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50
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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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.
When a superdecomposable pureinjective module over a serial ring exists
"... Abstract. We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are “tame ” according to the Klingler–Levy analysis in [4], [5] and [6], in particular over Dedekindlike rings and their homomorphic images. We show that both are undefined in most cases. 1 ..."
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Abstract. We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are “tame ” according to the Klingler–Levy analysis in [4], [5] and [6], in particular over Dedekindlike rings and their homomorphic images. We show that both are undefined in most cases. 1
Model theoretic properties of free, projective and flat acts
 Fund. Appl. Math
"... Abstract. This is the second in a series of articles surveying the body of work on the model theory of Sacts over a monoid S. The first concentrated on the theory of regular Sacts. Here we review the material on modeltheoretic properties of free, projective and (strongly, weakly) flat Sacts. We ..."
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Abstract. This is the second in a series of articles surveying the body of work on the model theory of Sacts over a monoid S. The first concentrated on the theory of regular Sacts. Here we review the material on modeltheoretic properties of free, projective and (strongly, weakly) flat Sacts. We consider questions of axiomatisability, completeness, model completeness and stability for these classes. Most but not all of the results have already appeared; we remark that the description of those monoids S such that the class of free left Sacts is axiomatisable, is new. 1.
Modeltheoretic imaginaries and coherent sheaves
, 2006
"... In this paper we attempt to bridge a gap or, perhaps, to strengthen some existing links. Model theory has evolved in two sharply different directions. One is setbased, centred ..."
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Cited by 3 (2 self)
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In this paper we attempt to bridge a gap or, perhaps, to strengthen some existing links. Model theory has evolved in two sharply different directions. One is setbased, centred
Pure injective envelopes of finite length modules over a Generalised Weyl Algebra
"... We investigate certain pureinjective modules over generalised Weyl algebras. We consider pureinjective hulls of finite length modules, the elementary duals of these, torsionfree pureinjective modules and the closure in the Ziegler spectrum of the category of finite length modules supported on a n ..."
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We investigate certain pureinjective modules over generalised Weyl algebras. We consider pureinjective hulls of finite length modules, the elementary duals of these, torsionfree pureinjective modules and the closure in the Ziegler spectrum of the category of finite length modules supported on a nondegenerate orbit of a generalized Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrasts with the behaviour of pure injective modules over finitedimensional algebras and hereditary orders.
Krull–Gabriel dimension and the modeltheoretic complexity of the category of modules over group rings of finite groups
 J. London Math. Soc
"... Abstract. We classify group rings of finite groups over a field F according to the modeltheoretic complexity of the category of their modules. For instance we prove that, if F contains a primitive cubic root of 1, then the Krull–Gabriel dimension of such rings is 0, 2, or undefined. 1. ..."
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Abstract. We classify group rings of finite groups over a field F according to the modeltheoretic complexity of the category of their modules. For instance we prove that, if F contains a primitive cubic root of 1, then the Krull–Gabriel dimension of such rings is 0, 2, or undefined. 1.
DECIDABILITY OF THE THEORY OF MODULES OVER COMMUTATIVE VALUATION DOMAINS
"... Abstract. We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all Vmodules is decidable. 1. ..."
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Abstract. We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all Vmodules is decidable. 1.
The nonstandard quantum plane
"... Let k be a field of char(k) ̸ = 2, and suppose that q ∈ k is not a root of unity. The associated quantum plane [5, §IV.1], denoted by kq[x, y], is defined to be the free kalgebra k{x, y} generated by x and y, modulo the relation yx = qxy. The set of monomials {x i y j}i, j≥0 is a basis for the unde ..."
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Let k be a field of char(k) ̸ = 2, and suppose that q ∈ k is not a root of unity. The associated quantum plane [5, §IV.1], denoted by kq[x, y], is defined to be the free kalgebra k{x, y} generated by x and y, modulo the relation yx = qxy. The set of monomials {x i y j}i, j≥0 is a basis for the underlying kvector space, and for every pair (i, j) of nonnegative integers,
Towards decidability of the theory of pseudofinite dimensional representations of sl2(k
"... In this paper, we refine the analysis begun in Ivo Herzog’s paper [7] on representations of the Lie algebra sl2(k), where k is an algebraically closed field of characteristic 0. Our principal contribution is to bring out a connection to fundamental problems in the diophantine geometry of curves. We ..."
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In this paper, we refine the analysis begun in Ivo Herzog’s paper [7] on representations of the Lie algebra sl2(k), where k is an algebraically closed field of characteristic 0. Our principal contribution is to bring out a connection to fundamental problems in the diophantine geometry of curves. We expect to show, in a subsequent paper, that the theory
HOW TO CONSTRUCT A ‘CONCRETE’ SUPERDECOMPOSABLE PUREINJECTIVE MODULE OVER A STRING ALGEBRA
"... Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective envelope of this element is a superdecomposable module. A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For an examp ..."
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Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective envelope of this element is a superdecomposable module. A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For an example, let R be the endomorphism ring of a countable dimensional vector space V, and let I be the ideal of R consisting of all endomorphisms with finite dimensional images. If R ′ = R/I and e is a projection on a subspace of V whose image and coimage are infinite dimensional, then R ′ ∼ = eR ′ ⊕ (1 − e)R ′ as a right module over itself and R ′ ∼ = eR ′ ∼ = (1 − e ′)R ′. Furthermore, every nontrivial decomposition of R ′ as a right module is of this form, therefore R ′ is superdecomposable. For more examples, let R = k〈X, Y 〉 be a free algebra over a field k, and let E = E(RR) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that RR has no nonzero uniform submodules. Therefore the same is true for E, hence E is a superdecomposable injective module. More generally, this is a common feature of finite dimensional wild algebras, that they usually (conjecturally always) have a superdecomposable pureinjective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pureinjective if M is a direct summand of a direct product of finite dimensional Amodules. If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pureinjective Amodule has an indecomposable direct summand. But recently Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pureinjective module (note that every string algebra is tame). The main