arXiv subjects

Neil Epstein

Publications and source records attributed to Neil Epstein.

At least 37 records · Page 2Linked to original sources

arXiv2021

The Ohm-Rush content function III: Completion, globalization, and power-content algebras

One says that a ring homomorphism $R \rightarrow S$ is Ohm-Rush if extension commutes with arbitrary intersection of ideals, or equivalently if for any element $f\in S$, there is a unique smallest ideal of $R$ whose extension to $S$ contains $f$, called the content of $f$. For Noetherian local rings, we analyze whether the completion map is Ohm-Rush. We show that the answer is typically `yes' in dimension one, but `no' in higher dimension, and in any case it coincides with the content map having good algebraic properties. We then analyze the question of when the Ohm-Rush property globalizes in faithfully flat modules and algebras over a 1-dimensional Noetherian domain, culminating both in a positive result and a counterexample. Finally, we introduce a notion that we show is strictly between the Ohm-Rush property and the weak content algebra property.

math.AC↗

arXiv2021

Regularity and intersections of bracket powers

Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.

math.AC↗

arXiv2020

The McCoy property in Ohm-Rush algebras

An Ohm-Rush algebra $R \rightarrow S$ is called *McCoy* if for any zero-divisor $f$ in $S$, its content $c(f)$ has nonzero annihilator in $R$, because McCoy proved this when $S=R[x]$. We answer a question of Nasehpour by giving an example of a faithfully flat Ohm-Rush algebra with the McCoy property that is not a weak content algebra. However, we show that a faithfully flat Ohm-Rush algebra is a weak content algebra iff $R/I \rightarrow S/I S$ is McCoy for all radical (resp. prime) ideals $I$ of $R$. When $R$ is Noetherian (or has the more general \emph{fidel (A)} property), we show that it is equivalent that $R/I \rightarrow S/IS$ is McCoy for all ideals.

math.AC↗

arXiv2020

Nakayama closures, interior operations, and core-hull duality

Exploiting the interior-closure duality developed by Epstein and R.G., we show that for the class of Matlis dualizable modules $\mathcal{M}$ over a Noetherian local ring, when cl is a Nakayama closure and i its dual interior, there is a duality between cl-reductions and i-expansions that leads to a duality between the cl-core of modules in $\mathcal{M}$ and the i-hull of modules in $\mathcal{M}^\vee$. We further show that many algebra and module closures and interiors are Nakayama and describe a method to compute the interior of ideals using closures and colons. We use our methods to give a unified proof of the equivalence of F-rationality with F-regularity, and of F-injectivity with F-purity, in the complete Gorenstein local case. Additionally, we give a new characterization of the finitistic tight closure test ideal in terms of maps from $R^{1/p^e}$. Moreover, we show that the liftable integral spread of a module exists.

math.AC↗

arXiv2018

The Ohm-Rush content function II. Noetherian rings, valuation domains, and base change

The notion of an Ohm-Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan-Hochster proof of Stillman's conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from our previous article of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Prüfer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions.

math.AC↗

arXiv2018

On monomial ideals and their socles

For a finite subset $M\subset [x_1,\ldots,x_d]$ of monomials, we describe how to constructively obtain a monomial ideal $I\subseteq R = K[x_1,\ldots,x_d]$ such that the set of monomials in $\text{Soc}(I)\setminus I$ is precisely $M$, or such that $\overline{M}\subseteq R/I$ is a $K$-basis for the the socle of $R/I$. For a given $M$ we obtain a natural class of monomials $I$ with this property. This is done by using solely the lattice structure of the monoid $[x_1,\ldots,x_d]$. We then present some duality results by using anti-isomorphisms between upsets and downsets of $(\mathbb Z^d,\preceq)$. Finally, we define and analyze zero-dimensional monomial ideals of $R$ of type $k$, where type $1$ are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in $\mathbb Z^d$.

math.AC↗

arXiv2017

Gaussian elements of a semicontent algebra

The connection between a univariate polynomial having locally principal content and the content function acting like a homomorphism (the so-called Gaussian property) has been explored by many authors. In this work, we extend several such results to the contexts of multivariate polynomials, power series over a Noetherian ring, and base change of affine $K$-algebras by separable algebraically closed field extensions. We do so by using the framework of the Ohm-Rush content function. The correspondence is particularly strong in cases where the base ring is approximately Gorenstein or the element of the target ring is regular.

math.AC↗

arXiv2016

A computation concerning relative Hilbert-Kunz multiplicities

In preparing the paper "Some extensions of Hilbert-Kunz multiplicity", we had occasion to perform an intricate set of computations pertaining to a single illustrative example. In the end, we have decided not to include the computations in the main paper, but instead to publish them here for later reference. In particular, the current preprint gives an example where the converse to one of our main theorems holds, even though it does not fit into any of the previously known cases where a converse holds.

math.AC↗

arXiv2016

Perinormality in pullbacks

We further develop the notion of perinormality from our last paper, showing that it is preserved by many pullback constructions. In doing so, we introduce the concepts of relative perinormality and fragility for ring extensions.

math.AC↗

arXiv2016

Hilbert-Kunz multiplicity of products of ideals

We give bounds for the Hilbert-Kunz multiplicity of the product of two ideals, and we characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations.

math.AC↗

arXiv2015

Perinormality -- a generalization of Krull domains

We introduce a new class of integral domains, the perinormal domains, which fall strictly between Krull domains and weakly normal domains. We establish basic properties of the class, and in the case of universally catenary domains we give equivalent characterizations of perinormality. (Later on, we point out some subtleties that occur only in the non-Noetherian context.) We also introduce and explore briefly the related concept of global perinormality, including a relationship with divisor class groups. Throughout, we provide illuminating examples from algebra, geometry, and number theory.

math.AC↗

arXiv2015

Continuous closure, axes closure, and natural closure

Let $R$ be a reduced affine $\mathbb C$-algebra, with corresponding affine algebraic set $X$. Let $\mathcal C(X)$ be the ring of continuous (Euclidean topology) $\mathbb C$-valued functions on $X$. Brenner defined the \emph{continuous closure} $I^{\rm cont}$ of an ideal $I$ as $I\mathcal C(X) \cap R$. He also introduced an algebraic notion of \emph{axes closure} $I^{\rm ax}$ that always contains $I^{\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \in I^{\rm ax}$ if its image is in $IS$ for every homomorphism $R \to S$, where $S$ is a one-dimensional complete seminormal local ring. We also introduce the \emph{natural closure} $I^\natural$ of $I$. One of many characterizations is $I^\natural = I + \{f \in R: \exists n >0 \text{ with } f^n \in I^{n+1}\}$. We show that $I^\natural \subseteq I^{\rm ax}$, and that when continuous closure is defined, $I^\natural \subseteq I^{\rm cont }\subseteq I^{\rm ax}$. Under mild hypotheses on the ring, we show that $I^\natural= I^{\rm ax}$ when $I$ is primary to a maximal ideal, and that if $I$ has no embedded primes, then $I = I^\natural$ if and only if $I = I^{\rm ax}$, so that $I^{\rm cont}$ agrees as well. We deduce that in the polynomial ring $\mathbb C[x_1, \ldots, x_n]$, if $f = 0$ at all points where all of the ${\partial f \over \partial x_i}$ are 0, then $f \in ( {\partial f \over \partial x_1}, \, \ldots, \, {\partial f \over \partial x_n})R$. We characterize $I^{\rm cont}$ for monomial ideals in polynomial rings over $\mathbb C$, but we show that the inequalities $I^\natural \subset I^{\rm cont}$ and $I^{\rm cont} \subset I^{\rm ax}$ can be strict for monomial ideals even in dimension 3. Thus, $I^{\rm cont}$ and $I^{\rm ax}$ need not agree, although we prove they are equal in $\mathbb C[x_1, x_2]$.

math.AC↗

arXiv2014

The Ohm-Rush content function

The content of a polynomial over a ring $R$ is a well understood notion. Ohm and Rush generalized this concept of a content map to an arbitrary ring extension of $R$, although it can behave quite badly. We examine five properties an algebra may have with respect to this function -- content algebra, weak content algebra, semicontent algebra (our own definition), Gaussian algebra, and Ohm-Rush algebra. We show that the Gaussian, weak content, and semicontent algebra properties are all transitive. However, transitivity is unknown for the content algebra property. We then compare the Ohm-Rush notion with the more usual notion of content in the power series context. We show that many of the given properties coincide for the power series extension map over a valuation ring of finite dimension, and that they are equivalent to the value group being order-isomorphic to the integers or the reals. Along the way, we give a new characterization of Prüfer domains.

math.AC↗

arXiv2014

A Dedekind-Mertens theorem for power series rings

We prove a power series ring analogue of the Dedekind-Mertens lemma. Along the way, we give limiting counterexamples, we note an application to integrality, and we correct an error in the literature.

math.AC↗

arXiv2014

Liftable integral closure

We develop the basic properties of an essentially new closure operation on submodules, the \emph{liftable integral closure} of a submodule, including its relationships with the two prevailing notions of integral closure of submodules. We show that for a quite general class of local rings, every finite length module may be represented as a quotient of the form $T/L$, where $T$ is torsionless and integrally dependent on $L$.

math.AC↗

arXiv2013

Strong Krull primes and flat modules

There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable replacement. In this paper, we show that the behavior of strong Krull primes most closely resembles that of associated primes over a Noetherian ring. We prove an analogue of a theorem of Epstein and Yao characterizing flat modules in terms of associated primes by replacing them with strong Krull primes. Also, we partly generalize a classical equational theorem regarding flat base change and associated primes in Noetherian rings. That is, when associated primes are replaced by strong Krull primes, we show containment in general and equality in many special cases. One application is of interest over any Noetherian ring of prime characteristic. We also give numerous examples to show that our results fail if other popular generalizations of associated primes are used in place of strong Krull primes.

math.AC↗

arXiv2013

Algebra retracts and Stanley-Reisner rings

In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field $k$ are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner rings over a field $k$ are again Stanley-Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over $k$ descend along graded algebra retracts.

math.AC↗

arXiv2013

Semistar operations and standard closure operations

Let $R$ be a commutative ring. It is shown that there is an order isomorphism between a popular class of finite type closure operations on the ideals of $R$ and the poset of semistar operations of finite type.

math.AC↗