Mike Prest

Covers in finitely accessible categories

We show that in a finitely accessible additive category every class of objects closed under direct limits and pure epimorphic images is covering. In particular, the classes of flat objects in a locally finitely presented additive category and of absolutely pure objects in a locally coherent category are covering.

Remarks on elementary duality

Abstract Elementary duality between left and right modules over a ring, especially its interpretation in terms of the relevant functor categories, is discussed, as is the relationship between these categories of functors and sorts in theories of modules. A topology on the set of indecomposable pure-injective modules over a ring is introduced. This topology is dual to the Ziegler topology and may be seen as a generalisation of the Zariski topology.

CLASSIFYING SERRE SUBCATEGORIES OF FINITELY PRESENTED MODULES

Given a commutative coherent ring R, a bijective correspondence between the thick subcategories of perfect complexes D per (R) and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective modules are used in an essential way.

The Ziegler and Zariski Spectra of Some Domestic String Algebras

It was a conjecture of the second author that the Cantor–Bendixson rank of the Ziegler spectrum of a finite-dimensional algebra is either less than or equal to 2 or is undefined. Here we refute this conjecture by describing the Ziegler spectra of some domestic string algebras where arbitrary finite values greater than 2 are obtained. We give a complete description of the Ziegler and Gabriel–Zariski spectra of the simplest of these algebras. The conjecture has been independently refuted by Schröer who, extending his work (1997) on these algebras, computed their Krull–Gabriel dimension.

Topological and Geometric Aspects of the Ziegler Spectrum

The aim here is to emphasise the topological and geometric structure that the Ziegler spectrum carries and to illustrate how this structure may be used in the analysis of particular examples. There is not space here for me to give a survey of what is known about the Ziegler spectrum so there are a number of topics that I will just mention in order to give some indication of what lies beyond what is discussed here.

Epimorphisms of rings, interpretations of modules and strictly wild algebras

Abstract It is shown that an epimorphism from a ring R to a ring S induces an embedding of the Ziegler spectrum of S as a closed subset of the Ziegler spectrum of R.

Rings of finite representation type and modules of finite Morley rank

This paper deals with rings whose modules have very pleasant decomposition properties: the right pure semisimple rings. The category of right modules over such a ring is, in many ways, similar to the category of modules over a semisimple artinian ring. Yet it is still an open question whether or not pure semisimplicity actually is a two-sided property (and hence [30] coincides with being of finite representation type). Most of the arguments in this paper are model-theoretic-algebraic in nature: it seems that ideas from model theory, and from the model theory of modules, find natural application in the context considered here. The paper is organized into three sections. The first is introductory and with it I attempt to make the other sections accessible to a reasonably wide readership. In the second section are gathered together various equivalents to right pure semisimplicity. Their equivalence is given a comparatively short, unified, essentially model-theoretic-algebraic proof. The main theorem, in the third section, is that a ring is of finite representation type if and only if all its (right) modules have finite Morley rank. A number of related results are developed. Most of these involve some model theory in their statement, but many are, in essense, algebraic. Since a good deal of information has been compressed into the introductory section, this section should, perhaps, first be read through quickly, and then referred back to as the need arises. This section also is meant to serve as background to a sequel, in which I consider pp formulas and types in terms of the corresponding matrices, and in which the main aim is more purely algebraic. The global conventions are: R denotes a ring with identity; “module”

Torsion classes of finite type and spectra

Given a commutative ring

Serial functors, Jacobson radical and representation type

R

Interpreting modules in modules

(respectively a positively graded commutative ring