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Model Theory and Modules
, 1988
"... 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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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
The Carlitz algebras
"... The Carlitz Fqalgebra C = Cν, ν ∈ N, is generated by an algebraically closed field K (which contains a nondiscrete locally compact field of positive characteristic p> 0, i.e. K ≃ Fq[[x,x −1]], q = p ν), by the (power of the) Frobenius map X = Xν: f ↦ → f q, and by the Carlitz derivative Y = Yν. ..."
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The Carlitz Fqalgebra C = Cν, ν ∈ N, is generated by an algebraically closed field K (which contains a nondiscrete locally compact field of positive characteristic p> 0, i.e. K ≃ Fq[[x,x −1]], q = p ν), by the (power of the) Frobenius map X = Xν: f ↦ → f q, and by the Carlitz derivative Y = Yν. It is proved that the Krull and global dimensions of C are 2, a classification of simple Cmodules and ideals are given, there are only countably many ideals, they commute (IJ = JI), and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple Cmodule is a sum of eigenspaces of the element Y X (the set of eigenvalues for Y X is given explicitly for each simple Cmodule). This fact is crucial in finding the group AutFq(C) of Fqalgebra automorphisms of C and in proving that two distinct Carlitz rings are not isomorphic (Cν ̸ ≃ Cµ if ν ̸ = µ). The centre of C is found explicitly, it is a UFD that contains countably many elements.
Futorny V., Weyl algebra modules
"... We investigate weight modules for finite and infinite Weyl algebras, classifying all such simple modules. We also study the representation type of the blocks of locallyfinite weight module categories and describe indecomposable modules in tame blocks. 1 Introduction. The nth Weyl algebra An is the ..."
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We investigate weight modules for finite and infinite Weyl algebras, classifying all such simple modules. We also study the representation type of the blocks of locallyfinite weight module categories and describe indecomposable modules in tame blocks. 1 Introduction. The nth Weyl algebra An is the unital associative algebra over a field K with generators xi,∂i, i = 1,2,...,n, which satisfy the defining relations [xi,xj] = 0 = [∂i,∂j] (1) [∂i,xj] = δi,j1, (2)
Rings of definable scalars of Verma modules
, 2006
"... Let M be a Verma module over the Lie algebra, sl2(k), of trace zero 2×2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U(sl2(k)) to RM is an epimorphism of rings. We also describe the ..."
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Let M be a Verma module over the Lie algebra, sl2(k), of trace zero 2×2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U(sl2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules. 1
Oystaeyen. Good reduction of good filtrations at places
 Algebr. Represent. Theory
"... We consider filtered or graded algebras A over a field K. Assume that there is a discrete valuation Ov of K with mv its maximal ideal and kv: = Ov/mv its residue field. Let Λ be Ovorder such that ΛK = A and Λ: = kv ⊗Ov Λ the Λreduction of A at the place K � kv. Using the filtration of A induced by ..."
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We consider filtered or graded algebras A over a field K. Assume that there is a discrete valuation Ov of K with mv its maximal ideal and kv: = Ov/mv its residue field. Let Λ be Ovorder such that ΛK = A and Λ: = kv ⊗Ov Λ the Λreduction of A at the place K � kv. Using the filtration of A induced by Λ we shall prove that for certain algebras A their properties are related to Λ.