Residue fields of valued function fields of conics
Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension of v0 to K having residue field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function field of a co...
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| Published in | Proceedings of the Edinburgh Mathematical Society Vol. 36; no. 3; pp. 469 - 478 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.10.1993
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| Online Access | Get full text |
| ISSN | 0013-0915 1464-3839 1464-3839 |
| DOI | 10.1017/S0013091500018551 |
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| Abstract | Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension of v0 to K having residue field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function field of a conic over a finite extension of k0. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory. |
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| AbstractList | Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension of v0 to K having residue field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function field of a conic over a finite extension of k0. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory. Suppose that K is a function field of a conic over a subfield K 0 . Let v 0 be a valuation of K 0 with residue field k 0 of characteristic ≠2. Let v be an extension of v 0 to K having residue field k . It has been proved that either k is an algebraic extension of k 0 or k is a regular function field of a conic over a finite extension of k 0 . This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory. |
| Author | Khanduja, Sudesh K. Garg, Usha |
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| Copyright | Copyright © Edinburgh Mathematical Society 1993 |
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| Notes | ArticleID:01855 PII:S0013091500018551 istex:D28735BE7A852D884244780D3D7B9BD5A2535E16 ark:/67375/6GQ-SB49RFML-S The research of the first author is supported partially by CSIR, New Delhi, vide grant No. 25/53/90-EMRII. |
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| References | Artin (S0013091500018551_ref001) 1967 S0013091500018551_ref008 S0013091500018551_ref009 S0013091500018551_ref004 S0013091500018551_ref005 S0013091500018551_ref006 S0013091500018551_ref007 Zariski (S0013091500018551_ref013) 1955; 1 S0013091500018551_ref003 Bourbaki (S0013091500018551_ref002) 1972 S0013091500018551_ref010 Ohm (S0013091500018551_ref011) 1990 Weil (S0013091500018551_ref012) 1962 |
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| Snippet | Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension... Suppose that K is a function field of a conic over a subfield K 0 . Let v 0 be a valuation of K 0 with residue field k 0 of characteristic ≠2. Let v be an... |
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| Title | Residue fields of valued function fields of conics |
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