Results 1  10
of
36
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 ..."
Abstract

Cited by 96 (23 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 60 (18 self)
 Add to MetaCart
(Show Context)
. 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 ..."
Abstract

Cited by 52 (6 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 26 (7 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 17 (10 self)
 Add to MetaCart
(Show Context)
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.
Good tilting modules and recollements of derived module categories
 Proc. Lond. Math. Soc. 104
, 2012
"... ar ..."
(Show Context)
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 ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
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.
THE REALIZATION PROBLEM FOR VON NEUMANN REGULAR RINGS
, 2008
"... 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. ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
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.