arXiv subjects
Publications and source records attributed to Neil Epstein.
Given an integral domain $D$ with fraction field $F$, its *reciprocal complement* is the subring of $F$ generated by all $1/d$ for nonzero $d$ in $D$. This paper serves doubly as a survey of the current state of the field and an update with new results and connections.
The theory of singularities defined by Frobenius has been extensively developed for $F$-finite rings and for rings that are essentially of finite type over excellent local rings. However, important classes of non-local excellent rings, such as Tate algebras and their quotients (affinoid algebras) do not fit into either setting. We investigate here a framework for moving beyond the $F$-finite setting, developing the theory of three related classes of regular rings defined by properties of Frobenius. In increasing order of strength, these are Frobenius Ohm-Rush (FOR), Frobenius intersection flat, and Frobenius Ohm-Rush trace (FORT). We show that Tate algebras are Frobenius intersection flat, from which it follows that reduced affinoid algebras have test elements using a result of Sharp. We also deduce new cases of the openness of the $F$-pure locus.
Using the fact that the structure sheaf of a resolution of singularities, or regular alteration, pushes forward to a Cohen-Macaulay complex in equal characteristic zero with a differential graded algebra structure, we introduce a tight-closure-like operation on ideals in equal characteristic zero using the Koszul complex, which we call KH (Koszul-Hironaka). We prove it satisfies various strong colon capturing properties, a substantial case of the Brian\c{c}on-Skoda theorem, and it behaves well under finite extensions. It detects rational singularities and is tighter than tight closure in equal characteristic zero. Furthermore, its formation commutes with localization and it can be computed effectively. On the other hand, the product of the KH closures of ideals is not always contained in the KH of the product, as one might expect. We also explore a related closure operation (canonical alteration closure), induced by canonical modules of regular alterations, which detects KLT-type singularities in equal characteristic zero and which is closely related to tight closure in characteristic $p > 0$. For parameter ideals we show both these closure operations coincide and reduce modulo $p \gg 0$ to tight closure. Finally, we explore an intermediate operation (Hironaka pre-closure) which which satisfies numerous desired properties, but for which we have not been able to prove idempotence.
The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be fundamental, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of local fundamentality is introduced and its behavior developed. On the algebraic side, the notions of locally, geometrically, and generically unit-additive rings are introduced, thus complementing unit-additivity as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout.
Let $T$ be a subset of a ring $A$, and let $M$ be an $A$-module. We study the additive subgroups $F$ of $M$ such that, for all $x \in M$, if $tx \in F$ for some $t \in T$, then $x \in F$. We call any such subset $F$ of $M$ a $T$-factroid of $M$, which is a kind of dual to the notion of a $T$-submodule of $M$. We connect the notion with the zero-divisors on $M$, various classes of primary and prime ideals of $A$, Euclidean domains, and the recent concepts of unit-additive commutative rings and of Egyptian fractions with respect to a multiplicative subset of a commutative ring. We also introduce a common generalization of local rings and unit-additive rings, called *sublocalizable* rings, and relate them to $T$-factroids.
Closure operations such as tight and integral closure and test ideals have appeared frequently in the study of commutative algebra. This articles serves as a survey of the authors' prior results connecting closure operations, test ideals, and interior operations via the more general structure of pair operations. Specifically, we describe a duality between closure and interior operations generalizing the duality between tight closure and its test ideal, provide methods for creating pair operations that are compatible with taking quotient modules or submodules, and describe a generalization of core and its dual. Throughout, we discuss how these ideas connect to common constructions in commutative algebra.
The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$ variables over a field. In particular, $R(D)$ is an $n$-dimensional, local, non-Noetherian, non-integrally closed, non-factorial, atomic G-domain, with infinitely many prime ideals at each height other than $0$ and $n$.
A ring is *unit-additive* if a sum of units is always either a unit or nilpotent. For example, $k[X]$ and $k[X]/(X^2)$ are unit-additive, but $\mathbb Z$ is not. We prove a wide-ranging theorem about unit-additivity in semigroup rings, showing among other things that an affine semigroup ring $A[M]$ is unit-additive if and only if $A$ is unit-additive and $M$ has no nontrivial invertible elements. Passing to algebraic geometry, we show that an irreducible affine variety $V$ over an algebraically closed field $k$ has unit-additive coordinate ring if and only if any polynomial mapping $V \rightarrow k$ has a root. This then places $\mathbb A^1_k$ into the class of varieties that satisfy a version of the Fundamental Theorem of Algebra. Specializing to elliptic curves, we show that the affine coordinate ring of an elliptic curve is always unit-additive. The concept of unit additivity leads to the related concept of unit dimension -- i.e. how far is an integral domain from being unit-additive? It turns out that rings of unit dimension 1 are of some interest, as they include the rings of integers of number fields, all power series rings, and most local rings. We construct rings of all unit dimensions and show that in the affine setting, unit dimension is bounded above by Krull dimension. We also construct the *unit-additive closure* of an integral domain $D$, being the smallest subring of the fraction field of $D$ that is unit-additive, as a localization at a certain multiplicative set in $D$. Throughout, we make connections with well-studied structures like PIDs, Euclidean domains, and the UU property.
Let D be a Euclidean domain, with fraction field K. Let R(D) be the subring of K generated by the reciprocals of the nonzero elements of D. The main theorem states that if R(D) is not equal to K, then R(D) is a rank 1 discrete valuation ring that contains a field consisting of the units of D along with 0. Connections are made to ideas from medieval Italian mathematics.
We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.
The notion of an \emph{Egyptian} integral domain $D$ (where every fraction can be written as a sum of unit fractions with denominators from $D$) is extended here to the notion that a ring $R$ is \emph{$W$-Egyptian}, with $W$ a multiplicative set in $R$. The new notion allows denominators just from $W$. It is shown that several results about Egyptian domains can be extended to the $W$-Egyptian context, though some cannot, as shown in counterexamples. In particular, being a sum of unit fractions from $W$ is \emph{not} equivalent to being a distinct sum of unit fractions from $W$, so we need to add the notion of the \emph{strictly} $W$-Egyptian ring. Connections are made with Jacobson radicals, power series, products of rings, factor rings, modular arithmetic, and monoid algebras.
The notion of an Egyptian domain (where the analogue of Egyptian fractions works appropriately), first explored by Guerrieri-Loper-Oman, is extended to the more general notions of generically and locally Egyptian domains. Results from the previous paper are extended, as well as reinterpreted in the new expanded context. It is shown that localizations of polynomial rings are typically Egyptian, that positive semigroup-graded domains are not Egyptian, and that affine semigroup rings are not Egyptian unless the semigroup is a group. However, any finitely generated algebra over a field or $\mathbb Z$ is shown to be locally Egyptian. It is further shown that if $R$ is a pullback of $S$, then $R$ is Egyptian if and only if $S$ is.
In this article we first compare the set of elements in the socle of an ideal of a polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ that are not in the ideal itself and Macaulay's inverse systems of such polynomial algebras in a purely combinatorial way for monomial ideals, and then develop some closure operational properties for the related poset ${{\nats}_0^d}$. We then derive some algebraic propositions of $\Gamma$-graded rings that then have some combinatorial consequences. Interestingly, some of the results from this part that uniformly hold for polynomial rings are always false when the ring is local. We finally delve into some direct computations, w.r.t.~a given term order of the monomials, for general zero-dimensional Gorenstein ideals and deduce a few explicit observations and results for the inverse systems from some recent results about socles.
There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize 3 ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including *hereditary*, *residual*, and *cofunctorial*, and see how they interact with other properties such as the *finitistic* property.
In this note, a condition (\emph{open persistence}) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme $X$ can be extended to a (pre)closure operation on sheaves of submodules of a coherent $\mathcal{O}_X$-module (resp. sheaves of ideals in $\mathcal{O}_X$). A second condition (\emph{glueability}) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (\emph{semi F-regularity}) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture.
We develop a duality for operations on nested pairs of modules that generalizes the duality between absolute interior operations and residual closure operations from [ER21], extending our previous results to the expanded context. We apply this duality in particular to integral and basically full closures and their respective cores to obtain integral and basically empty interiors and their respective hulls. We also dualize some of the known formulas for the core of an ideal to obtain formulas for the hull of a submodule of the injective hull of the residue field. The article concludes with illustrative examples in a numerical semigroup ring.
Broadening existing results in the literature to much wider classes of rings, we prove among other things: 1. Reduced quotients of excellent regular rings of characteristic $p$ admit big test elements, 2. The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is a discrete subset of $\mathbb Q$, and 3. If $R$ is a quotient of a locally excellent regular ring of prime characteristic, then there is a uniform upper bound on the Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue fields of $R$. To do so, we develop the parallel theories of Ohm-Rush and intersection flat algebras. We show that both properties can be checked locally in flat maps of Noetherian rings. We show that intersection-flatness admits a content theory parallel to that of Ohm-Rush content for Ohm-Rush algebras. We develop descent results for these properties. Using the descent result for intersection flatness, we obtain a local condition under which a faithfully flat map of Noetherian rings must be intersection-flat. The local condition for intersection-flatness allows us to conclude that finitely generated faithfully flat algebras over a Noetherian ring are intersection-flat. Combining the local condition for intersection flatness with results of Kunz and Radu yields the conclusion that the Frobenius endomorphism associated to a locally excellent regular ring of prime characteristic is intersection-flat, thus answering a question of Sharp. Applications of the latter result include the three enumerated results above. We also get applications to tight closure and parameter test ideals.
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.