54 research outputs found
On the geometry of Pr\"ufer intersections of valuation rings
Let be a field, let be a subring of and let be an irreducible
subspace of the space of all valuation rings between and that have
quotient field . Then is a locally ringed space whose ring of global
sections is . All rings between and that are
integrally closed in arise in such a way. Motivated by applications in
areas such as multiplicative ideal theory and real algebraic geometry, a number
of authors have formulated criteria for when is a Pr\"ufer domain. We give
geometric criteria for when is a Pr\"ufer domain that reduce this issue to
questions of prime avoidance. These criteria, which unify and extend a variety
of different results in the literature, are framed in terms of morphisms of
into the projective line Comment: 13 pages, to appear in Pacific Journal of Mathematic
Prescribed subintegral extensions of local Noetherian domains
We show how subintegral extensions of certain local Noetherian domains
can be constructed with specified invariants including reduction number,
Hilbert function, multiplicity and local cohomology. The construction behaves
analytically like Nagata idealization but rather than a ring extension of ,
it produces a subring of such that is subintegral.Comment: 25 pages; to appear in Journal of Pure and Applied Algebr
One-dimensional bad Noetherian domains
Local Noetherian domains arising as local rings of points of varieties or in
the context of algebraic number theory are analytically unramified, meaning
their completions have no nontrivial nilpotent elements. However, looking
elsewhere, many sources of analytically ramified local Noetherian domains have
been exhibited over the last seventy five years. We give a unified approach to
a number of such examples by describing classes of DVRs which occur as the
normalization of an analytically ramified local Noetherian domain, as well as
some that do not occur as such a normalization. We parameterize these examples,
or at least large classes of them, using the module of K\"ahler differentials
of a relevant field extension.Comment: To appear in Trans. Amer. Math. So
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