arXiv subjects

Rebecca R. G

Publications and source records attributed to Rebecca R. G.

5 recordsLinked to original sources

arXiv2024

Rationality for arbitrary closure operations and the test ideal of full extended plus closure

We extend the notion of F-rationality to other closure operations, inspired by the work of Smith, Epstein and Schwede, and Ma and Schwede, which describe F-rationality in terms of the canonical module and top local cohomology module. We give conditions for a closure operation cl on a Cohen-Macaulay complete local ring under which cl-rationality is equivalent to parameter ideals being cl-closed. We also demonstrate that full extended plus closure as defined by Heitmann and weak full extended plus closure as defined by the first named author have no big test elements.

math.AC↗

arXiv2023

Betti numbers of the Frobenius powers of the maximal ideal over a general hypersurface

The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of $p$. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an $\mathfrak{c}$-compressed Gorenstein Artinian graded algebra from its socle degree, where $\mathfrak{c}$ is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert-Kunz function of the hypersurface ring, as well as the Castelnuovo-Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal.

math.AC↗

arXiv2017

Closure operations that induce big Cohen-Macaulay algebras

We study closure operations over a local domain R that satisfy a set of axioms introduced by Geoffrey Dietz. The existence of a closure operation satisfying the axioms (called a Dietz closure) is equivalent to the existence of a big Cohen-Macaulay module for R. We give an additional axiom (the Algebra Axiom), such that the existence of a Dietz closure satisfying this axiom is equivalent to the existence of a big Cohen-Macaulay algebra. We prove that many closure operations satisfy the Algebra Axiom, whether or not they are Dietz closures. We discuss the smallest big Cohen-Macaulay algebra closure on a given ring, and show that every Dietz closure satisfying the Algebra Axiom is contained in a big Cohen-Macaulay algebra closure. This leads to proofs that in rings of characteristic p > 0, every Dietz closure satisfying the Algebra Axiom is contained in tight closure, and there exist Dietz closures that do not satisfy the Algebra Axiom. Several of these results rely on work from arXiv:1512.06843.

math.AC↗

arXiv2017

Big Cohen-Macaulay and seed algebras in equal characteristic zero via ultraproducts

Let $R$ be a commutative, local, Noetherian ring. In a past article, the first author developed a theory of $R$-algebras, termed seeds, that can be mapped to balanced big Cohen-Macaulay $R$-algebras. In prime characteristic $p$, seeds can be characterized based on the existence of certain colon-killers, integral extensions of seeds are seeds, tensor products of seeds are seeds, and the seed property is stable under base change between complete, local domains. As a result, there exist directed systems of big Cohen-Macaulay algebras over complete, local domains. In this work, we will show that these properties can be extended to analogous results in equal characteristic zero. The primary tool for the extension will be the notion of ultraproducts for commutative rings as developed by Schoutens and Aschenbrenner.

math.AC↗

arXiv2016

Closure operations that induce big Cohen-Macaulay modules and classification of singularities

Geoffrey Dietz introduced a set of axioms for a closure operation on a complete local domain R so that the existence of such a closure operation is equivalent to the existence of a big Cohen-Macaulay module. These closure operations are called Dietz closures. In complete rings of characteristic p > 0, tight closure and plus closure satisfy the axioms. In order to study these closures, we define module closures and discuss their properties. For many of these properties, there is a smallest closure operation satisfying the property. We discuss properties of big Cohen-Macaulay module closures, and prove that every Dietz closure is contained in a big Cohen-Macaulay module closure. Using this result, we show that under mild conditions, a ring R is regular if and only if all Dietz closures on R are trivial. Finally, we show that solid closure in equal characteristic 0, integral closure, and regular closure are not Dietz closures, and that all Dietz closures are contained in liftable integral closure.

math.AC↗