Galois structure of ideals in wildly ramified abelian \(p\)-extensions of a \(p\)-adic field, and some applications. (English) Zbl 0889.11040

Byott, Nigel P.

J. Théor. Nombres Bordx. 9, No. 1, 201-219 (1997).

It is shown that fairly frequently ideals in the ring of integers of on extension \(L\) of \(K\) fail to be free over their associated order. Here \(K\) is a \(p\)-adic field and \(L/K\) is a finite abelian \(p\)-extension with ramification index \(p^n\). An efficient criterion for an ideal \(I\) not to be free over its associated order \(\mathcal A\) is given in terms of ramification numbers: if \(I\) is free, then, under a very mild extra condition which we don’t spell out here, all ramification numbers of \(L/K\) have to be congruent to \(-1\) modulo \(p^n\).
The first step towards this result is a very simple, yet clever statement (Theorem 2.1), which is worth being quoted in full: if a module \(M\) over the local ring \(\mathcal A\) has a system of generators \(m_1,\ldots,m_n\) such that none of the \(m_i\) is a generator individually, then \(M\) is not free of rank one.
One other main source is S. Vostokov’s criterion for an ideal \(I\) to be indecomposable as a module over \({\mathcal O}_K[\)Gal\((L/K)]\) [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklov 57, 64-84 (1976; Zbl 0355.12012) and ibid. 46, 14-35 (1974; Zbl 0345.12009)]. This makes the associated order of \(I\) local. These ingredients are combined with some further ideas in order to prove the main result.
Several applications are given. Firstly, it is a direct consequence that all ramification numbers are congruent to \(-1\) modulo \(p^n\) if \(S={\mathcal O}_L\) is Hopf Galois over \(R={\mathcal O}_K\). This is a pleasant generalization of previous results in the case \([L:K]=p^2\) due to N. P. Byott [Math. Z. 220, No. 4, 495-522 (1995; Zbl 0841.16021)], and to C. Greither [Math. Z. 210, No. 1, 37-68 (1992; Zbl 0737.11038)].
Secondly, the author considers division fields \(K=k_r\) and \(L=k_{m+r}\) associated to a Lubin-Tate group. Then \(L/K\) carries a priori two Hopf Galois structures: the first is the structure of a classical Galois extension, coming simply from the fact that \(L\) is already Galois over \(k=k_0\); the second comes from the theory of formal groups, in fact Spec\((L)\) is principal homogeneous under the group scheme of \(p^m\)-torsion points of the given Lubin-Tate group. If \(m\leq r\), these two structures are the same (this is analogous to Kummer theory), but for \(m>r\) they differ. Indeed this paper shows that they lead to different behaviour at integral level: the second structure always leads to a Hopf Galois extension \(S/R\), whereas for the first structure the main result allows to show that \(S\) is not free over its associated order if \(m>r\) and \(k\) is bigger than \({\mathbb{Q}}_p\). This quite well-written paper is an important contribution to local Galois module theory.

Reviewer: C.Greither (Laval)

MSC:

11S23	Integral representations
11S15	Ramification and extension theory
11S31	Class field theory; \(p\)-adic formal groups
16W30	Hopf algebras (associative rings and algebras) (MSC2000)

Keywords:

associated orders; Galois modules; ramification; Hopf Galois extensions; \(p\)-adic field; finite abelian \(p\)-extension; Kummer theory

Citations:

Zbl 0841.16021; Zbl 0778.11068; Zbl 0345.12009; Zbl 0355.12012; Zbl 0737.11038

References:

[1]	Bergé, A.-M., Arithmétique d’une extension galoisienne à groupe d’inertie cyclique, Ann. Inst. Fourier, Grenoble28 (1978), 17-44. · Zbl 0377.12009
[2]	Bertrandias, F. and Ferton, M.-J., Sur l’anneau des entiers d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sc. Paris274 (1972), A1330-A1333. · Zbl 0235.12007
[3]	Bley, W. and Burns, D., Über arithmetische assoziierte Ordnungen, J. Number Theory58 (1996), 361-387. · Zbl 0857.11063
[4]	Burns, D., Factorisability and wildly ramified Galois extensions, Ann. Inst. Fourier, Grenoble41 (1991), 393-430. · Zbl 0727.11048
[5]	Burns, D., On the equivariant structure of ideals in Galois extensions of fields, Preprint, King’s CollegeLondon (1996).
[6]	Byott, N.P., Some self-dual rings of integers not free over their associated orders, Math. Proc. Camb. Phil. Soc.110 (1991), 5-10; Corrigendum116 (1994), 569. · Zbl 0836.11039
[7]	Byott, N., On Galois isomorphisms between ideals in extensions of local fields, Manuscripta Math.73 (1991), 289-311. · Zbl 0771.11047
[8]	Byott, N.P., Tame and Galois extensions with respect to Hopf orders, Math. Z.220 (1995), 495-522. · Zbl 0841.16021
[9]	Byott, N.P., Uniqueness of Hopf Galois structure for separable field extensions, Comm. Alg.24 (1996), 3217-3228; Corrigendum24 (1996), 3705. · Zbl 0878.12001
[10]	Byott, N.P., Associated orders of certain extensions arising from Lubin- Tate formal groups, to appear in J. de Théorie des Nombres de Bordeaux. · Zbl 0902.11052
[11]	Byott, N.P. and Lettl, G., Relative Galois module structure of integers in abelian fields, J. de Théorie des Nombres de Bordeaux8 (1996), 125-141. · Zbl 0859.11059
[12]	Chan, S.-P. and Lim, C.-H., The associated orders of rings of integers in Lubin-Tate division fields over the p-adic number field, Ill. J. Math.39 (1995), 30-38. · Zbl 0816.11061
[13]	Childs, L.N., Taming wild extensions with Hopf algebras, Trans. Am. Math. Soc.304 (1987), 111-140. · Zbl 0632.12013
[14]	Childs, L.N. and Moss, D.J., Hopf algebras and local Galois module theory, in Advances in Hopf Algebras, , Vol. 158 (J. Bergen and S. Montgomery, eds.), Dekker, 1994, pp. 1-14. · Zbl 0826.16035
[15]	Elder, G.G., Galois module structure of ideals in wildly ramified cyclic extensions of degree p2, Ann. Inst. Fourier, Grenoble45 (1995), 625-647. · Zbl 0820.11070
[16]	Elder, G.G. and Madan, M.L., Galois module structure of the integers in wildly ramified cyclic extensions, J. Number Theory47 (1994), 138-174. · Zbl 0801.11046
[17]	Ferton, M.-J., Sur les idéaux d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Paris276 (1973), A1483-A1486. · Zbl 0268.12006
[18]	Greither, C., Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring, Math. Z.210 (1992), 37-67. · Zbl 0737.11038
[19]	Greither, C. and Pareigis, B., Hopf Galois theory for separable field extensions, J. Algebra106 (1987), 239-258. · Zbl 0615.12026
[20]	Leopoldt, H.-W., Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine u. angew. Math.201 (1959), 119-149. · Zbl 0098.03403
[21]	Rzedowski Calderón, M., Villa Salvador, G.D. and Madan, M.L., Galois module structure of rings of integers, Math. Z.204 (1990), 401-424. · Zbl 0682.12003
[22]	Serre, J.-P., Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. · Zbl 1492.11158
[23]	Serre, J.-P., Local fields (Graduate Texts in Mathematics, Vol. 67), Springer, 1979. · Zbl 0423.12016
[24]	Taylor, M.J., Formal groups and the Galois module structure of local rings of integers, J. reine angew. Math.358 (1985), 97-103. · Zbl 0582.12008
[25]	Taylor, M.J., Hopf structure and the Kummer theory of formal groups, J. reine angew. Math.375/376 (1987), 1-11. · Zbl 0609.12015
[26]	Ullom, S., Integral normal bases in Galois extensions of local fields, Nagoya Math. J.39 (1970), 141-148. · Zbl 0199.08401
[27]	Vostokov, S.V., Ideals of an abelian p-extension of an irregular local field as Galois modules, Zap. Nauchn. Sem. Lening. Otdel. Math. Inst. Steklov. (LOMI) 46 (1974), 14-35; English transl. in J. Soviet Math.9 (1978), 299-317. · Zbl 0396.12016
[28]	Vostokov, S.V., Ideals of an abelian p-extension of a local field as Galois module, Zap. Nauchn. Sem. Lening. Otdel. Math. Inst. Steklov. (LOMI) 57 (1976), 64-84; English transl. in J. Soviet Math.11 (1979), 567-584. · Zbl 0403.12017
[29]	Waterhouse, W.C., Normal basis implies Galois for coconnected Hopf algebras, Preprint, Pennsylvania State University (1992).