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21
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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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
Failure Of Brown Representability In Derived Categories
"... Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural tr ..."
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Cited by 15 (0 self)
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Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [36], it was proved that Adams' theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis [5] made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and homological functors fT c g op ! Ab which are not restrictions of representables. Contents
Cohomological quotients and smashing localizations
 Amer. J. Math
"... Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the coh ..."
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Cited by 14 (2 self)
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Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic Ktheory and demonstrates the relevance of the telescope
Homological algebra in bivariant Ktheory and other triangulated categories
"... Abstract. Bivariant (equivariant) Ktheory is the standard setting for noncommutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, pro ..."
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Cited by 6 (2 self)
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Abstract. Bivariant (equivariant) Ktheory is the standard setting for noncommutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivariant Ktheory. An important observation of Beligiannis is that we can approximate our
On The Freyd Categories Of An Additive Category
, 2000
"... To any additive category C, we associate in a functorial way two additive categories A(C), B(C). The category A(C), resp. B(C), is the reflection of C in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration AB(C) i ..."
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Cited by 5 (1 self)
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To any additive category C, we associate in a functorial way two additive categories A(C), B(C). The category A(C), resp. B(C), is the reflection of C in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration AB(C) is the reflection of C in the category of abelian categories and exact functors. We call A(C) and B(C) the Freyd categories of C since the first systematic study of these categories was done by Freyd in the midsixties. The purpose of the paper is to study further the Freyd categories and to indicate their applications to the module theory of an abelian or triangulated category.
Triangulated categories and the Ziegler spectrum
 IN PROCEEDINGS OF KTHEORY AND NONCOMMUTATIVE GEOMETRY
, 2006
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C ∗ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED KTHEORY
, 810
"... Abstract. We define the filtrated Ktheory of a C ∗algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated Ktheory. For finite spaces with totally ordered lattice of open subsets, this spect ..."
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Cited by 2 (0 self)
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Abstract. We define the filtrated Ktheory of a C ∗algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated Ktheory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated Ktheory is not yet a complete invariant. We describe two C ∗algebras over a space X with four points that have isomorphic filtrated Ktheory without being KK(X)equivalent. For this space X, we enrich filtrated Ktheory by another Ktheory functor to a complete invariant up to KK(X)equivalence that satisfies a Universal Coefficient Theorem. 1.
QUOTIENTS OF CLUSTER CATEGORIES
"... Abstract. Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2cluster categories, the “ ..."
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Abstract. Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2cluster categories, the “lowest” type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2cluster categories. In contrast, the same is not true in Dynkin type E. This paper is about the connection between quotient categories and cluster categories, so let me start by explaining these two notions. Quotient categories come in a number of different flavours. The one
Parametrizing recollement data
"... Abstract. We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compa ..."
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Abstract. We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the