The Zariski–Riemann space of valuation domains associated to pseudo-convergent sequences
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- by G. Peruginelli and D. Spirito PDF
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Abstract:
Let $V$ be a valuation domain with quotient field $K$. Given a pseudo-convergent sequence $E$ in $K$, we study two constructions associating to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ has rank one. The first one has been introduced by Ostrowski, the second one more recently by Loper and Werner. We describe the main properties of these valuation domains, and we give a notion of equivalence on the set of pseudo-convergent sequences of $K$ characterizing when the associated valuation domains are equal. Then, we analyze the topological properties of the Zariski–Riemann spaces formed by these valuation domains.References
- Victor Alexandru and Nicolae Popescu, Sur une classe de prolongements à $K(X)$ d’une valuation sur un corps $K$, Rev. Roumaine Math. Pures Appl. 33 (1988), no. 5, 393–400 (French). MR 950136
- Victor Alexandru, Nicolae Popescu, and Alexandru Zaharescu, A theorem of characterization of residual transcendental extensions of a valuation, J. Math. Kyoto Univ. 28 (1988), no. 4, 579–592. MR 981094, DOI 10.1215/kjm/1250520346
- V. Alexandru, N. Popescu, and Al. Zaharescu, Minimal pairs of definition of a residual transcendental extension of a valuation, J. Math. Kyoto Univ. 30 (1990), no. 2, 207–225. MR 1068787, DOI 10.1215/kjm/1250520067
- V. Alexandru, N. Popescu, and A. Zaharescu, All valuations on $K(X)$, J. Math. Kyoto Univ. 30 (1990), no. 2, 281–296. MR 1068792, DOI 10.1215/kjm/1250520072
- N. Bourbaki, Algèbre commutative, Hermann, Paris, 1961.
- Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, Mathematical Surveys and Monographs, vol. 48, American Mathematical Society, Providence, RI, 1997. MR 1421321, DOI 10.1090/surv/048
- Alan Loper and Paul-Jean Cahen, Rings of integer-valued rational functions, J. Pure Appl. Algebra 131 (1998), no. 2, 179–193. MR 1637535, DOI 10.1016/S0022-4049(97)00120-5
- Jean-Luc Chabert, On the polynomial closure in a valued field, J. Number Theory 130 (2010), no. 2, 458–468. MR 2564907, DOI 10.1016/j.jnt.2009.09.016
- Klaas Pieter Hart, Jun-iti Nagata, and Jerry E. Vaughan (eds.), Encyclopedia of general topology, Elsevier Science Publishers, B.V., Amsterdam, 2004. MR 2049453
- M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. MR 251026, DOI 10.1090/S0002-9947-1969-0251026-X
- Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
- Christian Karpfinger and Heinz Wähling, Äquivalenz und Isomorphie bei Bewertungsfortsetzungen auf einfach-transzendente Erweiterungen, Math. Z. 238 (2001), no. 3, 461–481 (German, with English summary). MR 1869693, DOI 10.1007/s002090100261
- Franz-Viktor Kuhlmann, Value groups, residue fields, and bad places of rational function fields, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4559–4600. MR 2067134, DOI 10.1090/S0002-9947-04-03463-4
- K. Alan Loper and Nicholas J. Werner, Pseudo-convergent sequences and Prüfer domains of integer-valued polynomials, J. Commut. Algebra 8 (2016), no. 3, 411–429. MR 3546004, DOI 10.1216/JCA-2016-8-3-411
- Saunders MacLane, A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), no. 3, 363–395. MR 1501879, DOI 10.1090/S0002-9947-1936-1501879-8
- Bruce Olberding, A principal ideal theorem for compact sets of rank one valuation rings, J. Algebra 489 (2017), 399–426. MR 3686983, DOI 10.1016/j.jalgebra.2017.06.028
- Alexander Ostrowski, Untersuchungen zur arithmetischen Theorie der Körper, Math. Z. 39 (1935), no. 1, 321–404 (German). MR 1545506, DOI 10.1007/BF01201362
- Giulio Peruginelli, Prüfer intersection of valuation domains of a field of rational functions, J. Algebra 509 (2018), 240–262. MR 3812201, DOI 10.1016/j.jalgebra.2018.05.012
- Giulio Peruginelli, Transcendental extensions of a valuation domain of rank one, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4211–4226. MR 3690607, DOI 10.1090/proc/13574
- G. Peruginelli and D. Spirito, Extending valuations to the field of rational functions using pseudo-monotone sequences, preprint, arXiv:1905.02481.
- Paulo Ribenboim, Théorie des valuations, Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). Deuxième édition multigraphiée. MR 0249425
- P. Ribenboim, On the completion of a valuation ring, Math. Ann. 155 (1964), 392–396. MR 164960, DOI 10.1007/BF01350748
- Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR 507446, DOI 10.1007/978-1-4612-6290-9
- Michel Vaquié, Extension d’une valuation, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3439–3481 (French, with English summary). MR 2299463, DOI 10.1090/S0002-9947-07-04184-0
- Seth Warner, Topological fields, North-Holland Mathematics Studies, vol. 157, North-Holland Publishing Co., Amsterdam, 1989. Notas de Matemática [Mathematical Notes], 126. MR 1002951
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
- 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 0384768
Additional Information
- G. Peruginelli
- Affiliation: Dipartimento di Matematica “Tullio Levi-Civita”, University of Padova, Via Trieste 63, 35121 Padova, Italy
- MR Author ID: 891441
- ORCID: 0000-0001-7694-8920
- Email: gperugin@math.unipd.it
- D. Spirito
- Affiliation: Dipartimento di Matematica e Fisica, University of Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma
- Address at time of publication: Dipartimento di Matematica “Tullio Levi-Civita”, University of Padova, Via Trieste 63, 35121 Padova, Italy
- Email: spirito@math.unipd.it, spirito@mat.uniroma3.it
- Received by editor(s): September 14, 2018
- Received by editor(s) in revised form: March 6, 2020, and March 13, 2020
- Published electronically: September 9, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 7959-7990
- MSC (2010): Primary 12J20, 13A18, 13F30
- DOI: https://doi.org/10.1090/tran/8185
- MathSciNet review: 4169679