Forms of an affinoid disc and ramification
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, p. 1301-1347

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.

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.

DOI : https://doi.org/10.5802/aif.2957
Classification:  14G22,  13B02,  16W70
Keywords: twisted form, affinoid disc, ramification
@article{AIF_2015__65_3_1301_0,
     author = {Schmidt, Tobias},
     title = {Forms of an affinoid disc and ramification},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     pages = {1301-1347},
     doi = {10.5802/aif.2957},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_3_1301_0}
}
Schmidt, Tobias. Forms of an affinoid disc and ramification. Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1301-1347. doi : 10.5802/aif.2957. http://www.numdam.org/item/AIF_2015__65_3_1301_0/

[1] Auslander, M.; Buchsbaum, D. A. Homological dimension in local rings, Trans. Amer. Math. Soc., Tome 85 (1957), pp. 390-405 | Article | MR 86822 | Zbl 0078.02802

[2] Auslander, M.; Buchsbaum, D. A. Unique factorization in regular local rings, Proc. Nat. Acad. Sci. U.S.A., Tome 45 (1959), p. 733-734 | Article | MR 103906 | Zbl 0084.26504

[3] Baba, K. On p-radical descent of higher exponent, Osaka J. Math., Tome 18 (1981) no. 3, pp. 725-748 | MR 635730 | Zbl 0478.13001

[4] Bass, H. Algebraic K -theory, W. A. Benjamin, Inc., New York-Amsterdam (1968), pp. xx+762 | MR 249491 | Zbl 0174.30302

[5] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-archimedean fields, American Mathematical Society, Providence, Rhode Island, Math. Surveys and Monographs, Tome 33 (1990) | MR 1070709 | Zbl 0715.14013

[6] Bosch, S.; Güntzer, U.; Remmert, R. Non-Archimedean analysis, Springer-Verlag, Berlin (1984) | MR 746961 | Zbl 0539.14017

[7] Bourbaki, N. Commutative algebra. Chapters 1–7, Springer-Verlag, Berlin, Elements of Mathematics (Berlin) (1998) | MR 1727221 | Zbl 0719.12001

[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 302656 | Zbl 0203.23401

[10] Ducros, A. Toute forme modérément ramifiée d’un polydisque ouvert est triviale, Math. Z., Tome 273 (2013) no. 1-2, pp. 331-353 | Article | MR 3010163 | Zbl 1264.14035

[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 392906 | Zbl 0326.00001

[12] Kambayashi, T.; Miyanishi, M.; Takeuchi, M. Unipotent algebraic groups, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 414 (1974), pp. v+165 | MR 376696 | Zbl 0294.14022

[13] Kaplansky, I. Maximal fields with valuations, Duke Math. J., Tome 9 (1942), pp. 303-321 | Article | MR 6161 | Zbl 0063.03135

[14] Knus, M.-A.; Ojanguren, M. Théorie de la descente et algèbres d’Azumaya, Springer-Verlag, Berlin, Lecture Notes in Math., Vol. 389 (1974), pp. iv+163 | MR 417149 | Zbl 0284.13002

[15] Lang, Serge Algebra, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 211 (2002), pp. xvi+914 | MR 1878556 | Zbl 0984.00001

[16] Levi, F. W. Ordered groups, Proc. Indian Acad. Sci., Sect. A., Tome 16 (1942), pp. 256-263 | MR 7779 | Zbl 0061.03403

[17] Li, H.; Van Den Bergh, M.; Van Oystaeyen, F. Note on the K 0 of rings with Zariskian filtration, K-Theory, Tome 3 (1990) no. 6, pp. 603-606 | Article | MR 1071897 | Zbl 0709.16023

[18] Li, H.; Van Oystaeyen, F. Global dimension and Auslander regularity of Rees rings, Bull. Math. Soc. Belgique, Tome (serie A) XLIII (1991), pp. 59-87 | MR 1315771 | Zbl 0753.16003

[19] Li, H.; Van Oystaeyen, F. Zariskian filtrations, Kluwer Academic Publishers, Dordrecht, K-Monographs in Mathematics, Tome 2 (1996), pp. x+252 | Zbl 0862.16027

[20] Matsumura, H. Commutative ring theory, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 8 (1986), pp. xiv+320 | MR 879273 | Zbl 0603.13001

[21] Mcconnell, J. C.; Robson, J. C. Noncommutative Noetherian rings, John Wiley & Sons Ltd., Chichester, Pure and Applied Mathematics (New York) (1987), pp. xvi+596 | MR 934572 | Zbl 0644.16008

[22] Năstăsescu, C.; Van Oystaeyen, F. Methods of graded rings, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1836 (2004), pp. xiv+304 | MR 2046303 | Zbl 1043.16017

[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 338129 | Zbl 0292.18004

[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), Tome 43 (2010) no. 3, pp. 461-554 | Numdam | MR 2667022 | Zbl 1198.51006

[25] Russell, P. Forms of the affine line and its additive group, Pacific J. Math., Tome 32 (1970), pp. 527-539 | Article | MR 265367 | Zbl 0199.24502

[26] Samuel, P. Classes de diviseurs et dérivées logarithmiques, Topology, Tome 3 (1964) no. suppl. 1, pp. 81-96 | Article | MR 166213 | Zbl 0127.26002

[27] Serre, J.-P. Local fields, Springer-Verlag, New York, Graduate Texts in Math., Tome 67 (1979), pp. viii+241 | MR 554237 | Zbl 0423.12016

[28] Serre, J.-P. Galois cohomology, Springer-Verlag, Berlin, Springer Monographs in Math. (2002), pp. x+210 | MR 1867431 | Zbl 1004.12003

[29] Temkin, M. On local properties of non-Archimedean analytic spaces, Math. Ann., Tome 318 (2000) no. 3, pp. 585-607 | Article | MR 1800770 | Zbl 0972.32019

[30] Temkin, M. On local properties of non-Archimedean analytic spaces. II, Israel J. Math., Tome 140 (2004), pp. 1-27 | Article | MR 2054837 | Zbl 1066.32025

[31] Waterhouse, W. C. Introduction to affine group schemes, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 66 (1979), pp. xi+164 | MR 547117 | Zbl 0442.14017

[32] Weibel, C. An introduction to algebraic K-theory (http://www.math.rutgers.edu/~weibel/Kbook.html) | MR 3076731