Abstract
In the present paper it is shown that one can recover much of the inertia structure of (quasi) divisors of a function field K|k over an algebraically closed base field k from the maximal pro-ℓ abelian-by-central Galois theory of K. The results play a central role in the birational anabelian geometry and related questions.
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References
E. Artin, Geometric Algebra, Interscience, New York, 1957.
F. A. Bogomolov, On two conjectures in birational algebraic geometry, in Algebraic Geometry and Analytic Geometry, ICM-90 Satellite Conference Proceedings (A. Fujiki et al., eds.), Springer-Verlag, Tokyo, 1991, pp. 26–52.
F. A. Bogomolov and Y. Tschinkel, Commuting elements in Galois groups of function fields, in Motives, Polylogarithms and Hodge Theory (F.A. Bogomolov and L. Katzarkov, eds.), International Press, Somerville, MA, 2002, pp. 75–120.
N. Bourbaki, Algèbre commutative, Hermann, Paris, 1964.
I. Efrat, Valuations, Orderings and Milnor K-Theory, AMS Mathematical Surveys and Monographs, Vol. 124, American Mathematical Society, Providence, RI, 2006.
O. Endler and A. J. Engler, Fields with Henselian valuation rings, Mathematische Zeitschrift 152 (1977), 191–193.
A. J. Engler and A. Prestel, Valued Fields, Springer Monographs in Mathematics Series, Springer-Verlag, Berlin, 2005.
L. Schneps and P. Lochak (eds.), Geometric Galois Actions I, London Mathematical Society Lecture Notes, Vol. 242, Cambridge University Press, Cambridge, 1998.
A. Grothendieck, Letter to Faltings, June 1983; see [GGA].
A. Grothendieck, Esquisse d’un programme, 1984; see [GGA].
H. Koch, Die Galoissche Theorie der p-Erweiterungen, Math. Monogr. 10, Berlin, 1970.
J. Koenigsmann, Solvable absolute Galois groups are metabelian, Inventiones Mathematicae 144 (2001), 1–22.
L. Mahé, J. Mináč and T. L. Smith, Additive structure of multiplicative subgroups of fields and Galois theory, Documenta Mathematica 9 (2004), 301–355.
D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, Vol. 1358, 2nd edition, Springer-Verlag, Berlin, 1999.
J. Neukirch, Ü ber eine algebraische Kennzeichnung der Henselkörper, Journal für die Reine und Angewandte Mathematik 231 (1968), 75–81.
J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd edition, Grundlehren der Mathematischen Wissenschaften, Vol. 323, Springer-Verlag, Berlin, 2008.
A. N. Parshin, Finiteness Theorems and Hyperbolic Manifolds, in The Grothendieck Festschrift III (P. Cartier et al., eds.), PM Series, Vol. 88, Birkhäuser, Boston, Basel, Berlin, 1990.
F. Pop, On Grothendieck’s conjecture of birational anabelian geometry, Annals of Mathematics 138 (1994), 145–182.
F. Pop, Glimpses of Grothendieck’s anabelian geometry, in Geometric Galois Actions I, London Mathematical Society Lecture Notes, Vol. 242, (L. Schneps and P. Lochak eds.), Cambridge University Press, 1998, Cambridge, pp 133–126.
F. Pop, Pro-ℓ birational anabelian geometry over algebraically closed fields I, Manuscript, Bonn, 2003; see: http://arxiv.org/pdf/math.AG/0307076.
F. Pop, Pro-ℓ Galois theory of Zariski prime divisors, in Luminy Proceedings Conference, SMF No 13 (Débès et al., eds.), Hérmann, Paris, 2006.
F. Pop, Recovering fields from their decomposition graphs, Manuscript, 2007; see: http://www.math.upenn.edu/~pop/Research/Papers.html.
P. Roquette, Zur Theorie der Konstantenreduktion algebraischer Mannigfaltigkeiten, Journal für die Reine und Angewandte Mathematik 200 (1958), 1–44.
T. Szamuely, Groupes de Galois de corps de type fini (d’après Pop), Astérisque 294 (2004), 403–431.
J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, Berlin, 1965.
K. Uchida, Isomorphisms of Galois groups of solvably closed Galois extensions, The Tôhoku Mathematical Journal 31 (1979), 359–362.
R. Ware, Valuation Rings and rigid Elements in Fields, Canadian Journal of Mathematics 33 (1981), 1338–1355.
O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Springer-Verlag, New York, 1975.
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Pop, F. Pro-ℓ abelian-by-central Galois theory of prime divisors. Isr. J. Math. 180, 43–68 (2010). https://doi.org/10.1007/s11856-010-0093-y
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DOI: https://doi.org/10.1007/s11856-010-0093-y