Neil Epstein

A tight closure analogue of analytic spread

An analogue of the theory of integral closure and reductions is developed for a more general class of closures, called Nakayama closures. It is shown that tight closure is a Nakayama closure by proving a “Nakayama lemma for tight closure”. Then, after strengthening A. Vracius theory of *-independence and the special part of tight closure, it is shown that all minimal *-reductions of an ideal in an analytically irreducible excellent local ring of positive characteristic have the same minimal number of generators. This number is called the *-spread of the ideal, by analogy with the notion of analytic spread.

Some extensions of Hilbert–Kunz multiplicity

Let R be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do not assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures give truly different numerical results) and explore special cases where equivalence with matching tight closure can be shown. All of our measures derive ultimately from Hilbert–Kunz multiplicity.

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.

Algebra retracts and Stanley–Reisner rings

Abstract 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.

Reductions and special parts of closures

Abstract We provide an axiomatic framework for working with a wide variety of closure operations on ideals and submodules in commutative algebra, including notions of reduction, independence, spread, and special parts of closures. This framework is applied to tight, Frobenius, and integral closures. Applications are given to evolutions and special Briancon–Skoda theorems.

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 Prufer domains.

Progress in Commutative Algebra 2 : Closures, Finiteness and Factorization

This article is a survey of closure operations on ideals in commutative rings, with an emphasis on structural properties and on using tools from one part of the field to analyze structures in another part. The survey is broad enough to encompass the radical, tight closure, integral closure, basically full closure, saturation with respect to a fixed ideal, and the v-operation, among others.

Criteria for flatness and injectivity

Let R be a commutative Noetherian ring. We give criteria for flatness of R-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if R has characteristic p, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of R-modules in terms of coassociated primes and (h-)divisibility of certain Hom-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a Hom-module base change, and a local criterion for injectivity.

Zero-divisor graphs of nilpotent-free semigroups

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an Armendariz map between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.

Phantom depth and flat base change

We prove that if f: (R, m) → (S, n) is a flat local homomorphism, S/mS is Cohen-Macaulay and F-injective, and R and S share a weak test element, then a tight closure analogue of the (standard) formula for depth and regular sequences across flat base change holds. As a corollary, it follows that phantom depth commutes with completion for excellent local rings. We give examples to show that the analogue does not hold for surjective base change.