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Explicit Integral Galois Module Structure of Weakly Ramified Extensions of Local Fields

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Explicit Integral Galois Module Structure of Weakly Ramified Extensions of Local Fields PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 12, December 2015, Pages 5059–5071 http://dx.doi.org/10.1090/proc/12634 Article electronically published on May 7, 2015 EXPLICIT INTEGRAL GALOIS MODULE STRUCTURE OF WEAKLY RAMIFIED EXTENSIONS OF LOCAL FIELDS HENRI JOHNSTON (Communicated by Romyar T. Sharifi) Abstract. Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G =Gal(L/K). Let G1 and G2 be the first and second ramification groups. Thus L/K is tamely ramified when G1 is trivial and we say that L/K is weakly ramified when G2 is trivial. Let OL be the valuation ring of L and let PL be its maximal ideal. We show that if L/K ≡ | | n is weakly ramified and n 1mod G1 ,thenPL is free over the group ring OK [G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every free generator of PL over OK [G]isalsoafreegeneratorofOL over its associated order in the group algebra K[G]. Along the way, we prove a ‘splitting lemma’ for local fields, which may be of independent interest. 1. Introduction Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G =Gal(L/K). Let OL be the valuation ring of L and let PL be its maximal ideal. We recall that for i ≥−1 the ramification groups of L/K are { ∈ | − O ⊆ i+1} Gi := σ G (σ 1)( L) PL . Thus L/K is unramified if and only if G0 is trivial and is tamely ramified if and only if G1 is trivial. We say that L/K is weakly ramified if and only if G2 is trivial. In the case that L/K is weakly ramified we shall consider the structure of both n ≡ | | O O fractional ideals PL with n 1mod G1 over the group ring K [G], and of L over its associated order AL/K := {x ∈ K[G] | xOL ⊆OL}. A result often attributed to E. Noether is that if L/K is tamely ramified, then OL is free (of rank 1) as a module over the group ring OK [G]; in fact as noted in [Cha96, §1] she only stated and proved the result in the case that the residue characteristic of K does not divide |G| (see [Noe32]). Ullom [Ull70] proved the following: L/K is tamely ramified if and only if every non-zero fractional ideal in L is free over OK [G]; if any non-zero fractional ideal of L is free over OK [G], then L/K must be weakly ramified; and if L/K is totally and weakly ramified, then PL is free over OK [G]. K¨ock [K¨oc04, Th. 1.1] used cohomological methods to show the n O more general result that PL is a free K [G]-module (of rank 1) if and only if L/K is weakly ramified and n ≡ 1mod|G1|; this also follows from a minor variant of work of Erez [Ere91, Th. 1] on the square root of the inverse different or can be proved by the methods developed in Ullom’s papers [Ull69a, Ull69b, Ull70]. Received by the editors August 20, 2014 and, in revised form, September 15, 2014. 2010 Mathematics Subject Classification. Primary 11R33, 11S15. c 2015 American Mathematical Society 5059 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5060 HENRI JOHNSTON The results discussed above do not give explicit generators. However, Kawamoto [Kaw86] gave an elementary proof of the fact that if L/K is tamely ramified, then OL is free over OK [G], and constructed an explicit generator along the way; from this one easily obtains the analogous result for fractional ideals. (Chapman [Cha96] also gave a proof of the result for fractional ideals similar to that of Kawamoto.) The following theorem is a generalisation of these results to weakly ramified extensions. Theorem 1.1. Let L/K be a weakly ramified finite Galois extension of complete local fields with finite residue fields. Let G =Gal(L/K) and let n ∈ Z such that ≡ | | n n 1mod G1 . Then one can explicitly construct a free generator ε of PL over OK [G]. (The explicit description of ε is given in §4.) In §2 we cover some preliminary material, including the (well-known) construc- tions of generators for unramified extensions and for totally and tamely ramified extensions. In §3 we prove a ‘splitting lemma’ that says that for any finite Galois extension of complete local fields L/K with finite residue fields there exists a finite unramified extension L/L such that L/K is ‘doubly split’ (see Definition 3.1). Suppose that L/K is weakly ramified. Then L/K is also weakly ramified and we n O § give an explicit description of a free generator ε of PL over K [Gal(L /K)] in 4; n moreover, we show that the trace ε := TrL/L(ε )isafreegeneratorofPL over OK [G]. Thus we are reduced to verifying that ε is indeed a generator as claimed, which we do as follows. Let p>0 be the residue characteristic of K.In§5 we give a short and elementary proof of the fact that if L/K is a totally and weakly rami- fied p-extension, then any uniformizer πL is a free generator of PL over OK [G]; as explained in Remark 5.3, this particular result has already been proven by a num- ber of others. In §6 we treat the case of totally and weakly ramified extensions of arbitrary degree by carefully ‘glueing together’ generators from two subextensions: one that is totally and weakly ramified of p-power degree and another that is totally and tamely ramified. Finally, in §7 we perform a second glueing step that crucially depends on the fact that L/K is doubly split. We now consider the structure of OL over its associated order AL/K .Itiswell- known that AL/K coincides with the group ring OK [G] precisely when L/K is tamely ramified. Using the theory of Lubin-Tate extensions, Byott [Byo99, Th. 5] showed that if L/K is an abelian extension of p-adic fields that is weakly and wildly O O −1 ramified, then L is free over AL/K and, moreover, AL/K = K [G][πK TrG0 ]where π is any uniformizer of K and Tr = τ. Furthermore, by following the K G0 τ∈G0 proof one can construct an explicit generator. Byott also remarked [Byo99, §1] that if L/K is totally, weakly and wildly ramified (but not necessarily abelian) then it is straightforward to deduce the analogous statement from the fact that PL is free over OK [G] in this case, though one does not obtain an explicit generator in this way. The following theorem generalises these results by having weaker hypotheses and gives an explicit generator via Theorem 1.1; its elementary proof is given in §8. Theorem 1.2. Let L/K be a wildly and weakly ramified finite Galois extension of complete local fields with finite residue fields. Let G =Gal(L/K) and let πK be any O −1 uniformizer of K.ThenAL/K = K [G][πK TrG0 ] and any free generator of PL over OK [G] (e.g. as in Theorem 1.1) is also a free generator of OL over AL/K . 1.1. Conventions and notation. All modules are assumed to be left modules. However, if L/K is a Galois extension field and H ≤ Gal(L/K), then we let LH denote the subfield of L fixed by H.Forn ∈ N,weletζn denote a primitive nth root License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GALOIS MODULE STRUCTURE OF WEAKLY RAMIFIED EXTENSIONS 5061 of unity. By a ‘complete local field’ we mean a field that is complete with respect to a non-trivial discrete valuation; for such a field we fix the following notation: OK the ring of integers of K PK the maximal ideal of OK K the residue field OK /PK πK a uniformizer of K × vK the normalised valuation vK : K −→ Z We make no assumptions on the residue field K except where stated otherwise. 2. Preliminaries 2.1. A lemma on normal integral bases of ideals. We shall make frequent use of the following easy lemma. Lemma 2.1. Let L/K be a finite Galois extension of complete local fields with Galois group G.LetI be a non-zero fractional ideal of OL and let I = I/PK I. Let δ ∈ I and let δ be any lift to I. Then the following are equivalent: (i) I = K[G] · δ, (ii) I = OK [G] · δ, (iii) δ is a free generator of I over K[G], (iv) δ is a free generator of I over OK [G]. Proof. That (i) implies (ii) is a straightforward application of Nakayama’s Lemma once one notes that PK ·OK [G] is a two-sided ideal contained in the Jacobson radical of OK [G] (see e.g. [CR81, Prop. (5.22)(i)]); the converse is clear. Suppose (ii) holds; then the map OK [G] −→ I given by x → x · δ is surjective and a rank argument gives injectivity, so (iv) holds; the converse is clear. The proof of the equivalence of (i) and (iii) is similar. 2.2. Unramified extensions. The following result is well-known (see e.g.
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