Burch ideals and Burch rings

We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological, categorical. We relate them to other classes of ideals and rings in the literature.Comment: 23 pages, add Example 2.2, Prop 5.5 and Example 5.

When are KE-closed subcategories torsion-free classes?

Let $R$ be a commutative noetherian ring and denote by $\mathsf{mod} R$ the category of finitely generated $R$-modules. In this paper, we study KE-closed subcategories of $\mathsf{mod} R$, that is, additive subcategories closed under kernels and extensions. We first give a characterization of KE-closed subcategories: a KE-closed subcategory is a torsion-free class in a torsion-free class. As an immediate application of the dual statement, we give a conceptual proof of Stanley-Wang's result about narrow subcategories. Next, we classify the KE-closed subcategories of $\mathsf{mod} R$ when $\mathrm{dim} R \le 1$ and when $R$ is a two-dimensional normal domain. More precisely, in the former case, we prove that KE-closed subcategories coincide with torsion-free classes in $\mathsf{mod} R$. Moreover, this condition implies $\mathrm{dim} R \le 1$ when $R$ is a homomorphic image of a Cohen-Macaulay ring (e.g. a finitely generated algebra over a regular ring). Thus, we give a complete answer for the title.Comment: 16 pages, comments welcome

On the Apparent Activation Energy for Clustering in Dilute Al-Zn Alloys

The clustering phenomenon was observed when dilute Al-Zn alloys were annealed at temperatures higher than the solvus of the G.P.zones. In this report the apparent activation energy for clustering is estimated and compared with the experimental results. The estimated value of the apparent activation energy for clustering in several Al-Zn alloys comes to 0.51 eV, which is larger than the effective migration energy 0.43 eV of Zn atoms in Al-Zn alloys

Semidualizing Modules over Numerical Semigroup Rings

A semidualizing module is a generalization of Grothendieck's dualizing module. For a local Cohen-Macaulay ring $R$, the ring itself and its canonical module are always realized as (trivial) semidualizing modules. Reasonably, one might ponder the question; when do nontrivial examples exist? In this paper, we study this question in the realm of numerical semigroup rings and completely classify which of these rings with multiplicity at most 9 possess a nontrivial semidualizing module. Using this classification, we construct numerical semigroup rings in any multiplicity at least 9 possesses a nontrivial semidualizing module.Comment: 22 pages, comments welcom

On the projective dimension of tensor products of modules

In this paper we consider a question of Roger Wiegand, which is about tensor products of finitely generated modules that have finite projective dimension over commutative Noetherian rings. We construct modules of infinite projective dimension (and of infinite Gorenstein dimension) whose tensor products have finite projective dimension. Furthermore we determine nontrivial conditions under which such examples cannot occur. For example we prove that, if the tensor product of two nonzero modules, at least one of which is totally reflexive (or equivalently Gorenstein-projective), has finite projective dimension, then both modules in question have finite projective dimension.Comment: 14 page