On the geometry of Pr\"ufer intersections of valuation rings

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is $A = \bigcap_{V \in Z}V$. All rings between $D$ and $F$ that are integrally closed in $F$ 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 $A$ is a Pr\"ufer domain. We give geometric criteria for when $A$ 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 $Z$ into the projective line ${\mathbb{P}}^1_D$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 $S$ 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 $S$, it produces a subring $R$ of $S$ such that $R \subseteq S$ is subintegral.Comment: 25 pages; to appear in Journal of Pure and Applied Algebr

Generic formal fibers and analytically ramified stable rings

Let $A$ be a local Noetherian domain of Krull dimension $d$. Heinzer, Rotthaus and Sally have shown that if the generic formal fiber of $A$ has dimension $d-1$, then $A$ is birationally dominated by a one-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of $A$. 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

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