chap. VI, § . , Prop. 6].Thus, as a local version of the question of the finite generation of integral closures, it is natural to ask when O ω is the localization of a finite type O ν -algebra. Knaf proposed the following necessary and sufficient conditionin terms of classical invariants of the extension ω/ν [CN19, Conjecture 1.2]. Conjecture 1.1.
Let
L/K be a finite field extension, ω be a valuation of L and ν be the restriction of ω to K .Let L h (resp. K h ) denote the fraction field of the Henselization of O ω (resp. O ν ). Then O ω is essentially offinite type over O ν if and only if both the following conditions are satisfied: (1) [ L h : K h ] = [Γ ω : Γ ν ][ κ ω : κ ν ] . (2) ǫ ( ω | ν ) = [Γ ω : Γ ν ] , where ǫ ( ω | ν ) is the cardinality of { x ∈ Γ ω, ≥ : x < y for all y ∈ Γ ν,> } . Here for a totally ordered abelian group Γ , we define Γ ≥ := { x ∈ Γ : x ≥ } and Γ > := { x ∈ Γ : x > } . In ramification theory, [Γ ω : Γ ν ] is called the ramification index , [ κ ω : κ ν ] the inertia index and ǫ ( ω | ν ) the initialindex of the extension ω/ν . The first equality [ L h : K h ] = [Γ ω : Γ ν ][ κ ω : κ ν ] is the assertion that the extension ω/ν is defectless ; see Definition 4.8 for the notion of defect. The second equality ǫ ( ω | ν ) = [Γ ω : Γ ν ] just meansthat every element of the quotient Γ ω / Γ ν is the class of some element of Γ ω, ≥ .The question of the essential finite generation of extension of valuation rings arises naturally in approachesto the open problem of local uniformization of valuations using ramification theory. For example, an affirma-tive answer to this question for extensions of Abhyankar valuations is an important ingredient in Knaf andKuhlmann’s proof of the local uniformization of Abhyankar valuations [KK05]. In addition, the essential finitegeneration of extensions of valuation rings also features in Knaf and Kuhlmann’s valuation-theoretic argumentof the local uniformization of a valuation in a finite extension of its fraction field [KK09]. Conjecture 1.1 canbe viewed as a generalization of the beautiful ramification-theoretic characterization of the module-finiteness ofthe integral closure of a valuation ring in a finite extension of its fraction field [Bou98, Chap. VI, § . , Thm.2]. We refer the interested reader to [CN19, Cut19] for additional background on this problem. onjecture 1.1 is known in specific cases, often using different techniques. The necessity of conditions (1)and (2) for the essential finite generation of O ω over O ν was proved by Knaf; his argument is reproduced in[CN19, Thm. 4.1] (see also Remark 5.1 for a different approach using Zariski’s Main Theorem). The sufficiencyof conditions (1) and (2) for the essential finite generation of O ω over O ν is known when L/K is normal usingthe transitive action of
Gal(
L/K ) on the fibers of the integral closure of O ν in L [CN19, Cor. 2.2], when κ ω /κ ν is separable using the theory of Henselian elements [KN14, Thm. 1.3], when ω is the unique extension of ν to L using the theory of defect [CN19, Cor. 2.2], when ν is centered on an excellent local two dimensionaldomain with fraction field K using resolution of singularities for excellent surfaces [CN19, Thm. 1.4], when ν is an Abhyankar valuation of a function field K/k by [CN19, Thm. 1.5] and [Cut20, Thm. 1.7] using localuniformization of Abhyankar valuations, and when K is the function field over a field of characteristic usingan explicit form of embedded resolution of singularities [Cut19, Thm. 1.3].We will give a uniform argument that settles Conjecture 1.1 in full generality. Recall that the implicationthat remains to be shown is that if conditions (1) and (2) of Conjecture 1.1 hold, then O ω is essentially of finitetype over O ν . We take as our starting point the veracity of Conjecture 1.1 for unique extensions of valuations[CN19, Cor. 2.2]. Note that if ω h (resp. ν h ) is the valuation of L h (resp. K h ) whose valuation ring is theHenselization O hω (resp. O hν ), then ω h is the unique extension of ν h to L h up to equivalence of valuations by theHenselian property (see Corollary 4.2). Moreover, Henselizations do not alter value groups [Sta20, Tag 0ASK]and residue fields, which means that the ramification, residue and inertia indexes of ω h /ν h coincide with thoseof ω/ν . Thus, assuming (1) and (2) in Conjecture 1.1, it follows that O hω is essentially of finite type over O hν .Using this observation, we prove Conjecture 1.1 by establishing the following result. Theorem 1.2.
Let
L/K be a finite field extension. Suppose ω is a valuation of L with valuation ring ( O ω , m ω , κ ω ) and value group Γ ω . Let ν be the restriction of ω to K with valuation ring ( O ν , m ν , κ ν ) andvalue group Γ ν . Suppose the Henselization of O ω is O hω with fraction field L h and that of O ν is O hν with fractionfield K h . The following are then equivalent. (i) O ω is essentially of finite presentation over O ν . (ii) O ω is essentially of finite type over O ν . (iii) [ L h : K h ] = [Γ ω : Γ ν ][ κ ω : κ ν ] and ǫ ( ω | ν ) = [Γ ω : Γ ν ] . (iv) O hω is essentially of finite type over O hν . (v) O hω is a module finite O hν -algebra. Instead of using known cases of local uniformization or resolution of singularities, we prove Theorem 1.2 byanalyzing the behavior of integral maps under base change along Henselizations. This consideration turns outto be independent of valuation theory and is carried out in Section 3. This section is the heart of our paper anddoes most of the heavy lifting for the proof of Theorem 1.2. The key is the approximation result of Corollary3.7, which, combined the fact that valuation rings are maximal subrings of a field with respect to the partialorder induced by domination of local rings, leads to a proof of Theorem 1.2.The paper is structured as follows. In Section 2 we fix our conventions for the paper. Section 3 examinesbase change properties along Henselizations. In Section 4 we collect some basic results and definitions aboutextensions of valuation rings, Henselian valuation rings and the ramification theory of extensions of valuations.Finally, we prove Theorem 1.2 in Section 5.
Acknowledgments:
The author thanks Dale Cutkosky for helpful conversations.2.
Conventions and basic terminology
All rings are commutative with a multiplicative identity. For a ring A , MaxSpec( A ) will denote the set ofmaximal ideals of A . Note that rings in this paper will rarely be noetherian.The term local ring will mean a ring ( R, m , κ ) with a unique maximal ideal m and residue field κ . Pleasenote that local rings are not necessarily noetherian. We will use R h to denote the Henselization of the localring R with respect to the maximal ideal m . Recall that R h is a faithfully flat local extension of R whosemaximal ideal is the expansion of the maximal ideal of R and whose residue field is isomorphic to the residuefield of R [Sta20, Tag 07QM]. We will say that a local ring ( B, m B ) dominates a local ring ( A, m A ) if A ⊂ B and m A = m B ∩ A . Recall that a valuation ring of a field K is a local subring of K that is maximal in the ollection of local subrings of K under the partial order induced by domination of local rings. Whenever wetalk about an extension of valuation rings V ⊂ W , we will always assume W dominates V .Valuation rings also arise from valuations, which are denoted additively in this paper. We assume the readeris familiar with valuations and valuation rings, and we will skip their definitions.Let B be an A -algebra. We say B is essentially of finite type over A if B is the localization of a finite type A -algebra. We say B is essentially of finite presentation over A if B is the localization of a finitely presented A -algebra.While a finitely presented algebra is always of finite type, the converse is not true in a non-noetherian setting.However, the two notions often coincide in the valuative setting because of the following result and the factthat any torsion-free module over a valuation ring is free [Bou98, Chap. VI, § . , Lem. 1]. Lemma 2.1. [RG71, Cor. (3.4.7)]
Let A be a domain and B be a finite type A -algebra. If B is A -flat, then B is a finitely presented A -algebra. Henselization and base change
In this section we establish some base change properties of integral maps along Henselizations. The results donot use any valuation theory. The non-valuative considerations of this section will provide the main ingredientsfor the proof of Theorem 1.2. We will frequently use the fact that the Henselization of a local ring is flat [Sta20,Tag 07QM], that flat maps satisfy the Going-Down property [Sta20, Tag 00HS] and that the property of beingan integral ring map is preserved under base change [Sta20, Tag 02JK].We first recall a characterization of Henselian local domains that will important for the results that follow.
Lemma 3.1.
Let ( R, m ) be a local domain. The following are equivalent: (1) R is Henselian. (2) For every integral extension
R ֒ → A , if A is a domain then A is a local ring.If the equivalent conditions hold, then any integral extension of R that is also a domain is Henselian.Indication of proof. The equivalence follows from [Nag75, Chap. VII, Thm. (43.12)]. The fact that integralextension domains of R are Henselian follows by [Nag75, Chap. VII, Cor. (43.13)]. Note that in Nagata’sterminology, an integral extension of a domain is automatically a domain [Nag75, Chap. I, Pg. 30]. (cid:3) The next result highlights a key base change property along Henselizations.
Lemma 3.2.
Let ( R, m ) be a local ring, ϕ : R → A be a ring map and i : R → R h be the canonical map from R to its Henselization R h . Suppose P ∈ Spec( A ) contracts to m ∈ Spec( R ) , that is, ϕ − ( P ) = m . Consider theinduced map id A ⊗ i : A → A ⊗ R R h . (1) The fiber of
Spec(id A ⊗ i ) : Spec( A ⊗ R R h ) → Spec( A ) over P is a singleton. (2) If Q is the unique prime ideal of A ⊗ R R h that contracts to P , then (( A ⊗ R R h ) Q ) h ∼ = ( A P ) h .Proof. Consider the commutative diagram
A A ⊗ R R h R R h . id A ⊗ iiϕ ϕ ⊗ id Rh (3.2.1)Since i is faithfully flat, so is id A ⊗ i . Therefore, Spec(id A ⊗ i ) : Spec( A ⊗ R R h ) → Spec( A ) is surjective.(1) Let κ ( P ) (resp. κ ( m ) ) denote the residue field of P (resp. m ). Then the fiber of Spec(id A ⊗ i ) over P canbe identified with Spec( κ ( P ) ⊗ A ( A ⊗ R R h )) . Now, κ ( P ) ⊗ A ( A ⊗ R R h ) ∼ = κ ( P ) ⊗ R R h ∼ = κ ( P ) ⊗ κ ( m ) ( κ ( m ) ⊗ R R h ) . he maximal ideal m h of R h is m R h and that the induced map of residue fields κ ( m ) → κ ( m h ) is an isomorphism[Sta20, Tag 07QM]. Therefore κ ( m ) ⊗ R R h = κ ( m h ) and κ ( P ) ⊗ κ ( m ) ( κ ( m ) ⊗ R R h ) ∼ = κ ( P ) , that is, Spec( κ ( P ) ⊗ A ( A ⊗ R R h )) is the spectrum of a field. This proves (1).(2) There exists a unique prime ideal Q of A ⊗ R R h that contracts to P by (1). By the commutativity of (3.2.1), Q contracts to m h along ϕ ⊗ id R h : R h → A ⊗ R R h because m h is the unique prime ideal of R h that contracts m in R . The rest of (2) now follows from [Sta20, Tag 08HU]. (cid:3) We now focus on the base change properties of integral ring maps along Henselizations. When we use theterm ‘integral ring map’, we do not necessarily mean an integral extension.
Lemma 3.3.
Let ( R, m ) be a local ring, ϕ : R → A be an integral ring map and i : R → R h be the canonicalmap from ( R, m ) to its Henselization ( R h , m h ) . Then the map Spec(id A ⊗ i ) : Spec( A ⊗ R R h ) → Spec( A ) induced by id A ⊗ i : A → A ⊗ R R h has the following properties: (1) For every maximal ideal M of A , the fiber of Spec(id A ⊗ i ) over M is a singleton. (2) Let Q ∈ Spec( A ⊗ R R h ) . The following are equivalent: (2a) Q is a maximal ideal of A ⊗ R R h . (2b) Q contracts to a maximal ideal of Spec( A ) . (2c) Q contracts to m h . (3) Spec(id A ⊗ i ) induces a bijection MaxSpec( A ⊗ R R h ) ←→ MaxSpec( A ) .Proof. Consider the commutative diagram
A A ⊗ R R h R R h . id A ⊗ iiϕ ϕ ⊗ id Rh Since ϕ is integral, so is ϕ ⊗ id R h . Moreover, Spec(id A ⊗ i ) : Spec( A ⊗ R R h ) → Spec( A ) is surjective because id A ⊗ i is faithfully flat by base change.(1) ϕ − ( M ) = m since ϕ is integral and M is maximal. Then (1) follows by part (1) of Lemma 3.2.(2) Suppose Q is a maximal ideal of A ⊗ R R h . Then Q contracts to m h in R h along ϕ ⊗ id R h because this mapis integral. Thus, (2a) ⇒ (2c).Suppose Q contracts to m h in R h , and hence to m in R . Then the contraction Q c of Q to A must be maximalbecause Q c contracts to m along the integral map ϕ , and only maximal ideals can contract to maximal idealsalong integral maps. This proves (2c) ⇒ (2b).Finally, suppose Q contracts to a maximal ideal M of A . Since ϕ − ( M ) = m , it follows by the commutativityof the above diagram that Q contracts to m h along the integral ring map ϕ ⊗ id R h . Then Q must be maximal,thereby establishing (2b) ⇒ (2a).(3) The equivalent statements of part (2) tell us that the inverse image of MaxSpec( A ) under Spec(id A ⊗ i ) isprecisely MaxSpec( A ⊗ R R h ) , and part (1) shows that the induced map MaxSpec( A ⊗ R R h ) → MaxSpec( A ) isboth injective and surjective. (cid:3) An analogue of Lemma 3.3 exists for minimal primes.
Lemma 3.4.
Let ( R, m ) be an integrally closed domain, and ϕ : R ֒ → A be an integral extension of domains.Then we have the following: (1) R h is an integrally closed domain and Frac( R ) ⊗ R R h = Frac( R h ) . (2) Let Q ∈ Spec( A ⊗ R R h ) . The following are equivalent: (2a) Q is a minimal prime of A ⊗ R R h . Q lies over (0) in A . (2c) Q lies over (0) in R h . (3) There is a bijection { minimal prime of A ⊗ R R h } ←→ Spec(Frac( A ) ⊗ Frac( R ) Frac( R h )) . (4) If Frac( A ) is a finite extension of Frac( R ) , then A ⊗ R R h has finitely many minimal primes.Proof. (1) That R h is an integrally closed domain is a well-known permanence property of Henselization; see[Sta20, Tag 06DI]. Since R h is a colimit of local étale extensions, Frac( R h ) is an algebraic extension of Frac( R ) .So Frac( R ) ⊗ R R h is a field because it contains Frac( R ) and is contained in Frac( R h ) . Since Frac( R ) ⊗ R R h isa localization of R h , we get Frac( R ) ⊗ R R h = Frac( R h ) .(2) Let i : R → R h be the canonical map. Consider the commutative diagram A A ⊗ R R h R R h . id A ⊗ iiϕ ϕ ⊗ id Rh Since id A ⊗ i : A → A ⊗ R R h is flat, a minimal prime of A ⊗ R R h must contract to the unique minimal prime (0) of A by Going-Down. This proves (2a) ⇒ (2b).Suppose Q contracts to (0) in A . Then Q must contract to (0) in R because ϕ is injective. Let p := ( ϕ ⊗ id R h ) − ( Q ) . By the commutativity of the above diagram, p contracts to (0) in R . But (1) shows the generic fiber of i is asingleton, consisting of the unique minimal prime (0) of R h . Consequently, p = (0) , proving (2b) ⇒ (2c).Assume (2c). If Q is not a minimal prime of A ⊗ R R h , then we can find Q ′ ∈ Spec( A ⊗ R R h ) such that Q ′ ( Q . Then ( ϕ ⊗ id R h ) − ( Q ′ ) = (0) , which is a contradiction because ϕ ⊗ id R h is an integral extension, and integralextensions have zero dimensional fibers [Sta20, Tag 00GT]. Thus, (2c) ⇒ (2a).(3) By part (2), the set of minimal primes of A ⊗ R R h is precisely the generic fiber of id A ⊗ i : A → A ⊗ R R h ,which is in bijection with Spec(Frac( A ) ⊗ A ( A ⊗ R R h )) . The assertion now follows because Frac( A ) ⊗ A ( A ⊗ R R h ) ∼ = Frac( A ) ⊗ Frac( R ) (Frac( R ) ⊗ R R h ) ∼ = Frac( A ) ⊗ Frac( R ) Frac( R h ) , where the last isomorphism is a consequence of part (1).(4) If Frac( A ) is a finite extension of Frac( R ) , then Frac( A ) ⊗ Frac( R ) Frac( R h ) is a finite Frac( R h ) -algebra.Consequently, Spec(Frac( A ) ⊗ Frac( R ) Frac( R h )) is finite set. We are then done by part (3). (cid:3) The next result is well-known. We include a proof for the reader’s convenience.
Lemma 3.5.
Let A be a ring such that for all maximal ideals m of A , A m is a domain. (1) If p and q are distinct minimal primes of A , then p + q = A . (2) If A has finitely many distinct minimal primes p , . . . , p n , then the canonical map A → A/ p ×· · ·× A/ p n is an isomorphism.Proof. (1) Suppose for contradiction that p + q ( A . Then there exists a maximal ideal m of A such that p + q ⊆ m . Now both p A m and q A m are distinct minimal prime ideals of the domain A m , which is impossible.(2) The hypothesis implies that A is reduced, that is, p ∩ · · · ∩ p n = (0) . Since p i + p j = A for i = j , the resultnow follows by the Chinese Remainder Theorem [Sta20, Tag 00DT]. (cid:3) Proposition 3.6.
Let ( R, m ) be a local domain that is integrally closed in its fraction field K . Let L be a finitefield extension of K and let A be the integral closure of R in L . We have the following: (1) A has finitely many maximal ideals, that is, A is semi-local. (2) A ⊗ R R h is a semi-local ring. (3) If M is a maximal ideal of A ⊗ R R h , then ( A ⊗ R R h ) M is an integrally closed domain. (4) A ⊗ R R h has finitely many minimal primes. Each maximal ideal M of A ⊗ R R h contains a unique minimal prime p , and conversely, M is the uniquemaximal ideal that contains p . Moreover, the canonical map A ⊗ R R h → ( A ⊗ R R h ) M has kernel p andinduces an isomorphism A ⊗ R R h p ∼ = ( A ⊗ R R h ) M . Consequently, ( A ⊗ R R h ) M is Henselian. (6) If m , . . . , m n are the maximal ideals of A (assumed to be distinct), then A ⊗ R R h ∼ = ( A m ) h × · · · × ( A m n ) h . Moreover, if M i is the unique prime ideal of A ⊗ R R h that contracts to m i , then M i is maximal and ( A ⊗ R R h ) M i ∼ = ( A m i ) h . (7) The sets
MaxSpec( A ) , MaxSpec( A ⊗ R R h ) , Spec( L ⊗ K Frac( R h )) and { minimal prime of A ⊗ R R h } havethe same cardinality.Proof. (1) The integral closure of an integrally closed domain in a finite extension of its fraction field has finitefibers [Bou98, Chap. V, § . , Cor. 2]. Since MaxSpec( A ) is the closed fiber of R ⊂ A , it is finite.(2) MaxSpec( A ⊗ R R h ) is in bijection with the finite set MaxSpec( A ) by Lemma 3.3.(3) M contracts to a maximal ideal P in A by Lemma 3.3, and so, P contracts to m in R . Since A P is integrallyclosed, ( A P ) h , is also an integrally closed domain [Sta20, Tag 06DI]. By Lemma 3.2, ( A P ) h ∼ = (( A ⊗ R R h ) M ) h . Since ( A ⊗ R R h ) M → (( A ⊗ R R h ) M ) h is faithfully flat, descent of integral closedness [Sta20, Tag 033G] implies ( A ⊗ R R h ) M is an integrally closed domain.(4) This follows from part (4) of Lemma 3.4 because L is the fraction field of A .(5) A maximal ideal M ∈ A ⊗ R R h contains a unique minimal prime because ( A ⊗ R R h ) M is a domain by part(3). Let p , . . . , p k be the minimal primes of A ⊗ R R h . Then part (3) and Lemma 3.5 imply that A ⊗ R R h ∼ = ( A ⊗ R R h ) / p × · · · × ( A ⊗ R R h ) / p k . (3.6.1)Lemma 3.4 shows that a minimal prime of A ⊗ R R h contracts to (0) in R h . Thus, the composition R h ϕ ⊗ id Rh −−−−−→ A ⊗ R R h ։ ( A ⊗ R R h ) / p i is an integral extension for all i = 1 , . . . , k , and so, each ( A ⊗ R R h ) / p i is a Henselian local domain by Lemma3.1. Hence each minimal prime p i is contained in a unique maximal ideal, say M i . Since ( A ⊗ R R h ) M i is adomain by part (3), the kernel of A ⊗ R R h → ( A ⊗ R R h ) M i has to be p i . Moreover, ( A ⊗ R R h ) / p i is local withmaximal ideal M i / p i , so the induced injection ( A ⊗ R R h ) / p i ֒ → ( A ⊗ R R h ) M i is an isomorphism. In particular, ( A ⊗ R R h ) M i is a Henselian local domain.(6) The uniqueness and maximality of M i follow from Lemma 3.3, as does the fact that MaxSpec( A ⊗ R R h ) = { M , . . . , M n } . The decomposition (3 . . and part (5) then show that A ⊗ R R h ∼ = ( A ⊗ R R h ) M × · · · × ( A ⊗ R R h ) M n , and that each ( A ⊗ R R h ) M i is Henselian. Thus, (( A ⊗ R R h ) M i ) h ∼ = ( A ⊗ R R h ) M i . On the other hand, Lemma3.2 implies that (( A ⊗ R R h ) M i ) h ∼ = ( A m i ) h . Thus, A ⊗ R R h ∼ = ( A m ) h × · · · × ( A m n ) h . (7) All the sets are finite because of parts (1), (4) and the bijections of Lemma 3.3 and Lemma 3.4. It remainsto check that | MaxSpec( A ⊗ R R h ) | = |{ minimal prime of A ⊗ R R h }| . This follows by part (5). (cid:3) Corollary 3.7.
Let ( R, m ) be a local domain that is integrally closed in its fraction field K . Let L be a finitefield extension of K and A be the integral closure of R in L . Suppose MaxSpec( A ) = { m , . . . , m n } (the maximalideals are assumed to be distinct). (1) Let Σ be the collection of finite (equivalently, finitely generated) R -subalgebras B of A such that Frac( B ) =Frac( A ) = L and m i ∩ B = m j ∩ B , for i = j . Then Σ is filtered under inclusion and A = colim B ∈ Σ B. Let M ∈ MaxSpec( A ⊗ R R h ) and B ∈ Σ . If M B is the contraction of M to the subring B ⊗ R R h of A ⊗ R R h , then the induced map on local rings ( B ⊗ R R h ) M B → ( A ⊗ R R h ) M is injective, ( B ⊗ R R h ) M B is a Henselian domain, and ( A ⊗ R R h ) M = colim B ∈ Σ ( B ⊗ R R h ) M B . Proof. (1) Since every element of Σ is integral over R , finitely generated is equivalent to being module finite asan R -algebra. Note that Σ is non-empty. Indeed, since Frac( A ) = L is a finite extension of K , one can choosea K -basis of L consisting of elements b , . . . , b m ∈ A . By prime avoidance, for all i = 1 , . . . , n , choose a i ∈ m i such that a i is not contained in any of the other maximal ideals of A (here we need that A is semi-local). Thenby construction, the R -subalgebra R [ a , . . . , a n , b , . . . , b m ] of A is an element of Σ .If B ∈ Σ , then any finitely generated B -subalgebra C of A is also in Σ . Therefore if B , B ∈ Σ , then so is B [ B ] , that is, Σ is filtered under inclusion. Since A is the filtered union of finitely generated B -subalgebrasfor any B ∈ Σ and Σ = ∅ , we have A = colim B ∈ Σ B .(2) Fix B ∈ Σ . Since B ֒ → A is an integral extension, each m i ∩ B is a maximal ideal of B . Furthermore, sinceevery maximal ideal of B is contracted from a maximal ideal of A , we get MaxSpec( B ) := { m ∩ B, . . . , m n ∩ B } . Let M i ∈ Spec( A ⊗ R R h ) be the unique prime ideal that contracts to m i . Then Lemma 3.3 shows that MaxSpec( A ⊗ R R h ) = { M , . . . , M n } . If ( M i ) B is the contraction of M i to B ⊗ R R h , then using the integrality of the extension B ⊗ R R h ֒ → A ⊗ R R h we conclude that MaxSpec( B ⊗ R R h ) = { ( M ) B , . . . , ( M n ) B } . The defining property of Σ implies that for i = j , m i ∩ B = m j ∩ B . As ( M i ) B lies over m i ∩ B by thecommutativity of the diagram A A ⊗ R R h B B ⊗ R R h , we have ( M i ) B = ( M j ) B for i = j . In other words, B ⊗ R R h consists of n distinct maximal ideals ( M i ) B for i = 1 , . . . , n , and M i is the unique prime ideal of A ⊗ R R h that contracts to ( M i ) B .Since B is a finite extension of R , B ⊗ R R h is a finite extension of R h . The decomposition of finite extensionsof Henselian local domains [Sta20, Tag 04GH] gives us that B ⊗ R R h ∼ = ( B ⊗ R R h ) ( M ) B × · · · × ( B ⊗ R R h ) ( M n ) B , (3.7.1)and that each ( B ⊗ R R h ) ( M i ) B is a Henselian local ring. Moreover, B ⊗ R R h is a subring of A ⊗ R R h ,which is reduced because it decomposes as a finite product of domains by part (6) of Proposition 3.6. Thus, ( B ⊗ R R h ) ( M i ) B is reduced for all i .Applying part (3) of Lemma 3.4 to the integral extension R ֒ → B we see that the number of minimal primesof B ⊗ R R h equals the cardinality of Spec( L ⊗ K Frac( R h )) . But | Spec( L ⊗ K Frac( R h )) | = | MaxSpec( A ) | = n by part (7) of Proposition 3.6 and the fact that Frac( A ) = Frac( B ) = L . Consequently, each factor in thedecomposition (3.7.1) has exactly one minimal prime. Combined with reducedness, it follows that each ( B ⊗ R R h ) ( M i ) B is a domain.In particular, ( M i ) B contains a unique minimal prime (which expands to the zero ideal in ( B ⊗ R R h ) ( M i ) B ).Using part (5) of Proposition 3.6, if P i is the unique minimal prime of A ⊗ R R h contained in M i , then ( P i ) B := P i ∩ ( B ⊗ R R h ) ust be the unique minimal prime of B ⊗ R R h contained in ( M i ) B . Indeed, by part (2) of Lemma 3.4, P i contracts to (0) in R h . Thus ( P i ) B also contracts to (0) in R h . Applying part (2) of Lemma 3.4 again, but thistime to the integral extension R ֒ → B , then shows that ( P i ) B is a minimal prime of B ⊗ R R h .Using the commutative diagram B ⊗ R R h A ⊗ R R h ( B ⊗ R R h ) ( M i ) B ( A ⊗ R R h ) M i , it follows that P i ( A ⊗ R R h ) M i must contract to ( P i ) B ( B ⊗ R R h ) ( M i ) B in ( B ⊗ R R h ) ( M i ) B . Since ( A ⊗ R R h ) M i and ( B ⊗ R R h ) ( M i ) B are both domains (the former ring is a domain by Proposition 3.6), we have P i ( A ⊗ R R h ) M i = (0) and ( P i ) B ( B ⊗ R R h ) ( M i ) B = (0) . Thus, the bottom horizontal arrow is injective.A maximal ideal M of A ⊗ R R h coincides with some M i . Therefore the argument above shows that for any B ∈ Σ , ( B ⊗ R R h ) M B is a Henselian local domain and the induced local map ( B ⊗ R R h ) M B → ( A ⊗ R R h ) M is injective. As tensor product commutes with filtered colimits, we have A ⊗ R R h = colim B ∈ Σ B ⊗ R R h by part(1), and so, ( A ⊗ R R h ) M = colim B ∈ Σ ( B ⊗ R R h ) M B . In this case the filtered colimit is actually a filtered unionbecause ( B ⊗ R R h ) M B is a subring of ( A ⊗ R R h ) M , for all B ∈ Σ . (cid:3) Henselian valuation rings and some ramification theory
This section discusses some background from the ramification theory of extensions of valuations relevant toConjecture 1.1. We first recall how extensions of valuation rings arise in algebraic field extensions.
Proposition 4.1.
Let
L/K be an algebraic extension of fields. Let V be a valuation ring of K and A be theintegral closure of V in L . Then localization m A m induces a bijection MaxSpec( A ) ←→ { valuation rings of L that dominate V } . In particular, if
L/K is finite, then there are finitely many valuation rings of L that dominate V .Indication of proof. For the bijection see [Bou98, Chap. VI, § . , Prop. 6]. If L/K is finite, then A has finitelymany maximal ideals by part (1) of Proposition 3.6. That there are finitely many valuation rings of L thatdominate V now follows from the bijection of this Proposition. (cid:3) As a consequence of the Henselian property, one can now deduce:
Corollary 4.2.
Let V be a valuation ring of a field K . The following are equivalent: (1) V is Henselian. (2) If L is an algebraic extension of K and A is the integral closure of V in L , then A is the unique valuationring of L that dominates V .Proof. (1) ⇒ (2) By Lemma 3.1, A must be a local ring. By the bijection of Proposition 4.1 it follows that A must be the unique valuation ring of L that dominates V .Conversely, assume (2). Let B be a domain that is an integral extension of V . By Lemma 3.1 again, itsuffices to show that B is local. Let L = Frac( B ) . Then L/K is algebraic. If A is the integral closure of V in L , then B ⊂ A is integral. Since A is local by (2), Lying Over and the fact that only maximal ideals of A cancontract to maximal ideals of B imply that B must also be local. (cid:3) Remark 4.3.
In terms of valuations, Corollary 4.2 can be reinterpreted as saying that if ν is a valuation of afield K , then the valuation ring of ν is Henselian if and only if for every algebraic extension L/K , there existsa unique valuation ω of L (up to equivalence) that extends ν . he Henselization of a valuation ring admits a purely valuation theoretic description. However, for thepurposes of this paper, it is more helpful to think of Henselizations as filtered colimits of local étale extensionsthat induce isomorphisms on residue fields. One then has the following result. Lemma 4.4.
Let ν be a valuation of a field K with valuation ring O ν and value group Γ ν . Then the Henseliza-tion O hν of O ν is a valuation ring whose associated valuation ν h also has value group Γ ν .Proof. See [Sta20, Tag 0ASK]. The main points are that local étale extensions of valuation rings are valuationrings and a filtered colimit of valuation rings is a valuation ring. (cid:3)
Notation 4.5.
The fraction field of O hν will be denoted by K h . Thus, ν h is a valuation of K h whose valuationring is O hν .We record a descent result that we will need in the proof of Theorem 1.2. Lemma 4.6.
Let ϕ : V → W be a ring map and W be a valuation ring. The following are equivalent: (1) V is a valuation ring and ϕ is an injective local map. (2) ϕ is faithfully flat. (3) ϕ is cyclically pure, that is, for all ideals I of V , the induced map V /I → W/IW is injective.Proof.
Assume (1). If ϕ is injective, then W , being a domain, is a torsion-free V -module, hence flat [Bou98,Chap. VI, § . , Lem. 1]. Since ϕ is local, ϕ is faithfully flat. Thus, (1) ⇒ (2). Furthermore, (2) ⇒ (3) is aproperty of faithfully flat maps; see [Bou98, Chap. I, § . , Prop. 9].Assume (3). Taking I = (0) , we see that ϕ is injective. Thus, V is a domain because W is. To show that V is a valuation ring, it is enough to show that for all x, y ∈ V , xV ⊆ yV or yV ⊆ xV . Since W is a valuationring, we must have xW ⊆ yW or yW ⊆ xW . Cyclic purity of ϕ implies that ϕ − ( IW ) = I , for any ideal I of V . Thus, if xW ⊆ yW , then xV = ϕ − ( xW ) ⊆ ϕ − ( yW ) = yV . Similarly, yV ⊆ xV if yW ⊆ xW . Finally, ϕ is local because if m V is the maximal ideal of the valuation ring V , then injectivity of V / m V → W/ m V W shows m V W = W . (cid:3) Conjecture 1.1 relates essential finite generation of extensions of valuation rings to fundamental invariantsfrom the ramification theory of extensions of valuations. We now briefly introduce these invariants. Let
L/K be a field extension, ω be a valuation of L with value group Γ ω and ν be its restriction to K with value group Γ ν . Inclusion induces a local homomorphism of the corresponding valuation rings ( O ν , m ν , κ ν ) ֒ → ( O ω , m ω , κ ω ) . Note that Γ ν is a subgroup of Γ ω and κ ν is a subfield of κ ω .Fix a finite field extenion L/K . One has the fundamental inequality [Bou98, Chap. VI, § . , Lem. 2] [Γ ω : Γ ν ][ κ ω : κ ν ] ≤ [ L : K ] . (4.6.1)In particular, [Γ ω : Γ ν ] and [ κ ω : κ ν ] are finite invariants of ω/ν . This leads to the following definition. Definition 4.7.
Suppose
L/K is a finite extension and consider the extension of valuations ω/ν .(a) The ramification index of ω/ν , denoted e ( ω | ν ) , is [Γ ω : Γ ν ] .(b) The inertia index of ω/ν , denoted f ( ω | ν ) , is [ κ ω : κ ν ] .(c) The initial index of ω/ν , denoted ǫ ( ω | ν ) , is the cardinality of the set { x ∈ Γ ω, ≥ : x < Γ ν,> } .The finiteness of the initial index follows from the inequality ǫ ( ω | ν ) ≤ e ( ω | ν ) , which holds because if x, y ∈ Γ ω, ≥ are distinct elements such that x, y < Γ ν,> , then x + Γ ν = y + Γ ν in Γ ω / Γ ν .Indeed, assume without loss of generality that ≤ x < y . Then y − x ∈ Γ ω,> and y − x ≤ y < Γ ν,> , that is, y − x / ∈ Γ ν .By Lemma 4.4, if ω/ν is an extension of valuations, then for the extension of Henselizations ω h /ν h , we have e ( ω | ν ) = e ( ω h | ν h ) , f ( ω | ν ) = f ( ω h | ν h ) and ǫ ( ω | ν ) = ǫ ( ω h | ν h ) because Henselizations do not alter value groups nd residue fields. In addition, one can use the isomorphism of part (6) of Proposition 3.6 and Proposition 4.1to conclude that L ⊗ K K h is a finite product of fields, one of which coincides with L h , the fraction field of O hω .Thus, [ L h : K h ] ≤ [ L : K ] < ∞ . (4.7.1)Using these observations we introduce the notion of the defect of ω/ν . which measures to what extent equalityfails in (4.6.1), at least when ω is the unique extension of ν to L . Definition 4.8.
Let
L/K be a finite field extension and ν be a valuation of K . If ω is the unique extension of ν to L , then the defect of ω/ν , denoted d ( ω | ν ) , is defined to be d ( ω | ν ) = [ L : K ] e ( ω | ν ) f ( ω | ν ) . If the extension of valuations ω/ν is not necessarily unique, the defect of ω/ν is defined to be the defect of theextension of Henselizations ω h /ν h , that is, d ( ω | ν ) = [ L h : K h ] e ( ω | ν ) f ( ω | ν ) . We say ω/ν is defectless if d ( ω | ν ) = 1 , that is, if [ L h : K h ] = e ( ω | ν ) f ( ω | ν ) . Remark 4.9. (1) If
L/K is a finite extension, then ω h is the unique extension of ν h to L h by Corollary 4.2 and (4.7.1).Thus, the definition of the defect of an extension of valuations that is not necessarily unique in termsof the defect of the extension of henselizations makes sense.(2) If ω is the unique extension ν to L , then d ( ω | ν ) = d ( ω h | ν h ) . Thus, the two notions of defect areconsistent for unique extensions of valuations. The only thing we need to check is that [ L h : K h ] = [ L : K ] . By Proposition 4.1 and uniqueness of the extension ω/ν , the integral closure of O ν in L must be O ω . Then by part (6) of Proposition 3.6 applied to R = O ν and A = O ω , we get O ω ⊗ O ν O hν ∼ = O hω . Consequently, L ⊗ K K h ∼ = L h , and so, [ L h : K h ] = [ L : K ] .(3) If κ ν has characteristic , then we always have d ( ω | ν ) = 1 , and if κ ν has characteristic p > , then d ( ω | ν ) = p n , for some integer n ≥ [Kuh11, Pg. 280–281]. Thus, the notion of defect is only interestingin residue characteristic p > , that is, when O ν has prime or mixed characteristics. Furthermore, d ( ω | ν ) is always a positive integer.(4) Rephrased in terms of defect, Conjecture 1.1 asserts that O ω is essentially of finite type over O ν if andonly if ω/ν is defectless and e ( ω | ν ) = ǫ ( ω | ν ) . Example 4.10.
Let K be a field of characteristic p > for which [ K : K p ] < ∞ and ν be a valuation of K . If ν p denotes the restriction of ν to the subfield K p of K , then using pure inseparability of the extension K/K p one can verify that ν is the unique extension of ν p to K . Then d ( ν | ν p ) = [ K : K p ][Γ ν : p Γ ν ][ κ ν : κ pν ] . The defect of ν/ν p controls interesting properties of ν . For example, it is shown in [DS16, Proof of Thm. 5.1](see also [Dat18, Cor. IV.23]) that if K is a function field of a variety over a ground field k , and ν is a valuationof K/k , then ν/ν p is defectless if and only if ν is an Abhyankar valuation of K/k .5.
Proof of Theorem 1.2
We recall the statement of Theorem 1.2 for the reader’s convenience.
Theorem 1.2.
Let
L/K be a finite field extension. Suppose ω is a valuation of L with valuation ring ( O ω , m ω , κ ω ) and value group Γ ω . Let ν be the restriction of ω to K with valuation ring ( O ν , m ν , κ ν ) andvalue group Γ ν . Suppose the Henselization of O ω is O hω with fraction field L h and that of O ν is O hν with fractionfield K h . The following are equivalent. (i) O ω is essentially of finite presentation over O ν . (ii) O ω is essentially of finite type over O ν . iii) [ L h : K h ] = [Γ ω : Γ ν ][ κ ω : κ ν ] and ǫ ( ω/ν ) = [Γ ω : Γ ν ] . (iv) O hω is essentially of finite type over O hν . (v) O hω is a module finite O hν -algebra.Proof. (i) ⇒ (ii) is clear, and (ii) ⇒ (iii) was established by Knaf [CN19, Thm. 4.1] (see also Remark 5.1).Since ω h is the unique extension of ν h to L h by Corollary 4.2, the implication (iii) ⇒ (v) follows by [CN19,Cor. 2.2] because O hω is the integral closure of O hν in L h . Furthermore, clearly (v) ⇒ (iv).It remains to show that (v) ⇒ (i). Let A be the integral closure of O ν in L . By Proposition 4.1, let m bethe unique maximal ideal of A such that O ω = A m . Let Σ be the collection of finite O ν -subalgebras B of A as in Corollary 3.7, and we let m B := the maximal ideal m ∩ B of B .Then for all B ∈ Σ , A m dominates B m B and B m B dominates O ν .By Lemma 3.3, let M := the unique prime (equivalently, maximal) ideal of A ⊗ O ν O hν that contracts to m . For all B ∈ Σ , let M B denote the contraction of M to the O hν -subalgebra B ⊗ O ν O hν of A ⊗ O ν O hν (it is asubalgebra by flatness of O hν ). Then by the commutativity of the diagram A A ⊗ O ν O hν B B ⊗ O ν O hν , and Lemma 3.3 again, M B is the unique prime (equivalently, maximal) ideal of B ⊗ O ν O hν that contracts to themaximal ideal m B of B .By Corollary 3.7, for all B ∈ Σ , ( B ⊗ O ν O hν ) M B is a O hν -subalgebra of ( A ⊗ O ν O hν ) M , and ( A ⊗ O ν O hν ) M = colim B ∈ Σ ( B ⊗ O ν O hν ) M B . Note that the filtered colimit is a filtered union. By part (6) of Proposition 3.6, we have that ( A ⊗ O ν O hν ) M ∼ = ( A m ) h = O hω . Since O hω is a module-finite O hν -algebra by the hypothesis of (v), we can find B ∈ Σ such that ( A ⊗ O ν O hν ) M = ( B ⊗ O ν O hν ) M B . Therefore ( B ⊗ O ν O hν ) M B is a Henselian valuation ring, and by part (2) of Lemma 3.2, we conclude ( B m B ) h ∼ = (( B ⊗ O ν O hν ) M B ) h = ( B ⊗ O ν O hν ) M B . In other words, ( B m B ) h is a valuation ring, so by descent (Lemma 4.6), B m B is a valuation ring as well. By thedefinition of the collection Σ , we have Frac( B m B ) = Frac( B ) = L, that is, B m B is a valuation ring of L . Since O ω = A m is also a valuation ring of L that dominates B m B , wemust have O ω = B m B because valuation rings are maximal with respect to domination of local rings. Thus, O ω is the localization ofthe finite O ν -algebra B . But B is O ν -flat since it is a torsion-free O ν -module [Bou98, Chap. VI, § . , Lem. 1].Therefore, B is a finitely presented O ν -algebra by Lemma 2.1. This completes the proof of (v) ⇒ (i), hencealso of the Theorem. (cid:3) he proof of Theorem 1.2 establishes the stronger result that if conditions (1) and (2) of Conjecture 1.1 hold,then O ω is the localization of a finite O ν -algebra B contained in the integral closure of O ν in L . Since a finitelygenerated torsion-free module over a valuation ring is free, B is a free O ν -module of finite rank. Remark 5.1.
One can prove (i) ⇒ (v) (or, (ii) ⇒ (v)) using Zariski’s Main Theorem. Suppose O ω is thelocalization of a finite type O ν -algebra B at a prime ideal p . Then [ κ ( p ) : κ ν ] = [ κ ω : κ ν ] ≤ [ L : K ] < ∞ , where the first inequality follows from (4.6.1). Moreover, dim( B p / m ν B p ) = dim( O ω / m ν O ω ) = 0 because m ω is the only prime ideal of O ω that contracts to m ν (if not, a non-maximal prime of O ω that contractsto m ν will give a non-maximal prime of the integral closure A of O ν in L that contracts to the maximal ideal m ν ).Thus, B is quasi-finite at p by part (6) of [Sta20, Tag 00PK]. Then Zariski’s Main Theorem [Sta20, Tag 00QB]implies that there exists a finite O ν -subalgebra B ′ of B such that O ω is a localization of B ′ at a maximalideal q of B ′ ( q is maximal because it contracts to m ν ). By Lemma 3.2, O hω = ( B ′ ) h q is the Henselization of ( B ′ ⊗ O ν O hν ) Q , where Q is the unique prime ideal of B ′ ⊗ O ν O hν that contracts to q . But q is maximal, so Q is amaximal ideal as well by Lemma 3.3. Since B ′ ⊗ O ν O hν is a finite O hν -algebra, it decomposes as a finite productof finite Henselian local rings [Sta20, Tag 04GG]. Then ( B ′ ⊗ O ν O hν ) Q must coincide with one these local factors,that is, ( B ′ ⊗ O ν O hν ) Q is Henselian and finite. Consequently, O hω ∼ = (( B ′ ⊗ O ν O hν ) Q ) h = ( B ′ ⊗ O ν O hν ) Q is a finite O hν -algebra, proving (v). Once we know that O hω is a module-finite O hν algebra, part (iii) of Theorem 1.2 nowfollows by [Bou98, Chap. VI, § . , Thm. 2]. This gives a different proof of [CN19, Thm. 4.1]. References [Bou98] N. Bourbaki,
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