On the projectivity of proper normal curves over valuation domains

2021 ◽  
pp. 2250183
Author(s):  
Hagen Knaf
Keyword(s):  
Projective Dimension ◽  
Valuation Domain ◽  
Zariski Closure ◽  
Noetherian Rings ◽  
Regular Curve ◽  
Generic Fiber ◽  
Discrete Valuation Domain ◽  
Closed Point ◽  
Valuation Domains ◽  
Normal Formula

A theorem of Lichtenbaum states, that every proper, regular curve [Formula: see text] over a discrete valuation domain [Formula: see text] is projective. This theorem is generalized to the case of an arbitrary valuation domain [Formula: see text] using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring [Formula: see text] of a point [Formula: see text] is called regular, if every finitely generated ideal [Formula: see text] has finite projective dimension. The generalization is a particular case of a projectivity criterion for proper, normal [Formula: see text]-curves: such a curve [Formula: see text] is projective if for every irreducible component [Formula: see text] of its closed fiber [Formula: see text] there exists a closed point [Formula: see text] of the generic fiber of [Formula: see text] such that the Zariski closure [Formula: see text] meets [Formula: see text] and meets [Formula: see text] in regular points only.

scholarly journals On the Factorization of Polynomials Over Discrete Valuation Domains

2014 ◽  
Vol 22 (1) ◽  
pp. 273-280
Author(s):  
Doru Ştefănescu
Keyword(s):  
Valuation Domain ◽  
Discrete Valuation Domain ◽  
Valuation Domains ◽  
Univariate Polynomials ◽  
Factorization Of Polynomials

AbstractWe study some factorization properties for univariate polynomials with coefficients in a discrete valuation domain (A,v). We use some properties of the Newton index of a polynomial to deduce conditions on v(ai) that allow us to find some information on the degree of the factors of F.

scholarly journals Stacked submodules of torsion modules over discrete valuation domains

2003 ◽  
Vol 68 (3) ◽  
pp. 439-447 ◽  
Author(s):  
Pudji Astuti ◽  
Harald K. Wimmer
Keyword(s):  
Valuation Domain ◽  
Torsion Module ◽  
Discrete Valuation Domain ◽  
Valuation Domains ◽  
Torsion Modules

A submodule W of a torsion module M over a discrete valuation domain is called stacked in M if there exists a basis ℬ of M such that multiples of elements of ℬ form a basis of W. We characterise those submodules which are stacked in a pure submodule of M.

REGULAR LOCAL ALGEBRAS OVER VALUATION DOMAINS: WEAK DIMENSION AND REGULAR SEQUENCES

2008 ◽  
Vol 07 (05) ◽  
pp. 575-591
Author(s):  
HAGEN KNAF
Keyword(s):  
Projective Dimension ◽  
Regular Sequence ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Regular Sequences ◽  
Local Algebras ◽  
Weak Dimension ◽  
Valuation Domains ◽  
Finitely Presented

A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.

scholarly journals Pi-balanced torsion-free modules over a discrete valuation domain

2006 ◽  
Vol 295 (1) ◽  
pp. 269-288 ◽  
Author(s):  
David M. Arnold ◽  
K.M. Rangswamy ◽  
Fred Richman
Keyword(s):  
Valuation Domain ◽  
Discrete Valuation Domain ◽  
Free Modules ◽  
Torsion Free

On the Weak Global Dimension of Pseudovaluation Domains

1978 ◽  
Vol 21 (2) ◽  
pp. 159-164 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Integral Domain ◽  
Global Dimension ◽  
Valuation Domain ◽  
Valuation Domains ◽  
Dimension 2 ◽  
The Ideal

In [7], Hedstrom and Houston introduce a type of quasilocal integral domain, therein dubbed a pseudo-valuation domain (for short, a PVD), which possesses many of the ideal-theoretic properties of valuation domains. For the reader′s convenience and reference purposes, Proposition 2.1 lists some of the ideal-theoretic characterizations of PVD′s given in [7]. As the terminology suggests, any valuation domain is a PVD. Since valuation domains may be characterized as the quasilocal domains of weak global dimension at most 1, a homological study of PVD's seems appropriate. This note initiates such a study by establishing (see Theorem 2.3) that the only possible weak global dimensions of a PVD are 0, 1, 2 and ∞. One upshot (Corollary 3.4) is that a coherent PVD cannot have weak global dimension 2: hence, none of the domains of weak global dimension 2 which appear in [10, Section 5.5] can be a PVD.

scholarly journals The Ohm–Rush content function II. Noetherian rings, valuation domains, and base change

2019 ◽  
Vol 18 (05) ◽  
pp. 1950100
Author(s):  
Neil Epstein ◽  
Jay Shapiro
Keyword(s):  
Noetherian Ring ◽  
Artinian Ring ◽  
Base Change ◽  
Noetherian Rings ◽  
Prime Characteristic ◽  
Field Extensions ◽  
Dimension Formula ◽  
Valuation Domains ◽  
Prüfer Domains ◽  
Polynomial Extensions

The notion of an Ohm–Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan–Hochster proof of Stillman’s conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from [N. Epstein and J. Shapiro, The Ohm-Rush content function, J. Algebra Appl. 15(1) (2016) 1650009, 14 pp.] of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Prüfer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions.

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