Holomorphy rings of function fields

Intersections of valuation overrings of two-dimensional Noetherian domains

“…. This ring is also interesting in that it is a Hilbert ring such that for each nonzero proper finitely generated ideal I of H, there exist for each h = 1, 2 and d = 0, 1, infinitely many prime ideals minimal over I of height h and dimension d [20,Proposition 3.11 and Theorem 4.7]. All these claims remain true if D is assumed to be a two-dimensional affine Kdomain such that K is existentially closed in the quotient field of D; moreover, similar results hold in higher dimensions; see [20].…”

“…This ring is also interesting in that it is a Hilbert ring such that for each nonzero proper finitely generated ideal I of H, there exist for each h = 1, 2 and d = 0, 1, infinitely many prime ideals minimal over I of height h and dimension d [20,Proposition 3.11 and Theorem 4.7]. All these claims remain true if D is assumed to be a two-dimensional affine Kdomain such that K is existentially closed in the quotient field of D; moreover, similar results hold in higher dimensions; see [20]. See also [12] for a way to create similar examples with no restriction on whether K is algebraically closed, and using all the valuation overrings of D, not just those with residue field K: the caveat is that these valuation overrings must be extended to Since D m is a regular local ring, the mapping ord m extends to a rank one discrete valuation (the order valuation with respect to m) on the quotient field of D. Let E be a subset of K 2 .…”

Group-theoretic and topological invariants of completely integrally closed Prüfer domains

“…If in addition every invertible ideal is principal, then R is a Bézout domain. The class of completely integrally closed Prüfer domains includes a number of prominent examples in non-Noetherian commutative ring theory, such as the ring of integer-valued polynomials [9, Proposition VI.2.1, p. 129], the ring of entire functions [14, Proposition 8.1.1 (6), p. 276], the real holomorphy ring of a function field [39,Corollary 3.6], and the Kronecker function ring of a field extension of at most countable transcendence degree [26,Corollary 3.9]. For a completely integrally closed Prüfer domain R, the set Div(R) of nonzero divisorial fractional ideals of R is a lattice-ordered group (ℓ-group) with respect to ideal multiplication, as is the group Inv(R) of invertible fractional ideals of R. We prove in Proposition 3.1 that Div(R) is the completion of Inv(R), a consequence of which is that the group Div(R) can be calculated solely from the ℓ-group Inv(R).…”

On the geometry of Prüfer intersections of valuation rings

“…The notion of existential closure leads to more general results on Prüfer holomorphy rings in function fields. For references on this generalization, see [19].…”