The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

this article. Throughout, we restrict to studying finite-dimensional associative algebras (with 1) over an algebraically closed field K, and write D for duality with the field. Except where stated, all modules are left modules. We write A-mod for the category of finite-dimensional A-modules, and A-Mod for the category of all

this paper. Consider the exact functor

. For any associative algebra A over a field K we define a family of algebras \Pi (A) for 2 K\Omega Z K 0 (A). In case A is the path algebra of a quiver, one recovers the deformed preprojective algebra introduced by M. P. Holland and the author. In case A is the coordinate ring of a smooth curve, the family includes the ring of differential operators for A and the coordinate ring of the cotangent bundle for SpecA. In case A is quasi-free and\Omega 1 A is a finitely generated A-A-bimodule we prove that \Pi (A) is well-behaved under localization. We use this to prove a Conze embedding for deformations of Kleinian singularities. If K is an algebraically closed field of characteristic zero and \Gamma is a non-trivial finite subgroup of SL 2 (K) then the coordinate ring of the Kleinian singularity K 2 =\Gamma has a family of deformations O where 2 Z(K \Gamma). They have been defined and studied in work of M. P. Holland and the author [5]. If has trace zero on the regular ...

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. 1. The Ziegler spectrum 2. Various dimensions 3. These dimensions for artin algebras 4. These dimensions in general 5. Duality 6. The complexity of morphisms in mod-R 7. The Gabriel-Zariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pure-injectives Suppose that M is a submodule of N . Consider a finite system \Sigma n i=1 x i r ij = a j (j = 1; :::m) of R-linear equations over M : that is, the r ij are in R, the 1 a j are in M and the x i are indeterminates. Suppose that there is a solution b 1 ; ...

Abstract. We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag–Leffler ones. A right R–module M is called Baer if Ext1 R (M, T) = 0 for all torsion modules T, and M is Mittag–Leffler in case the canonical map M ⊗R�i∈I Qi →�i∈I (M ⊗R Qi) is injective where {Qi}i∈I are arbitrary left R–modules. We show that a module M is Baer iff M is p–filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag–Leffler modules. Since the fundamental work of Ringel [29], the study of infinite dimensional modules has become one of the challenging tasks of the representation theory of

We describe the Gabriel-Zariski spectrum and the sheaf of locally definable scalars for tame hereditary algebras.

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a ring R and tilting theory. We first provide constructions of tilting objects from given recollements, recovering several different results from the literature. Secondly, we show how to construct a recollement from a tilting module of projective dimension one. By [31], every recollement of D(ModR) is associated to a differential graded homological epimorphism λ: R → S. We will focus on the case where λ is a homological ring epimorphism or even a universal localization. Our results will be employed in a forthcoming paper in order to investigate stratifications of D(ModR).