×

zbMATH — the first resource for mathematics

The factoriality of Zariski rings. (English) Zbl 0631.13017
Soient k un corps algébriquement clôs de caractéristique \(p\neq 0,\) \(g\in k[X,Y]\) tel que \((g_ x,g_ y)=1\), alors l’anneau \(A=k[X^ p,Y^ p,g]\) est appelé anneau de Zariski. L’A. donne des conditions dans lesquelles A est un anneau factoriel. Le résultat principal: Soient k un corps de caractéristique \(p\geq 5\), \(g\in k[X,Y]\) de degré \(\geq 4\) et \(A=k[X^ p,Y^ p,g]\). Si g est générique, l’anneau A est factoriel.
Reviewer: N.Radu

MSC:
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] P. Blass : Zariski Surfaces . Dissertations Mathematicae 200 (1980). · Zbl 0523.14027
[2] P. Blass : Some geometric applications of a dinerential equation in characteristic p > 0 to the theory of algebraic surfaces . Contemp. Math. A.M.S. 13 (1982). · Zbl 0561.14018 · doi:10.1090/conm/013/37
[3] P. Blass : Picard groups of Zariski Surfaces I . Comp. Math. 54 (1985) 3-86. · Zbl 0624.14021 · numdam:CM_1985__54_1_3_0 · eudml:89680
[4] P. Blass and J. Lang : Picard groups of Zariski Surfaces II . Comp. Math. 54 (1985) 36-39. · Zbl 0624.14021 · numdam:CM_1985__54_1_3_0 · eudml:89680
[5] R. Fossum : The Divisor Class Group of a Krull Domain . Springer-Verlag, New York (1973). · Zbl 0256.13001
[6] H.W. Gould : Combinatorial Identities . Morgantown, W. Va (1972). · Zbl 0241.05011
[7] R. Hartshorne : Algebraic Geometry . Springer-Verlag, New York (1977). · Zbl 0367.14001
[8] I. Kaplansky : Commutative Rings . Allyn and Bacon, Boston (1970). · Zbl 0203.34601
[9] J. Lang : An example related to the affine theorem of Castelnuovo . Michigan Math. J. 28 (1981). · Zbl 0495.14021 · doi:10.1307/mmj/1029002568
[10] J. Lang : The divisor classes of the hypersurfaces zpn = G(x1, ..., xm) in characteristic p > 0 . Trans A.M.S. 278 2 (1983). · Zbl 0528.14018 · doi:10.2307/1999174
[11] J. Lang : The divisor class group of the surface zpn = G(x, y) over fields of characteristic p > 0 . J. Alg. 84, 2 (1983). · Zbl 0528.14017 · doi:10.1016/0021-8693(83)90084-4
[12] J. Lang : The divisor classes of the surface zp = G(x, y), a programmable problem . J. Alg. 100, (1986).
[13] J. Lang : Locally factorial generic Zariski surfaces are factorial . J. Alg., to appear. · Zbl 0643.14023 · doi:10.1016/0021-8693(87)90221-3
[14] M. Nagata : Local Rings . John Wiley & Sons, Inc. (1962). · Zbl 0123.03402
[15] M. Nagata : Field Theory . Marcel Dekker, Inc. (1977). · Zbl 0366.12001
[16] Stohr and Voloch : A formula for the Cartier operator on plane algebraic curves . Submitted for publication. · Zbl 0605.14023 · doi:10.1515/crll.1987.377.49 · crelle:GDZPPN002204517 · eudml:152926
[17] P. Samuel : Lectures on Unique Factorization Domains . Tata Lecture Notes (1964). · Zbl 0184.06601
[18] R. Walker : Algebraic Curves . Princeton University Press, Princeton, (1950). · Zbl 0039.37701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.