arXiv subjects

Bruce Olberding

Publications and source records attributed to Bruce Olberding.

43 records · Page 3Linked to original sources

arXiv2013

Idempotent generated algebras and Boolean powers of commutative rings

A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker R-algebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module. For an indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. For a totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is equivalent to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology.

math.RA↗

arXiv2013

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.

math.AC↗

arXiv2012

Integrally closed rings in birational extensions of two-dimensional regular local rings

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined locally by finitely many valuation overrings of $D$. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of $D$, and, when $D$ is analytically normal, this property holds for $D$ if and only if it holds for the completion of $D$. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where $D$ is a regular local ring and $f$ is a regular parameter of $D$, then $H$ is determined locally by a single valuation. As a consequence, we show that if $H$ is also the integral closure of a finitely generated $D$-algebra, then the exceptional prime ideals of the extension $H/D$ are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.

math.AC↗

arXiv2012

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 Kähler differentials of a relevant field extension.

math.AC↗

arXiv2012

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.

math.AC↗

arXiv2012

Derivations, generic formal fibers and bad Noetherian rings

We consider a circle of ideas involving differential algebra, local Noetherian rings, and their generic formal fibers. Connecting these ideas gives rise to what we term a "twisted" subring $R$ of a ring $S$. Each such subring $R$ arises as a pullback of a derivation taking values in an $S$-module $K$. The twisting relationship proves to be a kind of inversion of Nagata idealization: whereas idealization extends $S$ to the larger ring $S \star K$, twisting produces a subring of $S$ which behaves much like the ring $S \star K$. The rings produced in this manner exhibit pathological features, such as failing to have finite normalization, but in spite of this they are quite tractable and conceptually (if not practically) easy to locate, and in this way provide a rich but manageable source of non-standard Noetherian rings. The theory developed to support this construction involves extensive use of both Noetherian and non-Noetherian commutative ring theory, as well as differential algebra.

math.AC↗

arXiv2012

A counterpart to Nagata idealization

Idealization of a module $K$ over a commutative ring $S$ produces a ring having $K$ as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an $S$-module $K$ and derivation $D:S\rightarrow K$ a subring $R$ of $S$ that behaves like the idealization of $K$ but is such that when $S$ is a domain, so is $R$. The ring $S$ is contained in the normalization of $R$ but is finite over $R$ only when $R = S$. We determine conditions under which $R$ is Noetherian, Cohen-Macaulay, Gorenstein, a complete intersection or a hypersurface. When $R$ is local, then its ${\bf m}$-adic completion is the idealization of the ${\bf m}$-adic completions of $S$ and $K$.

math.AC↗