COLORADO STATE UNIVERSITY MATHEMATICS Higher Ramification Groups Dean Bisogno May 24, 2016 1 ABSTRACT Studying higher ramification groups immediately depends on some key ideas from valuation theory. With that in mind, this paper will outline the essential results from valuation the- ory before proceeding to higher ramification groups. Higher ramification groups arise when studying extensions of fraction fields of Dedekind rings. Ramification groups are also useful when studying wildly ramified extensions of function fields. The goal while studying rami- fication in a wildly ramified extension of function fields is to calculate the exponent of the different. By Hilbert’s different theorem, the exponent of the different can be calculated by finding orders of higher ramification groups and ramification jumps. Higher ramification groups can also be used to study the subgroup structure of a Galois group, as the higher ram- ification group filtration can contain information about Sylow p-subgroups. The aim of this paper is to understand at a general level what higher ramification groups are, and investigate a couple examples of their applications. 2 PRELIMINARIES 2.1 VALUATIONS Though valuations need not be discrete, many results discussed in this paper either require discrete valuations or are most easily understood with respect to a discrete valuation. Thus discrete valuations will be the focus of this section. Definition 1 (Discrete Valuation Ring (DVR)). A discrete valuation ring is a principal ideal domain O with a unique maximal ideal p 0. 6Æ Example 1. Consider the ring Z and prime p Z. The localization Z p is a local PID with unique maximal 2 ( ) ideal pZ(p). Then Z(p) satisfies the definition of a discrete valuation ring. Example 2. Consider k[[x]], the formal power series of the field k. It was shown in Dummit and Foote, 1 2004, 7.3 #3, that all proper ideals of k[[x]] are of the form (xn) (this is actually a corollary to what is shown in that problem). Further (x) is maximal and contains all other proper ideals. It follows then that (x) is the unique maximal ideal in k[[x]] and all other proper ideals are principal. Thus k[[x]] is a DVR. Definition 2 (Uniformizing Parameter). Let p be the unique maximal ideal of DVR O. Since O is a PID, there exists a prime ¼ in O such that p (¼). Such a ¼ is called a uniformizing parameter. Æ Theorem 7, section 16.2 in (Dummit and Foote, 2004) shows that ¼ is unique (up to multi- plication by a unit) and generates the unique maximal ideal of O. Further in problem 26, in section 7.1 of Dummit and Foote, 2004, (proven in math 566) it is shown that O \p is the set of units in O. Then for any a O \{0}, (a) is an ideal of O. Because (a) is a nonzero ideal of O,(¼) 2 is maximal, and (a) is in a local ring O; there exists some n Z such that (a) (¼)n. Notice if 2 Æ a is a unit, then n 0 because (a) O. Æ Æ Definition 3 (Discrete Valuation). The exponent n used above is the valuation of a, denoted vp(a), it is a function vp : K ¤ Z ! where K is the field of fractions of O. The valuation vp can be extended to K by setting vp(0) Æ . Further vp satisfies the following 1 I. vp(ab) vp(a) vp(b). Æ Å II. vp(a b) min{vp(a),vp(b)}. Å ¸ All valuations can be classified as either archimedean or nonarchimedean. A valuation v is nonarchimedean if v(n) is bounded for every n N. Proposition 3.6 in Neukirch, 1999, 2 is often incorporated in the definition of discrete valuations, as every discrete valuation is nonarchimedean. Proposition 1. A valuaion v(x) is nonarchimedean if and only if it satisfies v(x y) max{v(x),v(y)}. Å · Proof. ( ) The reverse direction is straight forward. Just notice ( v(n) v(1 ... 1) 1 n N. Æ Å Å · 8 2 ( ) Now, v(n) N for all n N for some N N. Then for arbitrary x, y K , without loss ) · 2 2 l n l 2 of generality, consider v(x) v(y). Choose l 0, then v(x) v(y) ¡ v(x). Now applying ¸ ¸ · binomial formula yields n n X ¡n¢ l n l n v(x y) v l v(x) v(y) ¡ N (n 1)(v(x)) Å · l 0 · Å Æ taking the nth root of both sides yields 1 1 1 1 v(x y) N n (1 n) n v(x) N n (1 n) n max{v(x),v(y)} Å · Å Æ Å The result then follows if letting n . The conclusion is then for any discrete valuation ! 1 (which is nonarchimedean) v(x y) max{v(x),v(y)}. // Å · 2 The following proposition is particularly useful because it provides a correspondence be- tween prime ideals and valuations in Dedeking rings. Proposition 2 (Neukirch, 1999). Let O be a noetherian integral domain. O is a Dedekind domain if and only if, for all prime ideals p 0, the localizations Op are discrete valuation rings. 6Æ When the previous proposition is applied to Z and the fraction field Q is consider, the follow- ing proposition is gained. Proposition 3 (Neukirch, 1999). Every valuation of Q is equivalent to one of ºp (x) (the nonarchimedean valuations) or º (x) 1 (the archimedean valuations). Finally, since ramification occurs via extensions or covers, it will be useful to know how valu- ations extend. Definition 4 (Prolongation of a valuation). If A B are rings with B integral over A, p a prime in A, and q prime in B dividing p with ½ ramification index eq then vq(x) eqvp(x) is a prolongation of vp with index eq (Serre, 1995). Æ 2.2 FIELD COMPLETION With a valuation º on a field K then for any real number a (0,1) an absolute value on K can 2 be defined by ( aº(x) x 0 x 6Æ jj jj Æ 0 x 0 Æ which satisfies the usual conditions of a metric as outlined by Serre, 1995, II.1. A topology is then induced on K via the absolute value metric, and the completion of K with respect to the valuation º is denoted by Kb. Note also that the metrics induced by different choices of a are topologically equivalent, so the completion is dependent only on º. Further, º extends in the completion of K to a valuation (which will continue to be called º) on Kb (Serre, 1995). The Galois group of Lb/Kb relates to the decomposition group of L/K in the following way. Proposition 4. For L/K Galois with group G, ºp(x) a valuation on K and !q(x) a prolongation of ºp, let L,b Kb be the completions of L and K with respect to !q and ºp respectively. If Dq(L/K ) is the decom- position group of q over p, then Lb/Kb is Galois with Galois group D(L/K ) (Serre, 1995). Proof. This is corollary 4 to theorem 1 in Serre, 1995, II.3, which shows that [Lb : Kb] e f . As Æ proven in class D(L/K ) e f . // j j Æ 3 HIGHER RAMIFICATION GROUPS Consider now K , the fraction field of a Dedekind ring OK , with valuation º and uniformizing parameter ¼K . For Galois extension L/K with Galois group G; prolongation of º, !, on L; 3 uniformizing parameter ¼L of !; and discrete valuation ring OL. The definitions of the higher ramification groups generalizes the definition of the inertia group. Definition 5 (Higher Ramification Groups, Serre, 1995). For every real number i 1 define the i th ramification group of L/K with respect to ! by ¸ ¡ G G (L/K ) {σ G !(σ(a) a) i 1 a O }. i Æ i Æ 2 j ¡ ¸ Å 8 2 L Note that because º corresponds to a prime p and ! corresponds to a prime q p, the rami- j fication groups are generalizations of the decomposition and inertia groups. The following is a proposition which helps characterize the higher ramification groups with respect to the presentation of valuations above. Proposition 5. i 1 For any a O and σ G, !(σ(a) a) i 1 if and only if σ(a) a mod (¼ ) Å . 2 L 2 ¡ ¸ Å ´ L Proof. ( ) ) t x ºL(σ(a) a) i 1 σ(a) a ¼ ¡ ¸ Å Æ) ¡ Æ L y where t i 1 and x, y,¼L OL all relatively prime. This follows from the definition of the ¸ Å 2 i 1 valuation in the first section. The equation above then implies σ(a) a 0 mod (¼ ) Å . ¡ ´ L ( ) ( i 1 t x σ(a) a 0 mod ¼ Å σ(a) a ¼ ¡ ´ L Æ) ¡ Æ L y with the same restrictions as above. It follows then that º (σ(a) a) i 1. // L ¡ ¸ Å The condition that º(σ(a) a) i 1 clearly induces the following filtration on G ¡ ¸ Å {Gi (L/K )}i 1 G 1 G0 G1 .... ¸¡ Æ ¡ ¶ ¶ ¶ This filtration can be investigated further due to the fact that i, G /G. 8 i i 1 i 1 Proof. Consider ¿ G. By membership of G, !(¿(a) a) 0 a OL, i.e. ¿((¼L) Å ) (¼L) Å 2 ¡ ¸ 8i 21 Æ for all i. Thus there is an induced automorphism ¿¯ of OL/(¼LÅ ). Now consider Ái 1 : G i 1 Å ! aut(OL/(¼ Å ) such that Ái 1(¿) ¿¯.