Purity Relative to Classes of Finitely Presented Modules

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Abstract

Any set of finitely presented left modules defines a relative purity for left modules and also apurity for right modules. Purities defined by various classes are compared and investigated,especially in the contexts of modules over semiperfect rings and over tame hereditary, andmore general, finite-dimensional algebras. Connections between the indecomposable relativelypure-injective modules and closure in the full support topology (a refinement of theZiegler spectrum) are described.Duality between left and right modules is used to define the concept of a class of leftmodules and a class of right modules forming an almost dual pair. Definability of suchclasses is investigated, especially in the case that one class is the closure of a set of finitelypresented modules under direct limits. Elementary duality plays an important role here.Given a set of finitely presented modules, the corresponding proper class of relativelypure-exact sequences can be used to define a relative notion of cotorsion pair, which weinvestigate.The results of this thesis unify and extend a wide range of results in the literature.

Keyword(s)

purity, pure-injective, pure projective, dual module, (m,n)-purity, finite-dimensional algebra, Auslander-Reiten translate, Ziegler closure, full support closure, tame hereditary algebra, adic module, generic module, Prüfer module, definable classes, almost dual pair, cotorsion pair, S-cotorsion pair.

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