aa r X i v : . [ m a t h . N T ] F e b A FUNDAMENTAL IDENTITY FOR EXTENSIONS OFDEDEKIND DOMAINS
CHIA-FU YU
Abstract.
The well-known fundamental identity in number theory expressesthe degree of an extension of global fields in terms of local information. Inthis article we modify the fundamental identity so that it holds for arbitraryDedekind domains. We further give necessary and sufficient conditions for theground Dedekind domain such that the fundamental identity holds true. Inparticular, we show that a Dedekind domain is excellent if and only if it isJapanese. Introduction
Let A be an S -ring of integers of a global field K , where S is a nonempty finiteset of places of K containing all Archimedean ones. The well-known fundamentalidentity in number theory states that for any finite field extension L/K and anynonzero prime ideal p of A , one has(1.1) r X i =1 e i f i = [ L : K ] , where e i and f i are the ramification index and residue class degree of the primeideals P i lying over p (for 1 ≤ i ≤ r ), respectively. If A is an arbitrary Dedekinddomain, then the fundamental identity is no longer true but instead one has thefundamental inequality P i e i f i ≤ [ L : K ] in general. It is well known that whenthe integral closure B of A is L is a finitely generated A -module, the equality holds;on the other hand the strict inequality can occur; see [19].One can rephrase the statement in terms of valuation theory cf. [14, Chap.II, Proposition 8.5, p. 165]. Under this setting, the fundamental inequality wasgeneralized to extensions of an arbitrary valuation v of a field K by Cohen andZariski [5]. In [18] Schmidt constructed a valuation v of a field K such that thevaluation ring A is not Japanese, that is, there is a finite field extension L of K so that the integral closure B of A is not finitely generated as an A -module.Furthermore, one can also show that Schmidt’s example realizes the strict inequalityin the fundamental inequality in terms of valuaiton theory, cf. [5].In this paper we would like to modify the fundamental identity so that it holdsfor arbitrary Dedekind domains. Furthermore, we also would like to find a necessaryand sufficient condition for A such that the original fundamental identity holds truefor any finite extension L/K . The main results of this paper are as follows.
Date : February 23, 2021.2010
Mathematics Subject Classification.
Primary: 11S15; Secondary: 12J20, 13F05.
Key words and phrases. ramification of extensions, excellent Dedekind domains, valuations.
Theorem 1.
Let A be a Dedekind domain with quotient field K , L a finite extensionfield of K with integral closure B of A . For any nonzero prime ideal p of A onehas the equality (1.2) X P | p e P f P = [( L ⊗ K K ∗ p ) ss : K ∗ p ] , where K ∗ p denotes the completion of K at p and ( L ⊗ K K ∗ p ) ss denotes the maximalsemi-simple quotient of L ⊗ K K ∗ p . Theorem 2.
Let A be a Dedekind domain with quotient field K . Then the followingstatements are equivalent. (1) For any finite extension
L/K and any nonzero prime ideal p of A , one has P P | p e P f P = [ L : K ] . (2) For any finite extension
L/K and any nonzero prime ideal p of A , thetensor product L ⊗ K K ∗ p is a semi-simple ring, or equivalently, L ⊗ K K ∗ p isa semi-simple K ∗ p -algebra. (3) A is excellent. (4) A is quasi-excellent. (5) A is a Nagata ring. (6) A is universally Japanese. (7) A is a Japanese ring. We shall recall (universally) Japanese rings, Nagata rings and (quasi-)excellentrings, and some of their properties ; see [11] for details. In particular, the followingimplications hold:(excellent rings) = ⇒ (quasi-excellent rings) = ⇒ (Nagata rings) ⇐⇒ (Noetherian universally Japanese) = ⇒ (Japanese rings) . If A is a Dedekind domain, then the above classes of rings are all equivalent.This paper is organized as follows. In Section 2 we recall several fundamentalrings mentioned above and some of their properties. In Section 3, we introducesome of valuation theory. The proofs of Theorems 1 and 2 are given in Section 4.2. Japanese, Nagata, (quasi-)excellent and G -rings In this section we recall the definition of several fundamental rings in commu-tative algebra. We will also introduce some of their properties and relationshipsamong them. Our references are Matsumura [11], and EGA IV [7, 8]. All rings andalgebras in this section are commutative with identity.2.1.
Japanese, universally Japanese and Nagata rings.Definition 3.
Let A be an integral domain with quotient field K .(1) We say that A is N-1 if the integral closure A ′ of A in its quotient field K is a finite A -module.(2) We say that A is N-2 if for any finite extension field L over K , the integralclosure A L of A in L is a finite A -module.Clearly any integrally closed domain is N-1, and any N-2 ring is N-1. If A is N-1(resp. N-2), then so is any localization of A as the construction of normalizationcommutes with localization. If A is a Noetherian domain of characteristic zero, UNDAMENTAL IDENTITY 3 then A is N-2 if and only if A is N-1. This follows immediately from a well-knowntheorem that if L is a finite separable field extension of K and A is a Noetherian normal domain, then the integral closure A L of A in L is finite over A (cf. [11,Proposition 31.B, p. 232]).The first non-N-1 one-dimensional integral domain was constructed by Akizuki[1], cf. [16]. K. Schmidt [18] and respectively Nagata (Appendix: Examples of badNoetherian rings of [13]) constructed different Dedekind domains which are notN-2. Definition 4.
A ring A is said to be Nagata if(1) A is Noetherian, and(2) A/ p is N-2 for any prime ideal p of A .A Nagata ring is not required to be an integral domain; A is Nagata if and only ifso is its associated reduced ring A red . If A is Nagata, then any localization of A andany finite A -algebra (finite as an A -module) are all Nagata. Nagata uses the term“pseudo-geometric rings” as the coordinate rings of algebraic varieties over any fieldall share this property. Nagata rings are the same as what are called Noetherianuniversally Japanese rings in EGA IV [7, 23.1.1, p. 213], which we recall now. Definition 5. (1) An integral domain A is said to be Japanese if it is N-2.(2) A ring A is said to be Japanese if for any minimal prime ideal p of A , thequotient domain A/ p is N-2.(3) A ring A is said to be universally Japanese if any finitely generated integraldomain over A is Japanese.From the definition, a universally Japanese ring or Japanese ring is not requiredto be an integral domain nor to be Noetherian, neither. Clearly, a universallyJapanese ring is Japanese. It follows from the definition that any Noetherian uni-versally Japanese ring is Nagata. Conversely, the following theorem [11, Theorem72, p. 240], due to Nagata, shows that any Nagata ring is also a Noetherian uni-versally Japanese ring. Theorem 6. If A is a Nagata ring, then so is any finitely generated A -algebra. The proof of this theorem is quite involved; the reader to referred to Matsumura[11, § Lemma 7.
If the localization A m is Japanese for all maximal ideal m ∈ Max( A ) of A , then A is Japanese. Proof.
Replacing A by A/ p for each minimal prime p , we may assume that A isa Japanese domain. Let L/K be a finite field extension with integral closure B of A in L . Our assumption implies that B m is finite over A m for every m ∈ Max( A ). Foreach m ∈ Max( A ), there is an element f m ∈ A and a finite subset S m of B such that m ∈ D ( f m ) = Spec A [1 /f m ] and S m generates B [1 /f m ] over A [1 /f m ]. Since the opensubsets D ( f m ) for m ∈ Max( A ) form an open covering of Spec A , there is a finitesubcovering D ( f m i ) for i = 1 , . . . , m . Take S to be the union of the finite subsets S m i . We show that S generates B over A . Let b ∈ B be an element. Then foreach i , there is a positive integer n i such that f n i m i b lies in the A -submodule h S m i i A generated by S m i . Since { D ( f n i m i ) } is an open covering, there are elements a i ∈ A CHIA-FU YU such that 1 = P mi =1 a i f n i m i . It follows that b = P i a i ( f n i m i b ) lies in the A -submodulegenerated by S .For any scheme X , let Nor( X ) denote the subset of X that consists of normalpoints. Lemma 8.
Let A be a Noetherian domain and X := Spec A . (1) If there is a nonzero element f ∈ A such that A f := A [1 /f ] is normal, then Nor( X ) is open in X . (2) If A is N-1, then Nor( X ) is open in X . Proof.
See [20, Lemma 2.6]; also see Matsumura [11, § Definition 9.
A Noetherian local (resp. semi-local) ring A is said to be analyticallyunramified if its A ∗ completion is reduced, and analytically ramified otherwise. Theorem 10. ( [11, Theorem 70, p. 236] ) Any Nagata semi-local domain is ana-lytically unramified.
Theorem 10 is useful to show a ring to be non-Nagata. In [11, Sec. 34.B, p. 260]Matsumura presents the following example. Let k be a field of characteristic p > k : k p ] = ∞ . Let A be the subring of k [[ t ]] consisting of power series P ∞ n =0 a n t n satisfying [ k p ( a , a , . . . ) : k p ] < ∞ . One can easily show that forevery c ∈ k [[ t ]] r A , the simple integral extension A [ c ] is analytically ramified;see [20, Example 2.22]. Therefore, both A [ c ] and A are not Nagata according toTheorem 10.2.2. G-rings and closedness of singular loci.
Recall that a Noetherian localring ( A, m , k ) is said to be a regular local ring if dim A = dim k m / m [11, p. 78]. ANoetherian ring A is said to be regular if all local rings A p are regular for p ∈ Spec A .One can show that if the local ring A m is regular for all maximal ideals m of A ,then A is regular [11, § Definition 11. ([11, §
33, p. 249]).(1) Let A be a Noetherian ring containing a field k . We say that A is geo-metrically regular over k if for any finite field extension k ′ over k , the ring A ⊗ k k ′ is regular. This is equivalent to saying that the local ring A m hasthe same property for all maximal ideals m of A .(2) Let φ : A → B be a homomorphism (not necessarily of finite type) ofNoetherian rings. We say that φ is regular if it is flat and for each p ∈ Spec A , the fiber ring B ⊗ A k ( p ) is geometrically regular over the residuefield k ( p ).(3) A Noetherian ring A is said to be a G-ring if for each p ∈ Spec A , the naturalmap φ p : A p → ( A p ) ∗ is regular, where ( A p ) ∗ denotes the completion of thelocal ring A p .Note that the natural map φ p : A p → ( A p ) ∗ is faithfully flat. The fibers ofthe natural morphism Spec ( A p ) ∗ → Spec A p are called formal fibers . To say aNoetherian ring A is a G -ring then is equivalent to saying that all formal fibers ofthe canonical map φ p for each prime ideal p of A are geometrically regular. It is UNDAMENTAL IDENTITY 5 clear that, if A is a G-ring, then any localization S − A of A and any homomorphismimage A/I of A are G-rings. Lemma 12.
Let
K/k be any field extension. Then K is geometrically regular over k if and only if K/k is separable.
The field extension
K/k is separable if and only if for any finite field extension k ′ /k , the tensor product k ′ ⊗ k K is reduced or equivalently that k ′ ⊗ k K is regular.This proves the lemma. Theorem 13. [11, Theorem 93, p. 279] . Let ( A, m ) be a Noetherian local ringcontaining a field k . Then A is geometrically regular over k if and only if A isformally smooth over k in the m -adic topology. Lemma 14. [11, Lemma 2, p. 251] . let φ : A → B be a faithfully flat, regularhomomorphism. Then (1) A is regular (resp. normal, resp. Cohen-Macaulay, resp. reduced) if andonly if B has the same property; (2) If B is a G-ring, then so is A . For a Noetherian scheme X , let Reg( X ) denote the subset of X that consists ofregular points, which is called the regular locus of X . Definition 15.
Let A be a Noetherian ring.(1) We say that A is J-0 if Reg(Spec A ) contains a nonempty open set of Spec A .(2) We say that A is J-1 if Reg(Spec A ) is open in Spec A .Clearly, if A is J-1 and Reg(Spec A ) is nonempty, then A is J-0. It is well knownthat a local Artinian ring is a field if and only if it is reduced. Indeed, since itsmaximal ideal m is nilpotent, the reducedness implies that m = 0 and hence it is afield. Therefore, the regular locus Reg(Spec A ) is nonempty if and only if there isa minimal prime ideal p of A such that its localization A p is reduced. Theorem 16. [11, Theorem 73, p. 246] . For a Noetherian ring A , the followingconditions are equivalent: (a) any finitely generated A -algebra B is J-1; (b) any finite A -algebra B is J-1; (c) for any p ∈ Spec A , and for any finite radical extension K ′ of k ( p ) , thereexists a finite A -algebra A ′ satisfying A/ p ⊆ A ′ ⊆ K ′ which is J-0 andwhose quotient field is K ′ . Definition 17.
A Noetherian ring A is said to be J-2 if it satisfies one of theequivalent conditions in Theorem 16.
Lemma 18.
Any Noetherian Japanese ring A of dimension one is J-2. Proof.
For each p ∈ Spec A , the quotient domain A/ p is either a field or aNoetherian Japanese domain of dimension one. In the first case, the condition (c)holds trivially by taking A ′ = K ′ . In the second case, the integral closure A ′ of A in K ′ is finite over A and is a Dedekind domain, which particularly J-0. Therefore, A is J-2. CHIA-FU YU
We gather some properties of G-rings.
Theorem 19. (1)
Any complete Noetherian local ring is a G-ring. (2)
If for any maximal ideal m of a Noetherian ring A , the natural map A m → ( A m ) ∗ is regular, then A is a G-ring (3) Let A and B be Noetherian rings, and let φ : A → B be a faithfully flat andregular homomorphism. If B is J-1, then so is A . (4) Any semi-local G-ring is J-1.
Proof. (1) See [11, Theorem 68, p. 225 and p. 250]. (2) See [11, Theorem 75,p. 251]. (3) and (4) See [11, Theorem 76, p. 252].
Theorem 20. (1)
Let A be a G-ring and B a finitely generated A -algebra. Then B is a G-ring. (2) Let A be a G-ring which is J-2. Then A is a Nagata ring. Proof. (1) See [11, Theorem 77, p. 254]. (2) See [11, Theorem 78, p. 257].2.3.
Excellent rings.
We recall the definition of catenary and universally catenaryrings [11, p. 84]:
Definition 21. (1) A ring A is said to be catenary if for any two prime ideals p ⊆ q , the relativeheight ht( q / p ) is finite and is equal to the length of any maximal chain ofprime ideals between them.(2) A Noetherian ring A is said to be universally catenary if any finitely gen-erated A -algebra is catenary.If A is catenary, then so are any localization of A and any homomorphism imageof A . To show a Noetherian ring A is universally catenary, it then suffices toshow that every polynomial ring A [ X , . . . , X n ], for n ≥
1, is catenary. If A isCohen-Macaulay, then A is catenary and A [ X ] is again Cohen-Macaulay cf. [6,Proposition 18.9 and Corollary 18.10]. Therefore, any Cohen-Macaulay ring isuniversally catenary. Since every regular ring is Cohen-Macaulay, every regularring is universally catenary. Definition 22. [11, §
34, p. 259]. Let A be a Noetherian ring.(1) We say that A is quasi-excellent if the following conditions are satisfied:(i) A is a G-ring;(ii) A is J-2.(2) We say that A is excellent if it satisfies (i), (ii) and the following condition(iii) A is universally catenary. Remark . (1) Each of the conditions (i), (ii), and (iii) is stable under the localization andpassage to a finitely generated algebra (Theorems 16 (1) and 20 (1)).(2) Note that (i), (ii), (iii) are conditions depending only on A/ p , for p ∈ Spec A . Thus a Noetherian ring A is (quasi-)excellent if and only if so is A red . UNDAMENTAL IDENTITY 7 (3) The conditions (i) and (iii) are of local nature (in the sense that if theyhold for A p for all p ∈ Spec A , then they hold for A ), while the condition(ii) is not.(4) Theorem 20 (2) states that any quasi-excellent ring is a Nagata ring.(5) It follows from Theorems 16 and 19 (4) that any Noetherian local G-ringis quasi-excellent.(6) Nagata’s example of a 2-dimensional Noetherian local ring that is catenarybut not universally catenary [11, (14.E), p. 87] is a G -ring, and is also aJ-2 ring as any local G -ring is a J-2 ring. So it is a quasi-excellent catenarylocal ring that is not excellent.(7) Rotthaus [17] constructed a regular local ring R of dimension three whichcontains a field and which is Nagata, but not quasi-excellent. Example 24. [11, § R or C are excellent.(3) Any Dedekind domain A of characteristic zero is excellent.3. Valuations, completions and extensions
In this section, a group will mean an abelian group unless stated otherwise.3.1.
Valuations and valuation rings.
For a totally ordered abelian group Γwritten additively, we denote by Γ ∞ = Γ ∪ {∞} the totally ordered commutativemonoid with γ ≤ ∞ for all γ ∈ Γ and γ + ∞ = ∞ + γ = ∞ for all γ ∈ Γ ∞ . Definition 25. (1) An integral domain A with quotient field K is called a valuation ring or a valuation ring of K if for any x ∈ K × either x ∈ A or x − ∈ A .(2) A valuation of a field K is a group homomorphism v : K × → Γ, where Γ isa totally ordered abelian group, such that(3.1) v ( x + y ) ≥ min { v ( x ) , v ( y ) } , ∀ x, y ∈ K × . We extend v to a function v : K → Γ ∞ by putting v (0) = ∞ . Clearly, thecondition (3.1) holds for all x, y ∈ K . The homomorphism image v ( K × ) iscalled the value group of the valuation v . Clearly, A := { x ∈ K : v ( x ) ≥ } is a valuation ring and m := { x ∈ A : v ( x ) > } is its maximal ideal. Wecall A and κ := A/ m the valuation ring and residue field of v , respectively.(3) A valuation v of K is said to be discrete if its value group is isomorphic to Z compatible with the orders.(4) Two valuations v and v of K with value groups Γ and Γ are said to be equivalent if there is an isomorphism α : Γ ∼ −→ Γ of ordered groups suchthat v = α ◦ v .The construction v A gives rise to a map from the set of equivalence classesof valuations of K to the set of valuation rings of K . The reverse construction is asfollows: For a given valuation ring ( A, m ), define Γ := K × /A × and P := m /A × theset of positive elements, then Γ is a totally ordered group and the natural projection v : K × → Γ is a valuation of K so that the valuation ring of v is equal to A . It iseasy to see the above map is bijection, cf. [4, VI, § CHIA-FU YU
Proposition 26.
Let A be a valuation ring of a field K . (1) The set of primes ideals of A is totally ordered by the order of inclusion. (2) If B ⊃ A is a subring of K , then B is a valuation ring and the maximalideal m ( B ) of B is a prime ideal of A . Moreover, the map B m ( B ) isa order-reversing bijection between the totally ordered set of subrings of K containing A and the totally ordered set of prime ideals of A . The inversemap is given by p A p , the localization of A at p . Proof. (1) See [4, VI, § § Definition 27.
A subgroup H of an ordered group G is said to be isolated if therelation 0 ≤ y ≤ x with x ∈ H implies y ∈ H . Proposition 28.
Let G be an ordered group and P the set of its positive elements. (1) The kernel of an increasing homomorphism of G to an ordered group is anisolated subgroup of G . (2) Conversely, let H be an isolated subgroup of G and g : G → G/H thecanonical homomorphism. Then g ( P ) is the set of positive elements of anordered group structure on G/H . Moreover, if G is totally ordered, so is G/H . If G is totally ordered, then the set of isolated subgroups of G are totally orderedby the order of inclusion. For otherwise, there is a positive element x in one isolatedsubgroup H but not in H ′ , and a positive element x ′ ∈ H ′ r H . Suppose for example x ≤ x ′ , then x ∈ H ′ , a contradiction. Definition 29. (1) Let G be a totally ordered group. If the number of isolated subgroups of G distinct from G is finite and is equal to n , G is said to be of height n . Ifthis number is infinite, G is said to be of infinite height . Denote by h ( G )the height of G .(2) The height of a valuation v of K is defined as the height of its value group.The height of the groups Z and R are of height 1. If G is a totally orderedgroup and H is an isolated subgroup, then h ( G ) = h ( H ) + h ( G/H ). In particular,if G is the lexicographic product of two totally ordered groups H and H ′ , then h ( G ) = h ( H ) + h ( H ′ ). Thus, the lexicographic product Z × Z is of height 2.Fix a valuation ring of A of K with the canonical valuation v A : K × → Γ A := K × /A × . For subring B of K containing A , B is a valuation ring and A × ⊂ B × . Let λ : Γ A → Γ B be the natural projection. As A ⊂ B , λ maps the positive elementsof Γ A to positive elements of Γ B . Thus, λ is a morphism of ordered groups andthe kernel H B of λ is an isolated subgroup of Γ A . The mapping B H B is anincreasing bijection of the set of subrings of of K containing A onto the set of isolatedsubgroups of Γ A cf. [4, VI, § A . Inparticular, h (Γ A ) = ht( m ( A )) = dim A , cf. [4, VI, § G is of height ≤ R , cf. [4, VI, § UNDAMENTAL IDENTITY 9
Topological fields and completions.
We first discuss the notion of com-pleteness of a Hausdorff commutative topological group and its completion. Ourreferences are [3] and [10, Chap. 10].
Definition 30.
Let X be a nonempty set.(1) A filter of X is a nonempty subset F ⊂ P ( X ) of the power set of X satisfyingF.1 If A, B ∈ F , then A ∩ B ∈ F ;F.2 If A ∈ F and A ⊂ A ′ ⊂ X , then A ′ ∈ F ;F.3 ∅ 6∈ F .(2) A filter base of X is a nonempty subset B ⊂ P ( X ) satisfyingFB.1 If A, B ∈ F , then there exists C ∈ B such that C ⊂ A ∩ B ;FB.2 ∅ 6∈ B .Condition F.3 implies that any finite intersection of members in F is nonempty.For a filter base B , the set F := { F ⊂ X : ∃ B ∈ B , B ⊂ F } is a filter of X ,called the filter generated by B , and B is called a base of F . A filter F ′ is called a refinement of F if F ⊂ F ′ . Example 31.
Let { x k } ∞ k =1 be a sequence in X . Let F be the set consisting of allsubsets E such that there exists N ≥ x k ∈ E for all k ≥ N . Then F isa filter. Let B be the set consisting of all subsets { x k , x k +1 , . . . , } for some k . Then B is a filter base which generates F . Example 32.
Let X be a topological space. For every point x ∈ X , let N x denotethe collection of all neighborhoods E of x (there exists an open subset U ∋ x contained in E ). Then N x is a filter. Any fundamental system of neighborhoods of x is a filter base of N x . Definition 33.
Let X be a topological space. We say a filter base F converges toa point x ∈ X , denoted by F → x , if every E ∈ N x contains a member F ∈ F . Inthis case, by F B. x is in the closure of every member F ∈ F . If F is a filter, thisis equivalent to that F is a refinement of N x .A topological space X is Hausdorff if and only if every filter converges to at mostone point. Let f : X → Y be a map of topological spaces and F a filter of X . Set f F := { F ⊂ F : ∃ E ∈ F , f ( E ) ⊂ F } , which is a filter as f ( A ∩ B ) ⊂ f ( A ) ∩ f ( B ). Then the map f is continuous if andonly if for every x ∈ X one has ( F → x ) = ⇒ ( f F → f ( x )). Definition 34.
Let ( A, +) be a Hausdorff commutative topological group.(1) A filter F of X is called a Cauchy filter if for any neighborhood U ∈ N there exists E ∈ F such that E − E := { x − y : x, y ∈ E } ⊂ U. (2) A is said to be complete if every Cauchy filter converges.(3) A completion of A is pair ( b A, ι ), where b A is a complete topological abeliangroup and ι : A → b A is a morphism of topological groups, satisfying thefollowing conditions.(a) ι : A → ι ( A ) is a homeomorphism.(b) ι ( A ) ⊂ b A is dense. The condition (a) says that ι is injective and topology of A is the same as thetopology induced by b A . It is proved [3, III, § § b A, ι ) exists and satisfies the functorial property: for any pair-ing (
B, f ) where B is a complete topological abelian group and f : A → B is amorphism of topological groups, then there exists a unique morphism g : b A → B such that g ◦ ι = f . In particular, a completion ( b A, ι ) is unique up to a uniqueisomorphism; such a pair ( b A, ι ) is called the completion of A . If A is a Hausdorfftopological ring, then the completion b A of ( A, +) is a complete topological ring(complete for the underlying topological group ( b A, +)), cf. [3, III, § v be a valuation of a field K with value group G . For all α ∈ G , let V α := { x ∈ K : v ( x ) > α } and V ≥ α := { x ∈ K : v ( x ) ≥ α } which are clearly additive subgroups of K . There exists a unique linear topology T v on K for which the sets V α form a fundamental system of neighborhoods of 0.If v is trivial, then T v is the discrete topology. Equipped with this topology K is aHausdorff topological field.Let K ∗ be the completion of K , which is a complete topological ring. Note thatif the v of height 1, or more generally there exists a countable fundamental systemof neighborhoods of 0. Then the notion of completeness and the construction ofcompletion can be made by Cauchy sequences, which is similar to the classicalconstruction of R from Q . Proposition 35.
Let v be a valuation of a field K with value group G and equip G with the discrete topology. (1) The complete ring K ∗ of K a topological field. (2) The continuous map v : K → G ∞ can be extended uniquely to a continuousmap v ∗ : K ∗ → G ∞ which is a valuation of K ∗ . (3) The topology on K ∗ is the topology defined by the valuation v ∗ . (4) For all α ∈ G , the closures V α and V ≥ α of V α and V ≥ α are the subsets of K ∗ defined by v ∗ ( x ) > α and v ∗ ( x ) ≥ α , respectively. (5) The valuation ring of v ∗ is the completion A ∗ of A ; its maximal ideal is thecompletion m ∗ of the maximal ideal m of A . (6) A ∗ = A + m ∗ ; the residue field of v ∗ is canonically identified with that of v . Proof.
See [4, VI, § Extensions of valuations and the fundamental inequality.
Let v bea valuation of a field K , A the valuation ring of v , m its maximal ideal, and Γ v the value group. Let L/K be a finite field extension and w be a valuation of L which extends v . Denote by Γ w the value group, A ′ the valuation ring and m ′ themaximal ideal of w , respectively. Write κ ( v ) and κ ( w ) for the residue fields of v and w , respectively. The completion of K at v (resp. of L at w ) is denoted by K ∗ (resp. L ∗ w ). The valuation of K ∗ extending v is denoted by v ∗ . Definition 36. (1) The ramification index of w over v is defined as e ( w/v ) := [Γ w : Γ v ].(2) The residue class degree of w over v is defined as f ( w/v ) := [ κ ( w ) : κ ( v )]. Lemma 37.
Let K , v , L and w be as above. UNDAMENTAL IDENTITY 11 (1)
The inequality e ( w/v ) f ( w/v ) ≤ [ L : K ] holds. In particular, e ( w/v ) and f ( w/v ) are finite. (2) The height of w is equal to that of v . (3) The w is trivial (resp. discrete) if and only if so is v . Proof.
See [4, VI, § Definition 38 ([4, VI, § . Two valuations v and v ′ of K are said to be indepen-dent if the subring generated by their valuation rings is equal to K ; and dependent otherwise.The trivial valuation is independent of any valuation of K . Two valuations aredependent if and only if there is a relation A ⊂ A ′ ( K among their valuation rings A and A ′ . If two non-trivial valuations v and v ′ are of same finite height, then theyare dependent if and only if they are equivalent. Indeed, if v and v ′ are equivalent,then A = A ′ ( K and they are dependent. Conversely if they are dependent thenup to switching the order one has A ⊂ A ′ ( K . Therefore, A = A ′ for otherwiseht( v ′ ) < ht( v ), contradiction.Let Σ v be a complete set of representatives of equivalence classes of extensionsof a valuation v of K on L . If v is trivial then Σ v consists of the trivial valuationof L . Also write w | v if w ∈ Σ v . If w , w ∈ Σ v with w = w (this implies that v must be non-trivial), then there is no inclusion relation for their valuation ringscf. [5, (A), p.2]. Thus, every two distinct valuations in Σ v are independent. Proposition 39.
Let v be a valuation of K and L/K a finite field extension. (1)
For every w ∈ Σ v , one has e ( w ∗ /v ∗ ) = e ( w/v ) , f ( w ∗ /v ∗ ) = f ( w/v ) , [ L ∗ w : K ∗ ] ≤ [ L : K ] and e ( w/v ) f ( w/v ) ≤ [ L ∗ w : K ∗ ] . (2) Every set of pairwise independent valuations of L extending a non-trivialvaluation v is finite. Let { w , . . . , w r } be a maximal set of pairwise in-dependent valuations of L extending a non-trivial valuation v . Then thecanonical mapping φ : K ∗ ⊗ K L → Q ri =1 L ∗ w i (extending by continuity thediagonal map L → Q ri =1 L ∗ w i ) is surjective, its kernel is the Jacobson radicalof K ∗ ⊗ K L and (3.2) r X i =1 [ L ∗ w i : K ∗ ] ≤ [ L : K ] . Proof.
See [4, VI, § Corollary 40 (The fundamental inequality) . Let v be a valuation of K and L/K a finite extension. We have (3.3) X w | v e ( w/v ) f ( w/v ) ≤ [ L : K ] . Proof. If v is trivial, then w is trivial and one has Γ v = Γ w = { } , κ ( v ) = K and κ ( w ) = L . Therefore, P w | v e ( w/v ) f ( w/v ) = [ L : K ]. Suppose that v is non-trivial.By the remark above Proposition 39, Σ v is a maximal set { w , . . . , w r } of pairwiseindependent valuations of L extending v . Then (3.3) follows from Proposition 39. Definition 41 ([4, VI, § § . (1) Let G be an ordered set. A subset M of G is called major if the relations x ∈ M and y ≥ x imply y ∈ M .(2) Let G be a totally ordered commutative group and H a subgroup of finiteindex. Denote by G > ⊂ G the subset of strictly positive elements. The initial index of H in G , denoted by ε ( G, H ), is the number of major subsets M of G such that H > ⊂ M ⊂ G > .If G = Z and H = m Z with m >
0, letting M ( x ) := { y ∈ G : y ≥ x } ,then M (1) , M (2) , . . . , M ( m ) are all major subsets of G satisfying the property inDefinition 41 and ε ( G, H ) = m . Proposition 42.
Let G be a totally ordered commutative group and H a subgroupof finite index. (1) If the set G > has no least element, then ε ( G, H ) = 1 ; (2) If the set G > has the least element x , then ε ( G, H ) = [ G : ( G ∩ H )] ,where G is the cyclic subgroup generated by x .In particular, ε ( G, H ) divides [ G : H ] . Proof.
See [4, VI, § Definition 43.
Let v be a valuation of K and w | v a valuation of a finite extension L/K with value groups Γ v and Γ w , respectively. The initial ramification index of w with respect to v (or w over v ) is defined as ε ( w/v ) := ε (Γ w , Γ v ). Theorem 44.
Let
L/K be a and Let v a valuation of K with valuation ring A andmaximal ideal m . Let L/K be a finite field extension with integral closure B of A in L . The following conditions are equivalent: (a) B is a finite A -module; (b) B is a free A -module; (c) [ B/ m B : κ ( m )] = [ L : K ] ; (d) P w | v ε ( w/v ) f ( w/v ) = [ L : K ] . Proof.
See [4, VI, § Remark . (1) The fundamental inequality (Corollary 40) was first proved by Cohen andZariski for an arbitrary valuation v [5].(2) The condition (d) is equivalent to (d’) P w | v e ( w/v ) f ( w/v ) = [ L : K ] and ε ( w/v ) = e ( w/v ) for all w | v . When v is discrete, one has ε ( w/v ) = e ( w/v )and the condition (d) is equivalent to P w | v e ( w/v ) f ( w/v ) = [ L : K ]. Inthis special case, Theorem 44 was first proved by Cohen and Zariski [5].4. Proofs of Main Theorems
Proof of Theorem 1.
Theorem 1 follows from the following two lemmas.
Lemma 46.
Let A be a Dedekind domain with quotient field K , and L a finiteextension field of K of degree n with integral closure B of A in L . For each nonzeroprime ideal p of A , one has (4.1) r X i =1 e i f i = dim k ( p ) B/ p B ≤ n, UNDAMENTAL IDENTITY 13 where P , . . . , P r are the prime ideals of B over p , e i and f i are the ramificationindex and the residue class degree of P i over p . Moreover, if B is a finite A -module,then the equality holds. Proof.
This is well-known (cf.[19]); we include a proof for the reader’s conve-nience. We localize the Dedekind domains A and B at p and get a discrete valuationring A p and a semi-local Dedekind domain B p with same number of maximal ideals P i B p . Then B p is the integral closure of A p in L and the numerical invariants r, e i , f i remain the same. Therefore, after replacing A by A p , we can assume that A is a discrete valuation ring with uniformizer π . By the Chinese Remainder Theorem, B/ p B ≃ Q ri =1 B/ P e i i . We filter each k ( p )-vector space B/ P e i i by the decreasingsubspaces P ji / P e i i and obtain(4.2) dim k ( p ) B/ p B = r X i =1 e i − X j =0 dim k ( p ) P ji / P j +1 i . The ideal P ji / P j +1 i generated by one element a j as the quotient ring B/J for anynonzero ideal J is a principal ideal ring. The map 1 a j induces an isomorphism A/ P i ≃ P ji / P j +1 i and hence dim k ( p ) P ji / P j +1 i = f i . It follows from (4.2) thatdim k ( p ) B/ p B = P ri =1 e i f i .We prove the inequality in (4.2) by showing that if ¯ x , . . . ¯ x s are k ( p )-linearlyindependent, then their liftings x , . . . x s are K -linearly independent. Suppose not,then a x + a x + . . . a r x r = 0for some nonzero element a i in K . Multiplying a suitable power of π , we can assumethat a i ∈ A for all i but a i p for some i . Modulo p we get a non-trivial linearrelation ¯ a ¯ x + . . . ¯ a s ¯ x s = 0, a contradiction.Suppose B is a finite A -module. Since B is torsion free and A is a principalideal domain, B is a free A -module of rank s . Then one has dim K L = rank A B =dim k ( p ) B/ p B . This proves the desired equality. Lemma 47.
Let the notation be as in Lemma 46. Then (4.3) dim k ( p ) B/ p B = r X i =1 rank A ∗ p B ∗ P i = r X i =1 dim K ∗ p L ∗ P i = [( L ⊗ K K ∗ p ) ss : K ∗ p ] , where K ∗ p and A ∗ p (resp. L ∗ P i and B ∗ P i ) denote the completion of K and A (resp. L and B ) at p (resp. P i ), respectively. Proof.
The ramification index and residue class degree remain the same afterthe completion, that is, e ( P i / p ) = e ( P ∗ i / p ∗ ) and f ( P i / p ) = f ( P ∗ i / p ∗ ), where p ∗ = p A ∗ p and P ∗ i = P i B ∗ P i . Since K ∗ p is complete and L ∗ P i is a finite extensionfield of K ∗ p , we have e i f i = e ( P ∗ i / p ∗ ) f ( P ∗ i / p ∗ ) = [ L ∗ P i : K ∗ p ] [19, Chap. II, § B ∗ P i is the integral closure of A ∗ p in L ∗ P i . Since B ∗ P i is afinite free A ∗ p -module say of rank m , we have L ∗ P i = B ∗ P i ⊗ A ∗ p K ∗ p ≃ ( K ∗ p ) m andrank A ∗ p B ∗ P i = [ L ∗ P i : K ∗ p ] = e i f i . This and Lemma 46 prove the first two equalities. The last equality follows from the isomorphism Q ri =1 L ∗ P i ≃ ( L ⊗ K K ∗ p ) ss ; seeProposition 39. Proof of Theorem 2.
Theorem 2 follows from Proposition 48, Corollary 49,and Theorems 20 and 6.
Proposition 48.
Let A be a Dedekind domain with quotient field K . Then thefollowing statements are equivalent: (1) For any finite field extension
L/K and any nonzero prime ideal p of A , onehas P P | p e P f P = [ L : K ] . (2) For any finite field extension
L/K and any nonzero prime ideal p of A , thetensor product L ⊗ K K ∗ p is a semi-simple K ∗ p -algebra. (3) A is a G -ring. (4) A is a Japanese ring. Proof.
The equivalence of (1) and (2) follows from Theorem 1.We prove the equivalence of (2) and (3). To show A is a G -ring, one needs toshow that every formal fiber of φ p : A p → A ∗ p is geometrically regular. If p = 0, theformal fiber K → K ∗ = K is clearly geometrically regular. Suppose p is a nonzeroprime ideal. The special formal fiber k ( p ) → k ( p ∗ ) = k ( p ) is clearly geometricallyregular and the generic formal fiber is given by K → K ∗ p . By Lemma 12, K ∗ p /K isgeometrically regular if and only if it is separable. Thus, A is a G -ring if and onlyif for any nonzero prime ideal p of A , the field extension K ∗ p /K is separable. Thisis equivalent to that for any finite extension L/K and for any nonzero prime ideal p of A , the tensor product L ⊗ K K ∗ p is semi-simple.We prove the equivalence of (1) and (4). The direction (4) = ⇒ (1) follows fromLemma 46 and we show the other direction. For each nonzero prime ideal p , since P ri =1 e i f i = [ L : K ], the integral closure B p of A p in L is a finite A p -module byTheorem 44. Thus, for any finite field extension L/K the integral closure of A p in L is a finite A -module, and hence A p is Japanese. By Lemma 7, A is a Japanesering. This completes the proof of the proposition. Corollary 49.
Any Dedekind domain A which is a G -ring is excellent. Proof.
It follows immediately from Proposition 48 and Lemma 18 that A isquasi-excellent. It is well-known that any Dedekind domain is universally catenary.Therefore, A is excellent. Acknowledgements
The author learned a lot from [11] and [4] while preparing the present articleand wish to acknowledge the authors for their tremendous efforts of writing thesegreat books. The author is partially support by the MoST grants 109-2115-M-001-002-MY3.
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