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13
Model Theory and Modules
, 2006
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ..."
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Cited by 64 (20 self)
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The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ModR, the full subcategory of finitely presented modules will be denoted modR, the
InfiniteDimensional Modules in the Representation Theory of FiniteDimensional Algebras
, 1998
"... this article. Throughout, we restrict to studying finitedimensional 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 Amod for the category of finitedimensional Amodules, and AM ..."
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Cited by 18 (0 self)
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this article. Throughout, we restrict to studying finitedimensional 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 Amod for the category of finitedimensional Amodules, and AMod for the category of all
Preprojective Algebras, Differential Operators And A Conze Embedding For Deformations Of Kleinian Singularities
 Comment. Math. Helv
, 1998
"... . 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 cur ..."
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Cited by 15 (6 self)
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. 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 quasifree and\Omega 1 A is a finitely generated AAbimodule we prove that \Pi (A) is wellbehaved 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 nontrivial 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 ...
Topological and Geometric aspects of the Ziegler Spectrum
, 1998
"... 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 ..."
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Cited by 6 (5 self)
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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 modR 7. The GabrielZariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pureinjectives 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 Rlinear 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 ; ...
Double affine Lie algebras and finite groups
 the Pacific Journal of Mathematics. NICOLAS GUAY
"... Abstract. We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A: in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toro ..."
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Abstract. We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A: in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v] ⋊Γ). We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra A1 instead of the polynomial ring C[u, v], and, more generally, a rank one rational Cherednik algebra. We study quasifinite highest weight representations of these Lie algebras. Contents
Baer and Mittag–Leffler modules over tame hereditary algebras. preprint
"... 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 ..."
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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
Sheaves of Definable Scalars Over Tame Hereditary Algebras
, 1998
"... We describe the GabrielZariski spectrum and the sheaf of locally definable scalars for tame hereditary algebras. ..."
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Cited by 1 (1 self)
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We describe the GabrielZariski spectrum and the sheaf of locally definable scalars for tame hereditary algebras.
Recollements and tilting objects
, 2009
"... 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) ..."
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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).