RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS

Transcription

1 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS KENZ KALLAL Abstract. In these notes, we introduce the theory of (nonarchimedian) valued fields and its alications to the local study of extensions of number fields. Particular attention is given to how the ramification of rime ideals in such an extension can be described locally. We will finish this discussion with a detailed roof of a famous result relating to this, namely that the different ideal of an extension of number rings is equal to the annihilator of the module of Kähler differentials corresonding to the same extension. Then, we discuss the somewhat more descritive notion of the ramification subgrous of a Galois extension of local fields and exlain the connection between the ramification subgrous and the ramification of rime ideals. The filtration of ramification grous can also be alied to the automorhism grou of the local field F ((X)). Through the identification between the grou of wild automorhisms of F ((X)) and the Nottingham grou, we rove an original result giving a concise criterion for wild automorhisms whose iterates have ramification of a certain tye. Finally, we describe the imlications that such a comutation has on the locations of eriodic oints of ower series alied to nonarchimedian fields of ostive characteristic. Contents 1. Introduction 2 2. Ramification in Number Fields: Ideal-Theoretic Methods The Relative Discriminant Ideal The Relative Different Ideal 4 3. Valuation Theory 6 4. Alications of Localization Basic Results Localization and Ramification Local Methods Comletions of Valued Fields Extensions of DVRs Extending Scalars by Comletions Kähler Differentials and the Different Ramification Grous Ramification in the Nottingham Grou Classifying Power Series by Ramification 25 Acknowledgments 33 References 33 Date: Last udated January 24,

2 2 KENZ KALLAL 1. Introduction Recall the familiar notion of ramification in an extension of number fields L/K: A nonzero rime ideal O K (which we will refer to as a rime in K) is called ramified in L if the ideal O L in O L factors into rimes as O L q e1 1 qen n where e i > 1 for some i. From this definition arises the classical question Question 1.1. Given L/K, which rimes of K are ramified in L? Question 1.1 is answered using a reasonably natural construction, namely the relative discriminant of L/K, an ideal in O K tyically denoted L/K. In articular, a rime in K ramifies in L if and only if L/K. More generally, for any rime q in L containing a rime in K, we must have q O L, i.e. q q i for some i. Conversely, all of the q i s contain O L and thus. So the information above is a secific case of the ramification index e(q ) : e i (namely is ramified in L if e(q ) > 1 for some rime q in L containing ). In the other direction, for any rime q in L, q contains exactly one rime in K, namely q K. So, we can ask the oosite of Question 1.1: Question 1.2. Which rimes q in L are such that e(q q O K ) > 1? The answer to Question 1.2, along with some additional ramification information, is encoded in the relative different ideal D L/K O L (just as before, this ideal is constructed so that a rime q in L has e(q q O K ) > 1 if and only if q D L/K ), a beautiful construction which will feature rominently in this aer. It is fair to say that the ramification data of L/K is encoded by L/K and D L/K, and these constructions will be the subject of Section 2 of this aer. However, Section 2 will only resent the roofs that these constructions answer their resective questions in the case of K Q (and O K Z). The general roofs of these statements are ostoned for Section 3, when the oeration of localization will be introduced. The basic idea is that localizing at a rime in A will reserve ramification of rimes lying over, while reducing to the case where A is a PID, guaranteeing the existence of a basis for B as an A-module. The ramification of an extension of number fields at a air of rimes q is just as easily written in terms of the discrete valuations of the two fields corresonding to the two rimes. Leveraging the roerties of the comletions of the two fields with resect to these valuations, this gives rise to the theory of local fields. As a result of this theory, in the setting of the comletion, we may assume that O L has a ower basis over O K. In Section 3, we use this to rove the desired roerties of the discriminant and different, and to rove a fascinating result which strengthens the notion that ramification is an imortant iece of local geometric data: Theorem 1.3. Let L/K be an extension of number fields. Then D L/K Ann OL (Ω OL /O K ) where Ω OL /O K denotes the O L -module of Kähler differentials of O L over O K. In Section 6, we consider the setting of a Galois extension L/K of comletions of number fields, where we introduce an imortant filtration of Gal(L/K) called the sequence of ramification grous, and we show that the different ideal of a Galois extension of number fields can be determined locally by the ramification grous.

3 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 3 In Section 7 we will use this to make an analogous definition of ramification for wild automorhisms of F ((X)) (thus introducing the Nottingham grou of ower series tangent to the identity). It turns out that the ramification of iterates of these automorhisms is a iece of essential information about the dynamics of certain classes of ower series acting on the unit disc of a nonarchimedian field. In our case, the ramification tye of a ower series f(x) a 1 X + a 2 X 2 + F [[X]] is the series of ramification numbers i n (f) : i(f n ) for n 0, where i(g) denotes the ramification number of g (i(g) is determined by where g falls in the sequence of ramification grous). Under certain conditions on b and, we will classify the ower series f (in terms of their coefficients) for which i n (f) b+b+ b n for all n N. The comutation in Section 8 yields an original result (in collaboration with Hudson Kirkatrick) that achieves this classification under certain hyotheses on and b. 2. Ramification in Number Fields: Ideal-Theoretic Methods The material in this section is mostly drawn from Keith Conrad s online handouts [4, 3]. Let L/K be an extension of number fields of degree n. As reviously mentioned, the ramification henomena of O L over O K are encoded by the relative discriminant and relative different of the extension. It will be the goal of this section to define these constructions, and to rove that they indeed contain the answers to Questions 1.1 and 1.2 in the case where K Q. Throughout this aer, we will work in a somewhat more general setting: where L/K is an arbitrary finite extension of characteristic zero, let A K be a Dedekind domain with field of fractions K, and B its integral closure in L. Since any element of L can be multilied by an element of A to get an element of B, we have in general L FracB The Relative Discriminant Ideal. Fortunately, defining the relative discriminant is not much more involved than the absolute discriminant of a number field. Definition 2.1 (Relative discriminant). The relative discriminant B/A is the ideal in A generated by the discriminants disc L/K (b 1,..., b n ) : det(tr L/K (b i b j )) where (b 1,..., b n ) runs over all K-bases of L contained in B. One can check that in the case A Z and B Q, the relative discriminant is the ideal in Z generated by the discriminant of any integral basis for O L. In this case, we can rove the desired result directly as in [4, Theorem 1.3] Proosition 2.2. Let Z be a rime. Then () ramifies in L if and only if OL /Z. Proof. Recall that O L has an integral basis ω 1,..., ω n O L. Suose that O L factorizes into rime ideals of O L as Recall, too, that O L q e1 1 qem m. OL /Z disc L/Q (ω 1,..., ω n )Z det((tr OL /Z(ω i ω j )) ij )Z

4 4 KENZ KALLAL Where the trace is taken with resect to {ω i }. Note that O L /O L has an Z/Z F -basis given by the ω i s, where ω i denotes the reduction of ω i modulo O L. Therefore, Tr (OL /O L )/F (ω i ω j ) Tr OL /Z(ω i ω j ) (mod ) since the n n matrix given by x ω i ω j x with resect to the Z-basis {ω i } for O L, when reduced modulo, is the n n matrix given by x ω i ω j x with resect to the F -basis {ω i } for O L /O L. Taking determinants, this means that disc (OL /O L )/F (ω 1,..., ω n ) OL /Z (mod ). Thus, OL /Z if and only if disc (OL /O L )/F (ω 1,..., ω n ) 0 By the Chinese Remainder Theorem, we have O L /O L O L /q e1 1 O L/q em m. e Choose bases B i for each O L /q i i, and let B be the basis of their roduct (which we know from above is equal to O L /O L ) given by the concatenation of the B i s. It is easy to see from the definition that m disc (OL /O L )/F (B) disc (OL /q e i i )/F (B i ). i1 Since the discriminants of bases are all related by unit factors, the left hand side is zero if and only if disc (OL /O L )/F (ω 1,..., ω n ) 0, i.e. if and only if OL /Z. So, it suffices to determine whether disc (OL /q e i i )/F (B i ) 0 given e i (for which we can relace B i with any arbitrarily chosen basis). In articular, suose e i > 1 for some i (i.e. is ramified in L). Then we can choose an x q i \ q ei i. Letting x be the modulo-qei i reduction of x, we are thus guaranteed that x 0 but that x ei 0 in O L /q ei i, i.e. x is nilotent and thus its multiles all have trace zero. Extending x to a basis of O L /q ei i, this makes it immediate that the discriminant with resect to the extension of rings (O L /q ei i )/(F ) of any basis of O L /q ei i is zero. Therefore, OL /Z if ramifies in L. Conversely, suose e i 1 for all i (i.e. is unramified in L). The revious analysis does not aly because q i \ q ei i is then emty. Instead, (since q i is a rime and is therefore maximal as oosed to its higher owers), we know that O L /q i is a (finite) field. Therefore, the extension of rings over which we wish to comute the discriminant of a basis, namely (O L /q i )/F, is actually an extension of finite fields. But it is a well-known roerty of discriminants over field extensions that the discriminant of a basis is always nonzero (see [23, Theorem 7]). Thus, OL /Z, as desired. The roof that rime in A ramifies in B if and only if B/A will be ostoned until the next section, where localizing will allow us to reduce to the case where A is a PID and thus B has a finite A-basis The Relative Different Ideal. In a somewhat more comlicated way than the relative discriminant, the relative different is also based on the trace form x, y : Tr L/K (xy), which is a symmetric nondegenerate bilinear form on L. To show that it is nondegenerate, it suffices to show that the trace does not vanish on

5 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 5 L (if Tr L/K (x) 0 then y, x/y 0 for every nonzero y L which means the ma y y, is injective and thus bijective). Define B : {x L : Tr L/K (xy) A for all y B}. Under the identification L L given by the trace form, B is identified with a fractional ideal of B that contains B, so its inverse is an ideal in B. This is how we define the relative different: Definition 2.3. The relative different D B/A of the ring extension B/A is the ideal (B ) 1 B. When the choice of A and B is obvious, i.e. A O K and B O L, we sometimes denote the relative different by D L/K. The general roof of the desired roerty of the relative different, like that of the discriminant, will be ostoned until local methods are available. For now, we resent a roof in the case of K Q, A Z. Proosition 2.4. Let be an ideal in O L. Then e(, Z) > 1 if and only if D L/Q. Proof. Just as in the roof of Proosition 2.2, this roof will rely on the fact that O L has an integral basis which is also a Q-basis for L, which allows us to write the field trace as the trace of the multilication ma on the Z-module O L with resect to a suitable basis. In this secial case, it is easily seen that the definition of OL also means that for any fractional ideal a O L, every element of a has integral trace if and only if a is contained in OL (in other words, O L is the largest fractional ideal of O L such that all of its elements have integral trace ). So D L/Q if and only if D L/Q, i.e. if and only if 1 OL, which is therefore equivalent to all of the elements of 1 having integral trace. Suose that e(, Z) > 1. Letting Z Z, this means O L b, where b is an ideal in O L which is divisible by. This means 1 1 b. By the revious discussion, it follows that D L/Q is equivalent to Tr L/Q (b) Z for all b b. Rewriting the trace as the trace of the multilication ma on the Z-module O L, we have for any b b, Tr L/Q (b) Tr OL /Z(b). Taking the quotient of both rings by (), we actually have Tr (OL /O L )/F (b) Tr L/Q (b) (mod ) where b denotes the reduction of b modulo O L. But O L b 2, which means that b 2 b 2 O L, i.e. multilication by b is a nilotent oerator on the F -vector sace O L /O L. Hence, it has zero trace and Tr L/Q (b) Z for all b b, which means D L/Q. Finally, suose e(, Z) 1. Then we can write O L b, where b is not divisible by. As before, D L/Q is still equivalent to Tr L/Q (b) Z for all b b. So, we still want to consider the trace of an arbitrary element b b (and we still

6 6 KENZ KALLAL know that this trace is congruent modulo to Tr (OL /O L )/F (b)). By the Chinese Remainder Theorem, O L /O L OL / O L /b. So Tr (OL /O L )/F (b) Tr (OL /)/F (b) + Tr (OL /b)/f (b), where b denotes the reduction of b modulo the aroriate ideal. Of course, b 0 modulo b, so multilication by b in O L /b is the zero ma, which has zero trace. Hence we can ignore the last term and write Tr (OL /O L )/F (b) Tr (OL /)/F (b). But (O L /)/F is an extension of finite fields, so there must be a b b such that Tr (OL /)/F (b) 0. In articular, this means that Tr L/Q (b) Z, so D L/Q, as desired. In fact, the roof above can be extended to give a simle exlicit descrition (in most cases) of e( Z) based on the number of times D L/Q. 3. Valuation Theory In this section, we introduce a secial case the owerful theory of local fields, which is a generalization of the theory of the -adic numbers; this will be drawn from two classic references on the toic, namely Serre s book Local Fields [29] and Neukirch s Algebraic Number Theory [25]. We begin the develoment of this theory with a chain of reasoning analogous to the definition of the -adic numbers. In articular, let R be Dedekind domain with field of fractions K (a generalization of the rincial ideal domain Z with field of fractions Q from which we might roduce the -adics). We can associate to each rime ideal R the -adic valuation ν : K R { } defined by ν (x) e for x R, where e denotes the exonent of in the factorization of the rincial ideal (x) into rimes in R, i.e. so that (x) factorizes as (x) ν(x). This extends to all of K in the natural way by taking ( a ) ν ν (a) ν (a). b Also, we define ν (0). It is easy to see that ν satisfies some nice roerties: i) For x K, ν (x) if and only if x 0. ii) For x, y K, ν (xy) ν (x) + ν (y) iii) For x, y K, ν (x + y) min(ν (x), ν (y)). A ma satisfying these roerties is called a nonarchimedian valuation on K. It is our objective to make K into a metric sace (e.g. to take its comletion, still in analogy to the definition of Q ), so we transform ν in the following way: Let be defined to be : K R 0 x : q ν(x)

7 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 7 for some fixed real number q > 1. The three roerties that make ν a nonarchimedian valuation corresond to the following three roerties for : i) x 0 if and only if x 0. ii) xy x y. iii) x + y max( x, y ). Since it satisfies these roerties, is a nonarchimedian absolute value on K. Note that the third roerty is a stronger version of the triangle inequality, and thus (K, ) is a metric sace and we can take its comletion ˆK. In fact, ˆK has an obvious field structure, obtained from taking the ring of Cauchy sequences and modding out by the maximal ideal of Cauchy sequences which converge to zero. The fact that satisfies the third roerty is the reason why ν and are called nonarchimedian. If a ma satisfies the triangle inequality but not (iii), it is instead called an archimedian absolute value, and the corresonding valuation is called an archimedian valuation. However, in this case, ˆK has a very restricted theory: Theorem 3.1 (Ostrowski). Let K be a field which is comlete under some archimedian absolute value. Then (K, ) is isomorhic as a toological sace to R or C with the usual toologies. Proof. See Neukirch [25, Ch. II, Theorem 4.2]. This result justifies restricting the general study of valued fields to the nonarchimedian case. In the case where K is a number field (which will obviously be the case of interest to us), Ostrowski roved another result on the general theory of valuations: Theorem 3.2 (Ostrowski). Let K be a number field. Then all nontrivial nonarchimedian absolute values on K must be of the form ν for some rime in O K. All archimedian absolute values on K must corresond to the absolute value on an embedding of K into R or C. Proof. See [5, Theorem 3] Thus, we can restrict our attention in this aer to only the valuations of the form ν, though much of the following discussion features in the general theory of valued fields as well. Letting : K R 0 be the absolute value corresonding to ν : K R { } as before, we can associate to (still in analogy to the -adics) the ring O : {x K : ν (x) 0}, which contains all x K such that the fractional ideal xo K has rime factorization with a nonnegative exonent on. Since ν only takes into account the behavior at the rime, the other rimes of O K may still have negative exonents in the factorization of (x), thus O is strictly larger than O K (in general it is called the valuation ring of K). From this intuition, we can finally observe the geometric notion of localization at a rime ideal. Lemma 3.3. O is equal to the localization S 1 O K : { a s K : a O K, s S },

8 8 KENZ KALLAL where S O K \. It is a DVR, and its uniformizing arameter can be taken to be any π O 2 O Proof. The main content of the lemma is actually in the second half, which almost immediately follows from the following observation: First, we show that S 1 O K is a Dedekind domain (this actually holds for any multilicative set S O K and follows from the fact that O K is a Dedekind domain). Since the ideals of S 1 O K are localizations of ideals in O K, which are finitely generated over O K, we know that the ideals of S 1 O K are finitely generated (an ideal S 1 I is generated over S 1 O K by the same generating set as the ideal I O K ). So, S 1 O K is Noetherian. The rime ideals of S 1 O K are (by localization) in inclusion-reserving bijection with the rime ideals of O K that do not intersect S. If a rime q S 1 is not maximal, then it is roerly contained in a maximal ideal, which must be the localization S 1 of some ideal O K, which must roerly contain (by the above bijection). This contradicts the fact that rime ideals in O K are maximal, so indeed all rime ideals in S 1 O K are maximal. Finally, suose that a K is integral over S 1 O K. Then a is the root of some monic olynomial f over S 1 O K. Multilying by s S given by the deg f ower of the roduct of all the denominators of the coefficients of f, we get that sa is integral over O K, so sa O K (since O K is integrally closed in K). Thus, a S 1 O K, i.e. S 1 O K is integrally closed in its field of fractions. All of this amounts to the observation that S 1 O K is a Dedekind domain. Since the rime (i.e. maximal) ideals of S 1 O K arise as localizations of rime ideals of O K contained in, of which there is exactly one (namely ), we know that S 1 O K is a local ring with maximal ideal m S 1. We can take π m m 2 (these are guaranteed to be distinct ideals since S 1 O K is a Dedekind domain), and we know that (π) m (it must be a ower of m by unique factorization of ideals, it cannot be any higher ower of m since π is not in any higher ower of m). Thus, the ideals of S 1 O K are all equal to m n (π n ) for some n 0, which means that S 1 O K is a DVR with uniformizing arameter π. K is the field of fractions of S 1 O K O K, so every element of K is of the form u π n for n Z, where u is a quotient of units in S 1 O K, i.e. ν (u) 0 (one can show that the units in S 1 O K are exactly the elements of the form a/s where neither a nor s is in ). Then, O is just the set of all u π n such that n 0, which coincides with our characterization of S 1 O K, as desired. This lemma and its roof are analogous to the fact that the valuation ring of Q under the -adic valuation is the localization of Z at. Of course, we had to be more careful to account for the fact that we cannot just write fractions in lowest form, since O K is not necessarily a PID. Everything we just did still holds in the more general setting in which A is an arbitrary Dedekind domain with field of fractions K and is a rime in A. The valuation ring of K with resect to ν is the localization of A at The general fact that the valuation ring of a valued field is a DVR holds as long as the valuation is discrete, i.e. the image of the valuation on K has a smallest nonzero element. Any element of the valuation ring with that smallest valuation is then a uniformizing arameter.

9 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 9 4. Alications of Localization 4.1. Basic Results. Without considering more about the valuation associated with, O is already a owerful gadget for studying the factors of in the decomosition of ideals in O K. For this reason, we devote this section to a detour exlaining how to use this tool to rove the generalized versions of Proositions 2.2 and 2.4 to the relative different and discriminant. Readers familiar with algebraic geometry will recognize the localization of O K at a rime ideal as being analogous to the local ring of an affine variety at a oint (see Fulton s Algebraic Curves, [8, Ch. 3]), and the fact that this is a DVR as some notion of non-singularity of the variety at a oint. In fact, we have seen that Sec(O ) 1, so the affine scheme Sec(O ) is homeomorhic to the closed oint {} Sec(O K ) (recall that all rimes are maximal, i.e. O K has Krull dimension 1). So, we can write Sec(O K ) SecOK Sec(O ), and we exect to be able to recover O K from its localizations at all the rimes. In fact, we can recover any ideal of O K. Note that the localizations of O K are all embedded in its field of fractions K, so we can take their intersections, arriving at Lemma 4.1. For any ideal I O K, we have I IO, where the intersection is over every rime in O K. Proof. I O for all Sec(O K ), so the inclusion I IO is immediate. For the oosite inclusion, let x I be an element of K. Then we can consider the ideal of ossible denominators for x, namely J {a O K : xa I}. If 1 J, then x I, a contradiction. So J is a roer ideal, and it is contained in some maximal (i.e. rime) ideal m O K. Then, if x IO m we know that x a/s where a I and s O K m O K J, which means sx O K where s O K J, a contradiction. It follows that and we have the desired equality. x I x IO, This gives a useful slogan that will reduce many roblems to the case of a DVR: in order to show an equality of ideals, it suffices to show the equality for their localizations at each rime. In articular, this brings us much closer to the situation in Section 2, in which the bottom ring is a PID. It is a general fact from algebra that this situation yields a basis, as was exloited in the roofs of Proositions 2.4 and 2.2. Proosition 4.2. Let L/K be a finite extension of characteristic zero, let A K be a Dedekind domain with field of fractions K, and B be its integral closure in L. Then B is a finitely-generated A-module. Proof. Recall from the discussion on the different ideal that we have a nondegenerate airing, : L L K

10 10 KENZ KALLAL given by x, y Tr L/K (xy). Let x 1,..., x n be a basis for L over K. Clearing denominators in the minimal olynomial for x i, if the least common denominator of the coefficients is a i A, we can observe that a i x i L is integral over A, i.e. a i x i B. So, we have a free A-module San A (x 1,..., x n) B. Taking duals, and using the identification V V given by the nondegenerate airing, we get inclusions San A (x 1,..., x n) B B San A (x 1,..., x n). But it is easily seen that San A (x 1,..., x n) is sanned over A by the dual basis to x 1,..., x n, so it is a free A-module of finite rank. Since A is Noetherian (it is Dedekind), B and B are submodule of a Noetherian A-module (direct sums of Noetherian modules are Noetherian), hence B is finitely generated as an A-module. This is also the reason why B is finitely generated and thus is a fractional ideal of B. From now on, A, B, L, K are assumed to be as in the hyotheses of Proosition 4.2. Corollary 4.3. If A is a PID, then B is a free A-module of finite rank Proof. This follows from the revious roosition and the structure theorem for modules over PIDs. This makes localization useful in a concrete way: it lets us use, as in the roof of Proositions 2.4 and 2.2, an A-basis for B Localization and Ramification. The machinery we have develoed so far allows us to rove an imortant relation between the relative different and discriminant in a general setting; see [25, Ch. III, Theorem 2.9]. Theorem 4.4. Let L/K be an extension of number fields, A K a Dedekind domain with field of fractions K, and B its integral closure in K. Then B/A N L/K (D B/A ), where N L/K (I) denotes the ideal in A generated by the norms of elements in B (a.k.a. the relative norm ideal ). Proof. Suose that A is a PID. Then by the corollary, B has an A-basis ω 1,..., ω n. What s more, the fractional ideal D 1 B/A in B has an A-basis ω 1,..., ωn dual to ω 1,..., ω n under the trace form. Since it is a Dedekind domain with finitely many rime ideals, B is a rincial ideal domain, so D 1 B/A is generated by some d K as a fractional ideal. In summary, the situation is that D 1 B/A bb Aω 1b Aω n b Aω 1 Aω n. Comuting discriminants, it follows that disc L/K (ω 1 b,..., ω n b) disc L/K (ω 1,..., ω n ), and we obtain from the definition of the discriminant disc L/K (ω 1 b,..., ω n b)a N L/K (b) 2 disc L/K (ω 1,..., ω n )A N L/K (b) 2 B/A.

11 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 11 Here we used the fact that ω 1,..., ω n is an A-basis for B to write the relative discriminant as the rincial ideal generated by the discriminant of the basis. Since D 1 B/A is rincially generated by b, the norm of b is just the relative norm of this fractional ideal, and we can write disc L/K (ω 1,..., ω n )A N L/K (D 1 B/A )2 B/A. Finally, the same analysis that allows us to write the discriminant in terms of the trace form, and the definition of the dual basis gives disc L/K (ω 1,..., ω n )disc L/K (ω 1,..., ω n ) 1, so we can multily both sides of the revious equation by disc L/K (ω 1,..., ω n ) B/A and rearrange to get N L/K (D B/A ) 2 2 B/A, which imlies the desired result by unique factorization. Now, we return to the original case of A O K and B O L. By the revious remarks, it suffices to show that S 1 OL /O K S 1 N L/K (D OL /O K ) for all S of the form O K \ where is a rime in O K. Of course, S 1 OL /O K S 1 O L /S 1 O K (both inclusions follow by looking at the generators) and S 1 D B/A D S 1 B/S 1 A which follows from the fact that localization is comatible with duals and inverses. Thus, it suffices to show S 1 O L /S 1 O K N L/K (D S 1 O L /S 1 O K ) since localization is obviously comatible with taking relative norms. By Lemma 3.3, S 1 O K is a DVR and thus a PID. This is what we showed in the secial case above, so we are done. Of course, we really want to be able to rove statements about divisibility of ideals by and ramification. First, we recall a basic fact about localizations of ideals [15, Ch. 1, Proosition 16] Proosition 4.5. Let S be a multilicative set in A. The ma I S 1 I is a homomorhism from the grou of fractional ideals of A to the grou of fractional ideals of S 1 A. Though we could make use of more general roerties of localization, e.g. its inclusion-reserving roerty on ideals, or even Proosition 4.1, it ends u being easiest to aly the above rincile and unique factorization of ideals in Dedekind domains. Lemma 4.6. Let I be an ideal in A and let n N. Then I is divisible by n if and only if I is divisible by n A. Proof. Since A is a Dedekind domain, we know I factors as I q νq, q Sec(A)

12 12 KENZ KALLAL and by the roosition, I q Sec(A) from which the statement of the lemma follows. q νq A ν A Of course, O is a DVR and its ideals are all of the form n O (π n ) for any choice of uniformizing arameter π 2 so the lemma just means that divisibility by n is equivalent to the localization being divisible by π n in O. Thus, divisibility by owers of (i.e. ramification) and equality of ideals are both comatible with localization at in some formal sense. Abusing this machinery, we can reduce any statement about divisibility or equalities of ideals in O K to the same statement in O, which has a much simler ideal theory. On the other hand, we want to be able to use this machinery to generalize Proositions 2.2 and 2.4 to the general case of the extension of Dedekind domains B/A with fields of fractions L/K such that B is the integral closure of A in L (whereas before we assumed K Q). To do this, we would want to examine the ramification at the airs of rimes q in O L and O K. So, it would make sense to localize O K at and to localize O L at q. While this localization might reserve the ramification e(q ) and divisibility of ideals by owers of rimes, we don t exect it to be as useful because it does not necessarily reserve the fact that B is integral over A. Instead, we consider one rime of A at a time, and localize both rings at S A. Lemma 4.7. Let be a rime in A, and consider the multilicative set S A \. If q is a rime in B lying over, then Proof. Observe that e(q ) e(s 1 q S 1 ). (S 1 )B (A )B B S 1 (B). Since B is a Dedekind domain, B factors into rimes as B q νq q νq. q SecB q SecB q A It follows from the roosition that (S 1 )B S 1 (B) q SecB q A (S 1 q) νq. Recall that if q does not lie over, then S 1 q B. Otherwise, S 1 q are distinct rimes of B for distinct rimes q in A. So the lemma follows from unique factorization of ideals in the Dedekind domain S 1 B. As a result, localization at S A reserves ramification of rime ideals in B lying over, it reserves divisibility of ideals in A by, and it reserves divisibility of ideals in B by rimes lying over. The desired results on the fundamental roerties of the discriminant and different ideals soon follow: Corollary 4.8. Let L/K an extension of number fields, A K a Dedekind domain with field of fractions K and B its integral closure in L. If is a rime in A, then B/A if and only if e(b ) > 1.

13 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 13 Solution. By Lemma 4.6, B/A if and only if S 1 S 1 B/A S 1 B/S 1 A, where S A. Moreover, by Lemma 4.7, e(s 1 (B) S 1 ) e(b ), so the conclusion of the lemma is equivalent to S 1 S 1 B/S 1 A e(s 1 (B) S 1 ), now a statement about rime ideals in the ring extension S 1 B/S 1 A. Recall that S 1 A A is a DVR, so in articular it is a PID. We have already shown this case of the above statement (by abusing the fact that there is a basis for B over A), so the result follows. Corollary 4.9. Let L/K be an extension of number fields, A K a Dedekind domain with field of fractions K and B its integral closure in L. If q is a rime in B, then e(q q A) > 1 if and only if q D B/A. Proof. By the above discussion, the result is equivalent to S 1 q D S 1 B/S 1 A e(s 1 q S 1 q S 1 A) > 1 where S A A q. Of course, S 1 A is a PID, so we showed earlier that this result holds (S 1 q is a rime ideal in S 1 B by the corresondence). 5. Local Methods Taking the valuation ring of K under ν allowed us to ass to a situation in which A is a DVR. As we will demonstrate, it will be useful to be able to ass to a comlete DVR Comletions of Valued Fields. To obtain a comlete valued field containing K, we can take the comletion of K with resect to the toology induced by. The easiest way to obtain a field structure for the toological comletion of K is to let R be the ring of Cauchy sequences in K, and m be the set of sequences in R which converge to zero. One can check that m is a maximal ideal, so that K : R/m is a field equied with an embedding K K taking a field element to the residue class of the corresonding constant sequence. Finally, one can check from the triangle inequality that takes any Cauchy sequence in K to a (therefore convergent) Cauchy sequence in R, so we can extend to an absolute value on K by taking {a n } lim n a n, noticing that this does not deend on the reresentative Cauchy sequence, verifying that it yields a valid non-archimedian absolute value on K, and that K is comlete under the toology induced by it. We write ˆ for this absolute value on the comletion, and νˆ for the corresonding non-archimedian valuation. Since νˆ is obtained by a continuous transformation of ˆ, we know that for all x K, νˆ (x) lim n ν (x n )

14 14 KENZ KALLAL where {x n } is a Cauchy sequence in K reresenting x. Since the image of νˆ is Z { }, it follows that the sequence {ν (x n )} is eventually stable, and thus ν (K) νˆ ( ˆK ) Z { }. Let A be the valuation ring of K. The valuation ring of ˆK is  : {x ˆK : νˆ (x) 0}. If x Â, then 1/x is, so  has field of fractions K (this is a general fact about valuation rings). Every x  is the limit of a sequence in K whose valuation is eventually equal to νˆ (x) 0, so it is a limit oint of the valuation ring A of K. Conversely any limit oint of A must have nonnegative valuation (since the elements of A do), so  is the closure of A in ˆK. In articular,  is a DVR (it is the valuation ring of the discrete valuation νˆ ) and since it is a closed subset of the comlete metric sace K it is also comlete under its valuation. Since the discrete valuation on  extends that on A, any uniformizing arameter π for A (i.e. an element of valuation equal to 1) is also a uniformizing arameter for Â. The maximal ideal of  is the set of elements of ositive valuation, which by the same argument as before is the closure (i.e. comletion) of in K. Note that this is equal to ˆ : πâ (πa)â Â, so the closure of is equal the ideal it generates in Â. Since A is a DVR, its ideals are all of the form π n A for some n 0. Thus, the same argument above alies: we know that the closure of I π n A in ˆK is Î {x ˆK : ν (x) n} π n  IÂ. Hence, the ideals of A are in natural bijection with those of  under the ma I Î, and Î is equal to the closure of I in Â, as well as the ideal in  generated by I. Our goal is tyically to show a result about the ideals of O K, or more generally a Dedekind domain A whose field of fractions is K. In that case, the valuation ring of K is S 1 A, where S A. For any ideal I A, we just showed that the closure of S 1 I is (S 1 I) I(S 1 A)(Â) IÂ. Let n 1 and a/s S 1 I. Since s, there exists a b A such that bs 1 (mod n ). Then we have ab I and ν ( a s ab ) ν ( a s (1 bs) ) n. It follows that I is dense in S 1 I, so the closure of I in ˆK is just IÂ. In summary: assing from A to S 1 A kees only the ideal-theoretic information related to divisibility by owers of, while assing to  loses no additional information about the ideals. Let B/A be an extension of Dedekind domains with field of fractions L/K such that B is the integral closure of A in L. We used in the revious section the fact that localization reserves integral closures: S 1 B/S 1 A is a ring extension with the same roerties as above, but S 1 A is now guaranteed to be a DVR. Taking comletions allows us to do the same thing, excet both rings are DVRs (recall that the localization of B at a rime q

15 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 15 instead of at S is not necessarily integral over S 1 A). In articular, from [29, Ch.2, Proosition 3], Proosition 5.1. Let L/K be a finite extension of number fields, let A be a Dedekind domain with field of fractions K, and let B be its integral closure in L. Let q be a rime of B lying over the rime of A. Then the integral closure of  in ˆL q is ˆB q. Proof. First, observe that [ˆL q : ˆK ] [L : K] <. This is because any element x ˆL q is the limit of some Cauchy sequence of elements in L, each of which can be exanded in terms of a K-basis l 1,..., l n for L. So, we can write x {a i,1 l a i,n l n } i 1 and verify that since this sequence is Cauchy, each sequence {a i,j } i 1 is a Cauchy sequence that haens to live in K. It follows that x {a i,1 } i 1 l {a i,n } i 1 l n which means l 1,..., l n san ˆL q over ˆK. Let B be the integral closure of  in ˆK q. It is our goal to show that B ˆB q. Since [ˆL q : ˆK ] <, clearing denominators in the minimal olynomial of any x ˆL q tells us that ax B for some a Â. Therefore, FracB ˆL q. Since  is a DVR, it has only one rime ˆ, and all the rimes q i of B lie over it. Each q i defines a (toologically) inequivalent valuation on B (and hence on its field of fractions ˆL q ), which can be made to extend νˆ by scaling by e(q i ). It follows that each q i defines an inequivalent norm on ˆL q as a finite-dimensional ˆK -vector sace. But ˆK is comlete, so all norms on this finite-dimensional ˆK -vector sace are equivalent. Hence, B has only one rime ideal. Since it is a Dedekind domain, this means B is already a DVR with maximal ideal q, and the toology on ˆL q defined by ν q is the same as the one defined by ν q. Hence, the valuation ring of ˆL q is the same with resect to both valuations (the valuation ring can be defined urely in terms of the toology as the set of elements whose inverses do not tend to zero as they are raised to a ower n ). By Lemma 3.3 and our original definition of ˆB q, the valuation ring of ˆL q is ˆB q (B q ) 1 B. But B q consists of the units of B since it is a DVR, so the result is B ˆB q as desired. Tyical exositions on extensions of comlete fields (cite Neukirch and Lang) exloit Hensel s lemma [cite it] to rove the above facts and to show that the unique absolute value extending ˆ is actually given by x N ˆLq/ ˆK q (x) 1/[ ˆL q: ˆK q]. We will not require any of these facts for our uroses, however. Finally, we observe that comletion reserves ramification indices and inertial degrees: If q SecB lies over SecA, then we have ν e(q )ν q

16 16 KENZ KALLAL on A. On the other hand, ˆq, the maximal ideal of ˆB q, lies over ˆ, the maximal ideal of Â. We know that νˆq agrees with ν q on B, and the same is true on A for νˆ and ν. Thus, on A we have ν e(ˆq ˆ)ν q from which it follows that e(ˆq ˆ) e(q ). For the inertial degree, we use the standard fact that taking comletions induces an isomorhism of residue fields A/ Â/ˆ, since every Cauchy sequence in A eventually has constant image in A/. The same holds for B, so we have a commutative diagram B/q ˆBq /ˆq A/ Â /ˆ which yields f(q ) [B/q : A/] [ ˆB q /ˆq : Â/ˆ] f(ˆq ˆ) Extensions of DVRs. By bringing us to the case of an extension of DVRs, assing to comletions allows us to exloit the following result from [29, Ch. III, Proosition 12]. Proosition 5.2. Let A, K, L, B be such that A and B are DVRs with searable residue field extension κ(b)/κ(a). Then there exists an α A such that B A[α]. Proof. Let f f(q ) and e e(q ). Since κ(b)/κ(a) is searable of degree f, it admits a rimitive element ˆx (take x B to be an arbitrary lift of the rimitive element). Take π B to be an arbitrary uniformizing arameter for B. Since q is the only rime lying over we know that [L : K] ef and since A is a PID, B is automatically a free A-module of rank ef. We claim that the elements x i π j B for 0 i < f and 0 j < e are an A-basis for B. Since there are ef of them, it suffices to show that they generate B as an A-module. By Nakayama s lemma, it suffices to show that their residues generate B/B as an A/-vector sace. We know B πb e B, and the owers of x generate B/q B/π BB as a A/vector sace, so by multilying by the first e owers of π B we obtain a sanning set, as desired. To show that the owers of x san B as an A-module, it therefore suffices to show that we can obtain a uniformizing arameter for B as a linear combination of owers of x. To do this, take P κ(a)[x] to be the minimal olynomial of x over κ(a), and take an arbitrary lift of the coefficients of P to A to obtain a olynomial Q A[X] with the roerty that Q(x). In other words, ν q (Q(x)) 1. If equality holds, then Q(x) is a uniformizing arameter for B, and we are done. Otherwise, note that x + π B x, so x + π B is also a lift of a rimitive element for κ(b)/κ(a). In this case, however, we can comute (by Taylor exansion of olynomials) Q(x + π B ) Q(x) + π B Q (x) + πbb 2

17 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 17 where b B here we have collased all of the terms of valuation greater than 1. Moreover, κ(b)/κ(a) is searable, so the minimal olynomial P of the rimitive element x cannot share a root with P. Hence, P ( x) 0, which means that Q (x) is not divisible by π B. It follows that ν q (Q(x + π B )) 1, so Q(x + π B ) is a uniformizing arameter for B. The resulting basis (consisting either of roducts of owers of x and Q(x) or of x + π B and Q(x + π B )) is a ower basis for B over A, as desired Extending Scalars by Comletions. To be able to use this roerty to analyze the different ideal, it still remains to show that the different is comatible with taking comletions. We showed already that L sans all of ˆL q over ˆK. In other words, the multilication ma L K ˆK ˆL q is a surjective ma of ˆK -vector saces. A careful analysis of L K ˆK yields a concrete way in which the data of L is related to that of its comletions, from [29, Ch. II, Théorème 1]: Proosition 5.3. The multilication ma ϕ : L K ˆK q ˆL q is an isomorhism of ˆK -vector saces. Proof. By extending scalars of a basis for L over K (i.e. tensoring each basis element with 1 ˆK ), we can see that L K ˆK is a K-vector sace of dimension equal to [L : K]. The fundamental identity involving ramification indices and inertial degress (see Marcus) and the above remarks about ramification being reserved by comletion tells us that that this dimension is [L : K] q e(q )f(q ) q e(ˆq ˆ)f(ˆq ˆ) which is the dimension over ˆK of q ˆL q. It therefore suffices to show that ϕ is surjective. Since ˆK is comlete, any (finite-dimensional) subsace of ˆL q q is comlete and therefore closed. So, it suffices to show that the image of ϕ is dense in the codomain (since the image must be closed). Since all norms on a finitedimensional vector sace over a comlete field are equivalent, we might as well take the norm on the codomain to be the su norm. The conclusion follows from an imortant result from the theory of valued fields: Lemma 5.4 (Weak Aroximation). Let 1,..., m be airwise inequivalent absolute values on a field K. For any choice of a 1,..., a m K and ε > 0, there exists an x K such that x a i < ε for each i. Proof. See [25, Ch. II, Theorem 3.4] Since L is dense in each ˆL q, any element of ˆL q q can be aroximated by elements whose coordinates are all in L, which can in turn (by weak aroximation) can be aroximated by elements of the form ϕ(l 1). So the image in ϕ is dense in the codomain, as desired.

18 18 KENZ KALLAL The same is true for the base change of B under some hyotheses on A (note that we can achieve these hyotheses by localizing at first). Proosition 5.5. Suose A is a PID. Then the multilication ma ϕ : B A  q ˆB q is an isomorhism of A-modules. Proof. Since A is a PID, B is a free A-module of rank [L : K]. By extending the scalars of a basis for B over A, we get that B A  is a free  Since  is a DVR (and thus a PID) for each q, we also know that ˆB q is a free Â-module of rank [ˆL q : ˆK ]. Thus, the codomain is a free Â-module of rank [ˆL q : ˆK ] [L : K] q as well. This means it suffices to show that ϕ is surjective, i.e. that the image of a generating set for B A  is a generating set for the codomain. Since ˆ is the only maximal ideal of Â, it suffices to show this modulo ˆ by Nakayama s lemma. We have a canonical identifications A/ n Â/ˆ n for all n 0 (the argument is the same as for n 0: all Cauchy sequences must eventually have constant image under the rojection ma). In other words, it suffices to show that the ma (B A  )/ˆ ˆB q /ˆ q is surjective. We have an isomorhism of A-modules and q (B A  )/ˆ B A (Â/ˆ) B A (A/) B/B ˆB q /ˆ q ˆB q /ˆ ˆB q q ˆB q /ˆq e(q ) B/q e(q ). Following these identifications, the ma we want to show is surjective is identified with the rojection B/B q B/q e(q ) q which is indeed surjective by the Chinese remainder theorem. Finally, can see that the different may be defined locally: Proosition 5.6. Suose q lies over. Then D B/A ˆBq D ˆBq/Â. Proof. First, suose that A is a DVR (we will reduce to this case by localizing). The revious roosition tells us that B A  ˆB q q

19 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 19 via the multilication ma. Finitely many rimes lie over so this roduct is a direct sum of  -modules embedded in L K ˆK q ˆL. Moreover, the trace form on L extends to a ˆK -bilinear form on L K x, y q Tr ˆLq/ ˆK (xy) ˆK given by so that the dual module of B A  in L K ˆK is identified with q ( ˆB q ). Since A is a DVR, there is a basis e 1,..., e n for B and a dual basis e 1,..., e n for B the dual of B in L. By comuting traces, we can observe that e 1 1,..., e n 1 is a basis for the dual module of B A Â. This yields another identification of  -modules ( ˆB q ) (B A  ) B A Â. q It follows that B generates the fractional ideal ( ˆB q ) in ˆB q. D B/A generates D in ˆB ˆBq/ q. In the general case, let S A. Then we have D B/A S 1 B D S 1 B/S 1 A. Alying our revious analysis to S 1 B/S 1 A, we get D S 1 B/S 1 A ˆB q D ˆBq/Â. Plugging in the revious identification yields the desired result. Taking inverses, 5.4. Kähler Differentials and the Different. The main theorem of this section characterizes the different in terms of another structure from algebraic geometry: Definition 5.7. Let A be a ring and B an A-algebra. The module of Kähler differentials of B/A, denoted by Ω B/A, is obtained by taking the free B-module generated by the symbols dx for x B and quotienting by the relations for all x, y B and a A. d(x + y) dx + dy d(xy) xdy + ydx da 0 We can view the elements of Ω B/A as images under the ma d : B Ω B/A with the three roerties above. Such a ma is called an A-derivation, and in fact Ω B/A is characterized by a universal roerty using A-derivations: Proosition 5.8. For any B-module M and A-derivation d : B M, there exists a unique B-linear ma α : Ω B/A M such that d α d. Using the universal roerty, one can show a few basic roerties of the module of differentials: Proosition 5.9. Let A be an A-algebra. Then as B-modules. Ω (B A A )/A ΩB/A B (B A )

20 20 KENZ KALLAL Proof. See [6, Proosition 16.4]. Proosition Let S be a multilicative subset of B. Then Ω S 1 B/A S 1 Ω B/A Ω B/A B S 1 B. Proof. See [6, Proosition 16.9]. Proosition Let {B α } α I be a collection of A-algebras. Then Ω α I Bα/A α I Ω Bα/A Proof. See [6, Proosition 16.10]. Proosition The module of differentials is comatible with taking comletions and localization: If q and S A then and S 1 Ω B/A Ω B/A B S 1 B Ω S 1 B/S 1 A Ω B/A B ˆBq Ω ˆBq/Â. Proof. The first art of the roosition follows from Proosition 5.9, since S 1 B B A S 1 A. For the second art, assume first that A is a DVR, so that Proosition 5.5 alies. In this case, by Proosition 5.11 we have Ω B A Â Ω /Â q ˆB q/â Ω ˆBq/Â and Proosition 5.9 along with the fact that finite direct roducts are the same as direct sums gives ΩB/A B ˆB q Ω B/A B ˆBq. Ω B A Â /Â Putting this all together, we have an isomorhism Ω B/A B ˆBq q q Ω ˆBq/Â q which clearly mas comonentwise, giving the desired isomorhism q q Ω B/A B ˆBq Ω ˆBq/Â. Finally, in the general case we aly the above reasoning to S 1 B/S 1 A to get Ω ˆBq/Â Ω S 1B /S 1 A S 1 B ˆB q Ω B/A B S 1 B S 1 B ˆB q Ω B/A B ˆBq by the first art of the roosition. Theorem Let L/K be an extension of number fields. Then D OL /O K Ann OL (Ω OL /O L ). Proof. By Lemma 4.1, it suffices to show that the desired equality holds when both sides are localized at all rimes of q of B. So we fix a rime q lying over a rime of A. Let S B q. It suffices to show that S 1 D B/A S 1 Ann B (Ω B/A ).

21 RAMIFICATION IN ALGEBRAIC NUMBER THEORY AND DYNAMICS 21 Since the ideals of S 1 B are in bijection with the ideals of ˆB q (see the beginning of this section), it suffices to show that D B/A ˆBq Ann B (Ω B/A ) ˆB q. Moreover, ˆB q is a flat B-module (see Eisenbud 7.2) so Ann B (Ω B/A ) ˆB q Ann ˆBq (Ω B/A B ˆB q ) which means the by Proosition 5.12 it suffices to show D ˆBq/Â Ann ˆBq (Ω ˆBq/Â), i.e. we may assume that A and B are DVRs, so by Proosition 5.2 B A[β] for some β B with minimal olynomial f(x) A[x]. By comuting a dual basis (see [25, Ch. III, Proosition 2.4]) one sees that both sides of the desired equation are the ideal generated by f (β). 6. Ramification Grous Let L/K be a finite Galois extension of comlete valued fields of characteristic zero (actually searable is enough but we never roved that the trace form is nondegenerate in searable extensions of ositive characteristic), let B/A be their valuation rings with maximal ideals q, and suose they have searable residue field extension. By Proosition 5.2, there is some β B such that B A[β]. The grou G Gal(L/K) is equied with a decreasing series of subgrous: Definition 6.1. Let i 1. The i-th ramification subgrou of G is defined by G i : {σ G : ν q (b σ(b)) i + 1 for all b B}. Since B is the integral closure of A in L, note that G acts on B. Note that for σ G, the following are equivalent: ν q (b σ(b)) i + 1 for all b B ν q (β σ(β)) i + 1. σ acts trivially on B/q i+1. For σ G, take i G (σ) : ν q (β σ(β)). It is clear from the second characterization that i G (σ) 1 su{i 1 : σ G i }. If σ id, β σ(β) 0, so i G (σ) <. Since G is finite, we can take the maximum of i G over all non-identity elements to get that all σ id are eventually not in G i, i.e. the G i s are eventually trivial. One connection between the ramification grous and ramification comes from the different ideal. Since B is a DVR, D B/A is determined by the q-adic valuation of its generating element. Abusing notation, we denote this by ν q (D B/A ). Theorem 6.2. The different is given by ν q (D B/A ) σ G σ id i G (σ) i 0( G i 1). Proof. Recall that since B A(β), the different is generated f (β) where f A[X] is the minimal olynomial of β. Thus, it suffices to comute ν q (f (β)). By Galois theory, f(x) σ G(X σ(β))