On the projectivity of proper normal curves over valuation domains

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.

The Extension of the GVW Algorithm to Valuation Domains

The GVW algorithm is an effective algorithm to compute Gröbner bases for polynomial ideals over a field. Combined with properties of valuation domains and the idea of the GVW algorithm, we propose a new algorithm to compute Gröbner bases for polynomial ideals over valuation domains in this study. Furthermore, we use an example to demonstrate the improvement of our algorithm.

Prime avoidance property

Let [Formula: see text] be a commutative ring, we say that [Formula: see text] has prime avoidance property, if [Formula: see text] for an ideal [Formula: see text] of [Formula: see text], then there exists [Formula: see text] such that [Formula: see text]. We exactly determine when [Formula: see text] has prime avoidance property. In particular, if [Formula: see text] has prime avoidance property, then [Formula: see text] is compact. For certain classical rings we show the converse holds (such as Bezout rings, [Formula: see text]-domains, zero-dimensional rings and [Formula: see text]). We give an example of a compact set [Formula: see text], where [Formula: see text] is a Prufer domain, which has not prime avoidance property. Finally, we show that if [Formula: see text] are valuation domains for a field [Formula: see text] and [Formula: see text] for some [Formula: see text], then there exists [Formula: see text] such that [Formula: see text].