Forms of an affinoid disc and ramification

Schmidt, Tobias

doi:10.5802/aif.2957

Forms of an affinoid disc and ramification
[Formes d’un disque affinoïde et ramification]

Schmidt, Tobias ¹

¹ Institut für Mathematik Humboldt-Universität zu Berlin Rudower Chaussee 25, D-12489 Berlin (Germany)

Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1301-1347.

Résumé
Abstract

Soit $k$ un corps non archimédien complet et soit $X$ un disque $k$ -affinoïde fermé. Nous classifions les formes modérément ramifiées de $X$ . Nous généralisons quelques résultats classiques de P. Russell sur les formes inséparables d’une droite affine et nous construisons des familles explicites des formes sauvagement ramifiées de $X$ . Finalement, nous déterminons le groupe des classes et le groupe de Grothendieck de quelques formes de $X$ .

Let $k$ be a complete nonarchimedean field and let $X$ be an affinoid closed disc over $k$ . We classify the tamely ramified twisted forms of $X$ . Generalizing classical work of P. Russell on inseparable forms of the affine line we construct explicit families of wildly ramified forms of $X$ . We finally compute the class group and the Grothendieck group of forms of $X$ in certain cases.

DOI : 10.5802/aif.2957

Classification : 14G22, 13B02, 16W70
Keywords: twisted form, affinoid disc, ramification
Mots-clés : form twisté, disque affinoïde, ramification

Affiliations des auteurs :

Schmidt, Tobias ¹

¹ Institut für Mathematik Humboldt-Universität zu Berlin Rudower Chaussee 25, D-12489 Berlin (Germany)

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Schmidt, Tobias. Forms of an affinoid disc and ramification. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1301-1347. doi : 10.5802/aif.2957. https://aif.centre-mersenne.org/articles/10.5802/aif.2957/

Bibliographie
Cité par

[1] Auslander, M.; Buchsbaum, D. A. Homological dimension in local rings, Trans. Amer. Math. Soc., Volume 85 (1957), pp. 390-405 | DOI | MR | Zbl

[2] Auslander, M.; Buchsbaum, D. A. Unique factorization in regular local rings, Proc. Nat. Acad. Sci. U.S.A., Volume 45 (1959), pp. 733-734 | DOI | MR | Zbl

[3] Baba, K. On $p$ -radical descent of higher exponent, Osaka J. Math., Volume 18 (1981) no. 3, pp. 725-748 | MR | Zbl

[4] Bass, H. Algebraic $K$ -theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968, pp. xx+762 | MR | Zbl

[5] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-archimedean fields, Math. Surveys and Monographs, 33, American Mathematical Society, Providence, Rhode Island, 1990 | MR | Zbl

[6] Bosch, S.; Güntzer, U.; Remmert, R. Non-Archimedean analysis, Springer-Verlag, Berlin, 1984 | MR | Zbl

[7] Bourbaki, N. Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998 | MR | Zbl

[8] Conrad, B.; Temkin, M. Descent for non-archimedean analytic spaces. (http://math.huji.ac.il/~temkin/papers/Descent.pdf)

[9] Demazure, M.; Gabriel, P. Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris, 1970, pp. xxvi+700 (Avec un appendice ıt Corps de classes local par Michiel Hazewinkel) | MR | Zbl

[10] Ducros, A. Toute forme modérément ramifiée d’un polydisque ouvert est triviale, Math. Z., Volume 273 (2013) no. 1-2, pp. 331-353 | DOI | MR | Zbl

[11] Jacobson, N. Lectures in abstract algebra. III, Springer-Verlag, New York, 1975, pp. xi+323 (Theory of fields and Galois theory, Graduate Texts in Math., No. 32) | MR | Zbl

[12] Kambayashi, T.; Miyanishi, M.; Takeuchi, M. Unipotent algebraic groups, Lecture Notes in Mathematics, Vol. 414, Springer-Verlag, Berlin, 1974, pp. v+165 | MR | Zbl

[13] Kaplansky, I. Maximal fields with valuations, Duke Math. J., Volume 9 (1942), pp. 303-321 | DOI | MR | Zbl

[14] Knus, M.-A.; Ojanguren, M. Théorie de la descente et algèbres d’Azumaya, Lecture Notes in Math., Vol. 389, Springer-Verlag, Berlin, 1974, pp. iv+163 | MR | Zbl

[15] Lang, Serge Algebra, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002, pp. xvi+914 | MR | Zbl

[16] Levi, F. W. Ordered groups, Proc. Indian Acad. Sci., Sect. A., Volume 16 (1942), pp. 256-263 | MR | Zbl

[17] Li, H.; Van den Bergh, M.; Van Oystaeyen, F. Note on the $K_{0}$ of rings with Zariskian filtration, $K$ -Theory, Volume 3 (1990) no. 6, pp. 603-606 | DOI | MR | Zbl

[18] Li, H.; van Oystaeyen, F. Global dimension and Auslander regularity of Rees rings, Bull. Math. Soc. Belgique, Volume (serie A) XLIII (1991), pp. 59-87 | MR | Zbl

[19] Li, H.; van Oystaeyen, F. Zariskian filtrations, K-Monographs in Mathematics, 2, Kluwer Academic Publishers, Dordrecht, 1996, pp. x+252 | Zbl

[20] Matsumura, H. Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1986, pp. xiv+320 | MR | Zbl

[21] McConnell, J. C.; Robson, J. C. Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons Ltd., Chichester, 1987, pp. xvi+596 | MR | Zbl

[22] Năstăsescu, C.; Van Oystaeyen, F. Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer-Verlag, Berlin, 2004, pp. xiv+304 | MR | Zbl

[23] Quillen, D. Higher algebraic $K$ -theory. I, Algebraic $K$ -theory, I: Higher $K$ -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, p. 85-147. Lecture Notes in Math., Vol. 341 | MR | Zbl

[24] Rémy, B.; Thuillier, A.; Werner, A. Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings, Ann. Sci. Éc. Norm. Supér. (4), Volume 43 (2010) no. 3, pp. 461-554 | Numdam | MR | Zbl

[25] Russell, P. Forms of the affine line and its additive group, Pacific J. Math., Volume 32 (1970), pp. 527-539 | DOI | MR | Zbl

[26] Samuel, P. Classes de diviseurs et dérivées logarithmiques, Topology, Volume 3 (1964) no. suppl. 1, pp. 81-96 | DOI | MR | Zbl

[27] Serre, J.-P. Local fields, Graduate Texts in Math., 67, Springer-Verlag, New York, 1979, pp. viii+241 | MR | Zbl

[28] Serre, J.-P. Galois cohomology, Springer Monographs in Math., Springer-Verlag, Berlin, 2002, pp. x+210 | MR | Zbl

[29] Temkin, M. On local properties of non-Archimedean analytic spaces, Math. Ann., Volume 318 (2000) no. 3, pp. 585-607 | DOI | MR | Zbl

[30] Temkin, M. On local properties of non-Archimedean analytic spaces. II, Israel J. Math., Volume 140 (2004), pp. 1-27 | DOI | MR | Zbl

[31] Waterhouse, W. C. Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Springer-Verlag, New York, 1979, pp. xi+164 | MR | Zbl

[32] Weibel, C. An introduction to algebraic K-theory (http://www.math.rutgers.edu/~weibel/Kbook.html) | MR

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