435 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
Generic formal fibers and analytically ramified stable rings
Let be a local Noetherian domain of Krull dimension . Heinzer,
Rotthaus and Sally have shown that if the generic formal fiber of has
dimension , then is birationally dominated by a one-dimensional
analytically ramified local Noetherian ring having residue field finite over
the residue field of . We explore further this correspondence between prime
ideals in the generic formal fiber and one-dimensional analytically ramified
local rings. Our main focus is on the case where the analytically ramified
local rings are stable, and we show that in this case the embedding dimension
of the stable ring reflects the embedding dimension of a prime ideal maximal in
the generic formal fiber, thus providing a measure of how far the generic
formal fiber deviates from regularity. A number of characterizations of
analytically ramified local stable domains are also given.Comment: To appear in Nagoya J. Mat
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
IT’S NOT THEM, IT’S YOU: A CASE STUDY CONCERNING THE EXCLUSION OF NON-WESTERN PHILOSOPHY
My purpose in this essay is to suggest, via case study, that if Anglo-American philosophy is to become more inclusive of non-western traditions, the discipline requires far greater efforts at self-scrutiny. I begin with the premise that Confucian ethical treatments of manners afford unique and distinctive arguments from which moral philosophy might profit, then seek to show why receptivity to these arguments will be low. I examine how ordinary good manners have largely fallen out of philosophical moral discourse in the west, looking specifically at three areas: conditions in the 18th and 19th centuries that depressed philosophical attention to manners; discourse conventions in contemporary philosophy that privilege modes of analysis not well fitted to close scrutiny of manners; and a philosophical culture that implicitly encourages indifference or even antipathy toward polite conduct. I argue that these three areas function in effect to render contemporary discourse inhospitable to greater inclusivity where Confucianism is concerned and thus, more broadly, that greater self-scrutiny regarding unexamined, parochial western commitments and practices is necessary for genuine inclusivity
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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