The valuation theory of deeply ramified fields and its connection with defect extensions

Kuhlmann, Franz-Viktor; Rzepka, Anna

doi:10.1090/tran/8790

The valuation theory of deeply ramified fields and its connection with defect extensions
HTML articles powered by AMS MathViewer

by Franz-Viktor Kuhlmann and Anna Rzepka;

Trans. Amer. Math. Soc. 376 (2023), 2693-2738

DOI: https://doi.org/10.1090/tran/8790

Published electronically: January 24, 2023

HTML | PDF | Request permission

Abstract:

We study in detail the valuation theory of deeply ramified fields and introduce and investigate several other related classes of valued fields. Further, a classification of defect extensions of prime degree of valued fields that was earlier given only for the equicharacteristic case is generalized to the case of mixed characteristic by a unified definition that works simultaneously for both cases. It is shown that deeply ramified fields and the other valued fields we introduce only admit one of the two types of defect extensions, namely the ones that appear to be more harmless in open problems such as local uniformization and the model theory of valued fields in positive characteristic. We use our knowledge about such defect extensions to give a new, valuation theoretic proof of the fact that algebraic extensions of deeply ramified fields are again deeply ramified. We also prove finite descent, and under certain conditions even infinite descent, for deeply ramified fields. These results are also proved for two other related classes of valued fields. The classes of valued fields under consideration can be seen as generalizations of the class of tame valued fields. Our paper supports the hope that it will be possible to generalize to deeply ramified fields several important results that have been proven for tame fields and were at the core of partial solutions of the two open problems mentioned above.

References

Anna Blaszczok, Distances of elements in valued field extensions, Manuscripta Math. 159 (2019), no. 3-4, 397–429. MR 3959269, DOI 10.1007/s00229-018-1100-6
Anna Blaszczok and Franz-Viktor Kuhlmann, On maximal immediate extensions of valued fields, Math. Nachr. 290 (2017), no. 1, 7–18. MR 3604618, DOI 10.1002/mana.201500073
Anna Blaszczok and Franz-Viktor Kuhlmann, Counting the number of distinct distances of elements in valued field extensions, J. Algebra 509 (2018), 192–211. MR 3812199, DOI 10.1016/j.jalgebra.2018.05.010
N. Bourbaki, Commutative algebra, Paris, 1972.
J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), no. 1-3, 129–174. MR 1369413, DOI 10.1007/s002220050048
Artem Chernikov, Itay Kaplan, and Pierre Simon, Groups and fields with $\textrm {NTP}_2$, Proc. Amer. Math. Soc. 143 (2015), no. 1, 395–406. MR 3272764, DOI 10.1090/S0002-9939-2014-12229-5
Steven Dale Cutkosky and Olivier Piltant, Ramification of valuations, Adv. Math. 183 (2004), no. 1, 1–79. MR 2038546, DOI 10.1016/S0001-8708(03)00082-3
Samar ElHitti and Laura Ghezzi, Dependent Artin-Schreier defect extensions and strong monomialization, J. Pure Appl. Algebra 220 (2016), no. 4, 1331–1342. MR 3423451, DOI 10.1016/j.jpaa.2015.09.005
Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 357379, DOI 10.1007/978-3-642-65505-0
Antonio J. Engler and Alexander Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2183496
Ofer Gabber and Lorenzo Ramero, Almost ring theory, Lecture Notes in Mathematics, vol. 1800, Springer-Verlag, Berlin, 2003. MR 2004652, DOI 10.1007/b10047
Yatir Halevi and Assaf Hasson, Eliminating field quantifiers in strongly dependent Henselian fields, Proc. Amer. Math. Soc. 147 (2019), no. 5, 2213–2230. MR 3937695, DOI 10.1090/proc/14203
William Andrew Johnson, Fun with Fields, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–University of California, Berkeley. MR 3564042
Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
Hagen Knaf and Franz-Viktor Kuhlmann, Abhyankar places admit local uniformization in any characteristic, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 6, 833–846 (English, with English and French summaries). MR 2216832, DOI 10.1016/j.ansens.2005.09.001
Hagen Knaf and Franz-Viktor Kuhlmann, Every place admits local uniformization in a finite extension of the function field, Adv. Math. 221 (2009), no. 2, 428–453. MR 2508927, DOI 10.1016/j.aim.2008.12.009
Franz-Viktor Kuhlmann, Valuation theoretic and model theoretic aspects of local uniformization, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 381–456. MR 1748629
Franz-Viktor Kuhlmann, Elementary properties of power series fields over finite fields, J. Symbolic Logic 66 (2001), no. 2, 771–791. MR 1833477, DOI 10.2307/2695043
Franz-Viktor Kuhlmann, A classification of Artin-Schreier defect extensions and characterizations of defectless fields, Illinois J. Math. 54 (2010), no. 2, 397–448. MR 2846467
F.-V. Kuhlmann, Defect, in Commutative Algebra - Noetherian and non-Noetherian perspectives, Fontana, M., Kabbaj, S.-E., Olberding, B., Swanson, I. (Eds.), Springer 2011.
Franz-Viktor Kuhlmann, Elimination of ramification I: the generalized stability theorem, Trans. Amer. Math. Soc. 362 (2010), no. 11, 5697–5727. MR 2661493, DOI 10.1090/S0002-9947-2010-04973-6
Franz-Viktor Kuhlmann, The algebra and model theory of tame valued fields, J. Reine Angew. Math. 719 (2016), 1–43. MR 3552490, DOI 10.1515/crelle-2014-0029
Franz-Viktor Kuhlmann, Elimination of ramification II: Henselian rationality, Israel J. Math. 234 (2019), no. 2, 927–958. MR 4040849, DOI 10.1007/s11856-019-1940-0
Franz-Viktor Kuhlmann, Valued fields with finitely many defect extensions of prime degree, J. Algebra Appl. 21 (2022), no. 3, Paper No. 2250049, 18. MR 4391814, DOI 10.1142/S0219498822500499
F.-V. Kuhlmann, Valuation Theory, In preparation. Preliminary versions of several chapters are available at: http://math.usask.ca/~fvk/Fvkbook.htm.
Franz-Viktor Kuhlmann, Matthias Pank, and Peter Roquette, Immediate and purely wild extensions of valued fields, Manuscripta Math. 55 (1986), no. 1, 39–67. MR 828410, DOI 10.1007/BF01168612
F.-V. Kuhlmann and A. Rzepka, Valuation theory of deeply ramified fields, II, In preparation.
Franz-Viktor Kuhlmann and Izabela Vlahu, The relative approximation degree in valued function fields, Math. Z. 276 (2014), no. 1-2, 203–235. MR 3150200, DOI 10.1007/s00209-013-1194-1
Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
Alexander Prestel and Peter Roquette, Formally $p$-adic fields, Lecture Notes in Mathematics, vol. 1050, Springer-Verlag, Berlin, 1984. MR 738076, DOI 10.1007/BFb0071461
A. Rzepka and P. Szewczyk, Defect extensions and a characterization of tame fields, Submitted.
Michael Temkin, Inseparable local uniformization, J. Algebra 373 (2013), 65–119. MR 2995017, DOI 10.1016/j.jalgebra.2012.09.023
Seth Warner, Topological fields, North-Holland Mathematics Studies, vol. 157, North-Holland Publishing Co., Amsterdam, 1989. Notas de Matemática [Mathematical Notes], 126. MR 1002951
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. 1, Graduate Texts in Mathematics, No. 28, Springer-Verlag, New York-Heidelberg-Berlin, 1975. With the cooperation of I. S. Cohen; Corrected reprinting of the 1958 edition. MR 384768