12 research outputs found
Krull-gabriel dimension and the ziegler spectrum
We provide an introduction to the the Krull-Gabriel dimension of a ring, as well as many related ideas. In particular, we outline how Krull-Gabriel dimension relates to the Cantor-Bendixson rank of the Ziegler spectrum and with the Jacobson radical of the module category. We also include a list of examples of rings and categories where the Krull-Gabriel dimension has been calculated
Some model theory over hereditary noetherian domains
Questions in the model theory of modules over hereditary noetherian domains are investigated with particular attention being paid to differential polynomial rings and to generalized Weyl algebras. We prove that there exists no isolated point in the Ziegler spectrum over a simple hereditary generalized Weyl algebra A of the sort considered by Bavula [Algebra i Analiz 4(1) (1992), 75–97] over a field k with char(k) = 0 (the first Weyl algebra A1(k) is one such) and the category of finite length modules over A does not have any almost split sequence. We show that the theory of all modules over a wide class of generalized Weyl algebras and related rings interprets the word problem for groups, and in the case that the field is countable there exists a superdecomposable pure-injective module over A. This class includes, for example, the universal enveloping algebra Usl2(k)
Purity in categories of sheaves
We consider categorical and geometric purity for sheaves of modules over a
scheme satisfying some mild conditions, both for the category of all sheaves
and for the category of quasicoherent sheaves. We investigate the relations
between these four purities and compute a number of examples, in particular
describing both the geometric and categorical Ziegler spectra for the category
of quasicoherent sheaves over the projective line over a field
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Model Theory and Groups
The aim of the workshop was to discuss the connections between model theory and group theory. Main topics have been the interaction between geometric group theory and model theory, the study of the asymptotic behaviour of geometric properties on groups, and the model theoretic investigations of groups of finite Morley rank around the Cherlin-Zilber Conjecture
Ziegler spectra of valuation rings
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