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On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
Ideals in triangulated categories: Phantoms, ghosts and skeleta
 Adv. in Math
, 1998
"... ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral seq ..."
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ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for lowdimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A ∞ modules over an A ∞ ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim1 ←− group. The last two sections focus
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."
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The generating hypothesis in the derived category of Rmodules
 Journal of Pure and Applied Algebra
"... Abstract. In this paper, we prove a version of Freyd’s generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S ∈ D is an object whose endomorphism ring is graded commutative and concentrated in degree zero, then S generates (in the sense of Freyd) the thic ..."
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Abstract. In this paper, we prove a version of Freyd’s generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S ∈ D is an object whose endomorphism ring is graded commutative and concentrated in degree zero, then S generates (in the sense of Freyd) the thick subcategory determined by S if and only if the endomorphism ring of S is von Neumann regular. As a corollary, we obtain that the generating hypothesis is true in the derived category of a commutative ring R if and only if R is von Neumann regular. We also investigate alternative formulations of the generating hypothesis in the derived category. Finally, we give a characterization of the Noetherian stable homotopy categories in which the generating hypothesis is true.
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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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
Equivalences of derived categories for symmetric algebras
 J. Algebra
"... It is about a decade since Broué made his celebrated conjecture [2] on equivalences of derived categories in block theory: that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect group is abeli ..."
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It is about a decade since Broué made his celebrated conjecture [2] on equivalences of derived categories in block theory: that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect group is abelian. Since then, charactertheoretic evidence for the conjecture
Noncommutative Localization And Chain Complexes I. Algebraic K And LTheory
, 2001
"... The noncommutative (Cohn) localization # 1 R of a ring R is defined for any collection # of morphisms of f.g. projective left Rmodules. We exhibit # 1 R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if Tor 1 then every bounded f. ..."
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The noncommutative (Cohn) localization # 1 R of a ring R is defined for any collection # of morphisms of f.g. projective left Rmodules. We exhibit # 1 R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if Tor 1 then every bounded f.g.
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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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
Sheaves on local CalabiYau varieties
"... Abstract. We investigate sheaves supported on the zero section of the total space of a locallyfree sheaf E on a smooth, projective variety X when E satisfies ∧ rank E E ∼ = ωX. We rephrase this construction using the language of A∞algebra and provide a simple characterisation of the case E = ωX. ..."
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Abstract. We investigate sheaves supported on the zero section of the total space of a locallyfree sheaf E on a smooth, projective variety X when E satisfies ∧ rank E E ∼ = ωX. We rephrase this construction using the language of A∞algebra and provide a simple characterisation of the case E = ωX. 1.
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