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14
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
Smashing Subcategories And The Telescope Conjecture  An Algebraic Approach
 Invent. Math
, 1998
"... . We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a cl ..."
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Cited by 25 (6 self)
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. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pureinjective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...
Invariance and localization for cyclic homology of DG algebras
 J. PURE APPL. ALGEBRA
, 1998
"... We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyc ..."
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Cited by 24 (6 self)
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We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of ThomasonTrobaugh’s [51] and Yao’s [58] localization theorems in Ktheory (cf. also [55]).
Analytic versions of the zero divisor conjecture. In Geometry and cohomology in group theory
 of London Math. Soc. Lecture Note Ser
, 1994
"... This is an expanded version of the three lectures I gave at the Durham conference. The material is mainly expository, though there are a few new results, and for those I have given complete proofs. While the subject matter involves analysis, it is written from an algebraic point of view. Thus hopefu ..."
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Cited by 18 (7 self)
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This is an expanded version of the three lectures I gave at the Durham conference. The material is mainly expository, though there are a few new results, and for those I have given complete proofs. While the subject matter involves analysis, it is written from an algebraic point of view. Thus hopefully algebraists will find the
Homotopy decomposition of a group of symplectomorphisms
 of S 2 × S 2 . Topology 43 (2004
"... Abstract. We continue the analysis started by Abreu, McDuff and Anjos [Ab, AM, McD, An] of the topology of the group of symplectomorphisms of S 2 × S 2 when the ratio of the area of the two spheres lies in the interval (1, 2]. We express the group, up to homotopy, as the pushout (or amalgam) of cert ..."
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Cited by 10 (5 self)
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Abstract. We continue the analysis started by Abreu, McDuff and Anjos [Ab, AM, McD, An] of the topology of the group of symplectomorphisms of S 2 × S 2 when the ratio of the area of the two spheres lies in the interval (1, 2]. We express the group, up to homotopy, as the pushout (or amalgam) of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations. 1.
Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings
"... Abstract. We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence ..."
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Cited by 10 (7 self)
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Abstract. We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively, that L has a largest element. This extends several known results of G.M. Bergman, A.P. Huhn, J. T˚uma, and of a joint work of G. Grätzer, H. Lakser, and the author, and it solves Problem 2 of the survey paper [10]. The main tool used in the proof of our result is an amalgamation theorem for semilattices and algebras (over a given division ring), a variant of previously known amalgamation theorems for semilattices and lattices, due to J. T˚uma, and G. Grätzer, H. Lakser, and the author.
THE REALIZATION PROBLEM FOR VON NEUMANN REGULAR RINGS
, 802
"... Abstract. We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right Rmodules over a von Neumann regular ring R. This survey consi ..."
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Cited by 4 (3 self)
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Abstract. We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right Rmodules over a von Neumann regular ring R. This survey consists of four sections. Section 1 introduces the realization problem for von Neumann regular rings, and points out its relationship with the separativity problem of [7]. Section 2 surveys positive realization results for countable conical refinement monoids, including the recent constructions in [5] and [4]. We analyze in Section 3 the relationship with the realization problem of algebraic distributive lattices as lattices of twosided ideals over von Neumann regular rings. Finally we observe in Section 4 that there are countable conical monoids which can be realized by a von Neumann regular Kalgebra for some countable field K, but they cannot be realized by a von Neumann regular Falgebra for any uncountable field F. 1. The problem All rings considered in this paper will be associative, and all the monoids will be commutative. For a unital ring R, let V(R) denote the monoid of isomorphism classes of finitely generated projective right Rmodules, where the operation is defined by [P] + [Q] = [P ⊕ Q]. This monoid describes faithfully the decomposition structure of finitely generated projective modules. The monoid V(R) is always a conical monoid, that is, whenever x + y = 0, we have x = y = 0. Recall that an orderunit in a monoid M is an element u in M such that for every x ∈ M there is y ∈ M and n ≥ 1 such that x + y = nu. Observe that [R] is a canonical orderunit in V(R). By results of Bergman [11, Theorems 6.2 and 6.4] and Bergman and Dicks [12, page 315], any conical monoid with an orderunit appears as V(R) for some unital hereditary ring R.
Noncommutative Splitting Fields
, 1987
"... For commutative fields K, L with L generated over K by an algebraic element with separable minimal polynomial p the following facts are well known: (1) There exists an extension N of L such that N/K is a Galois extension ..."
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Cited by 1 (0 self)
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For commutative fields K, L with L generated over K by an algebraic element with separable minimal polynomial p the following facts are well known: (1) There exists an extension N of L such that N/K is a Galois extension
PUREPERIODIC MODULES AND A STRUCTURE OF PUREPROJECTIVE RESOLUTIONS
"... Dedicated to Stanis̷law Balcerzyk on the occasion of his seventieth birthday We investigate the structure of puresyzygy modules in a pureprojective resolution of any right Rmodule over an associative ring R with an identity element. We show that a right Rmodule M is pureprojective if and only i ..."
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Dedicated to Stanis̷law Balcerzyk on the occasion of his seventieth birthday We investigate the structure of puresyzygy modules in a pureprojective resolution of any right Rmodule over an associative ring R with an identity element. We show that a right Rmodule M is pureprojective if and only if there exists an integer n ≥ 0 and a pureexact sequence 0 → M → Pn → ·· · → P0 → M → 0 with pureprojective modules Pn,...,P0. As a consequence we get the following version of a result in Benson and Goodearl, 2000: A flat module M is projective if M admits an exact sequence 0 → M → Fn → ·· · → F0 → M → 0 with projective modules Fn,...,F0. 1. Introduction. Throughout this paper R is an associative ring with an identity element. We denote by Mod(R) the category of all right Rmodules. We recall (see [12]) that an exact sequence ·· · → Xn−1 → Xn → Xn+1 →... in Mod(R)