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Generic Modules Over Artin Algebras
 Proc. London Math. Soc
, 1995
"... this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a n ..."
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this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a new characterization of the pureinjective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pureinjective. Next we consider indecomposable endofinite modules. Recall that a module is endofinite if it is of finite length when regarded in the natural way as a module over its endomorphism ring. Changing slightly the original definition, we say that a module is generic if it is indecomposable endofinite but not finitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the nonfinitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PIrings. Existence results for generic modules over dualizing rings follow in Section 5. Several results in this paper depend on the fact that a functor f : Mod(\Gamma) ! Mod() which commutes with direct limits and products, preserves certain finiteness conditions. For example, if a \Gammamodule M is endofinite then f(M) is endofinite. If in addition End \Gamma (M) is a PIring, then End (N) is a PIring for every indecomposable direct summand N of f(M ). This material is collected in Section 6 and 7. In Section 8 we introduce an effective method to construct generic modules over artin algebras from socalled generalized tubes. The special case of a tube in the AuslanderReiten quiver is discussed in t...
Vaught's Conjecture for Superstable Theories of Finite Rank
, 1993
"... In this paper we prove Vaught's conjecture for superstable theories in which each complete type has finite U \Gamma rank. The general idea is to associate with the theory an V \Gamma definable group G (called the structure group) which controls the isomorphism types of the models. ..."
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Cited by 7 (2 self)
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In this paper we prove Vaught's conjecture for superstable theories in which each complete type has finite U \Gamma rank. The general idea is to associate with the theory an V \Gamma definable group G (called the structure group) which controls the isomorphism types of the models.
Pureinjective Modules over Tubular Algebras and String Algebras
, 2011
"... We show that, for any tubular algebra, the lattice of ppdefinable subgroups of the direct sum of all indecomposable pureinjective modules of slope r has mdimension 2 if r is rational, and undefined breadth if r is irrational and hence that there are no superdecomposable pureinjectives of ration ..."
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Cited by 4 (0 self)
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We show that, for any tubular algebra, the lattice of ppdefinable subgroups of the direct sum of all indecomposable pureinjective modules of slope r has mdimension 2 if r is rational, and undefined breadth if r is irrational and hence that there are no superdecomposable pureinjectives of rational slope, but there are superdecomposable pureinjectives of irrational slope, if the underlying field is countable. We determine the pureinjective hull of every direct sum string module over a string algebra. If A is a domestic string algebra such that the width of the lattice of ppformulas has defined breadth, then classify “almost all” of the pureinjective indecomposable Amodules.
Pure injectivity and model theory for Gsets
 Journal of Symbolic Logic
"... In the model theory of modules the Ziegler spectrum, the space of indecomposable pureinjective modules, has played a key role. We investigate the possibility of defining a similar space in the context of Gsets where G is a group. 1 ..."
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In the model theory of modules the Ziegler spectrum, the space of indecomposable pureinjective modules, has played a key role. We investigate the possibility of defining a similar space in the context of Gsets where G is a group. 1
Finitistic dimension and Ziegler spectrum
"... : Given a twosided artinian ring , it is shown that the Ziegler spectrum of forms a test class for certain homological properties of . We discuss the finitistic dimension of , Nunke's condition, and also the relation between the big and the little finitistic dimension. Let be a twosided artin ..."
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: Given a twosided artinian ring , it is shown that the Ziegler spectrum of forms a test class for certain homological properties of . We discuss the finitistic dimension of , Nunke's condition, and also the relation between the big and the little finitistic dimension. Let be a twosided artinian ring. Denote by Mod the category of (right) modules and by mod the full subcategory of all finitely presentedmodules. Given amodule M , we denote by pdM its projective dimension, and the finitistic dimension Fd of is the supremum of the projective dimensions of themodules with finite projective dimension. The Ziegler spectrum Zsp of is by definition the set of isomorphism classes of indecomposable pureinjectivemodules. The aim of this note is to show that the Ziegler spectrum forms a test class for certain homological properties of . Furthermore, the Ziegler spectrum carries a topology and we shall use its compactness to obtain an equivalent formulation of Nunke's condition. In the fin...
Pureinjectivity and model theory for Gsets
, 2008
"... In the model theory of modules the Ziegler spectrum, the space of indecomposable pureinjective modules, has played a key role. We investigate the possibility of defining a similar space in the context of Gsets where G is a group. ..."
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In the model theory of modules the Ziegler spectrum, the space of indecomposable pureinjective modules, has played a key role. We investigate the possibility of defining a similar space in the context of Gsets where G is a group.