159 research outputs found
Noncommutative localisation in algebraic K-theory II
No full textIn [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385-1425] we proved a localisation theorem in the algebraic K-theory of noncommutative rings. The main purpose of the current article is to express the general theorem of the previous paper in a more user-friendly fashion, in a way more suitable for applications. In the process we compare our result to the existing theorems in the literature, showing how the previous paper improves all the existing results. It should be pointed out that there have been two very interesting recent preprints on related topics. The reader is referred to the beautiful papers of Krause [Henning Krause, Cohomological quotients and smashing localizations, http://wwwmath.upb.de/~hkrause/publications.html. [8]] and Dwyer [William G. Dwyer, Noncommutative localization in homotopy theory, preprint, http://www.nd.edu/~wgd/. [4]]. Krause studies the lifting of chain complexes and the relation with the telescope conjecture, and Dwyer generalises to the homotopy theoretic framework
Noncommutative localisation in algebraic K-theory II
No full textIn [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385–1425] we proved a localisation theorem in the algebraic K-theory of noncommutative rings. The main purpose of the current article is to express the general theorem of the previous paper in a more user-friendly fashion, in a way more suitable for applications. In the process we compare our result to the existing theorems in the literature, showing how the previous paper improves all the existing results. It should be pointed out that there have been two very interesting recent preprints on related topics. The reader is referred to the beautiful papers of Krause [Henning Krause, Cohomological quotients and smashing localizations
Weakly approximable triangulated categories and enhancements: a survey
No full textThis paper surveys some recent results, concerning the intrinsicness of natural subcategories of weakly approximable triangulated categories. We also review the results about uniqueness of enhancements of triangulated categories, with the aim of showing the fruitful interplay. In particular, we show how this leads to a vast generalization of a result by Rickard about derived invariance for schemes and rings.
of weakly approximable triangulated categories. We also review the results about uniqueness
of enhancements of triangulated categories, with the aim of showing the fruitful interplay.
In particular, we show how this leads to a vast generalization of a result by Rickard about
derived invariance for schemes and rings
Uniqueness of enhancements for derived and geometric categories
No full textWe prove that the derived categories of abelian categories have unique enhancements—all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a Grothendieck abelian category are also shown to have unique enhancements. Finally, we show that the derived category of complexes with quasi-coherent cohomology and the category of perfect complexes have unique enhancements for quasi-compact and quasi-separated schemes
An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves
No full textOrlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor. In this paper we show that this result is false without the fully faithfulness hypothesis. We also show that our functor does not lift to the homotopy category of spectral categories if the ground field is
An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves
No full textOrlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor. In this paper we show that this result is false without the fully faithfulness hypothesis. We also show that our functor does not lift to the homotopy category of spectral categories if the ground field is {\mathbb Q}.This research started at the Mathematical Sciences Research Institute in 2013 with the support
of the National Science Foundation under Grant No. 0932078 000. A number of results were
obtained during research visits of the first author to the University of Hasselt, supported by
ESF Exchange Grant 4498 in the framework of the Project “Interactions of Low-Dimensional
Topology and Geometry with Mathematical Physics (ITGP)” and by the FWO Grant
1503512N “Non-commutative algebraic geometry” and by the second author to the
International School for Advanced Studies (SISSA) at Trieste. The research by Amnon Neeman was partly supported by the Australian Research Council
The passage among the subcategories of weakly approximable triangulated categories
No full textIn this article we prove that all the inclusions between the 'classical' and
naturally defined full triangulated subcategories of a weakly approximable
triangulated category are intrinsic (in one case under a technical condition).
This extends all the existing results about subcategories of weakly
approximable triangulated categories. Together with a forthcoming paper about
uniqueness of enhancements, our result allows us to generalize a celebrated
theorem by Rickard which asserts that if and are left coherent rings,
then a derived equivalence of and is "independent of the decorations".
That is, if and are equivalent
as triangulated categories for some choice of decorations and ,
then they are equivalent for every choice of decorations. But our theorem is
much more general, and applies also to quasi-compact and quasi-separated
schemes -- even to the relative version, in which the derived categories
consist of complexes with cohomology supported on a given closed subscheme with
quasi-compact complement.Comment: Main paper by Alberto Canonaco, Amnon Neeman and Paolo Stellari with
an appendix by Christian Haesemeyer. 65 page
DUALIZING COMPLEXES—THE MODERN WAY
No full textAbstract. We give a survey of some recent results on Grothendieck duality. We begin with a brief reminder of the classical theory, and then launch into an overview of some of the striking developments since 2005. Contents 1. Grothendieck duality done classically
The homotopy category of injectives
No full textKrause studied the homotopy category K.(Inj A) of complexes of injectives in a locally noetherian Grothendieck abelian category A. Because A is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category K.InjA/ has coproducts. It turns out that K. (Inj A) is compactly generated, and Krause studies the relation between the compact objects in K. (Inj A)/, the derived category D.A/, and the category Kac. (Inj A) of acyclic objects in K. (Inj A). We wish to understand what happens in the nonnoetherian case, and this paper begins the study. We prove that, for an arbitrary Grothendieck abelian category A, the category K. (Inj A) has coproducts and is μ-compactly generated for some sufficiently large μ. The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of K. (Inj A) into K.A has a left adjoint and the existence of coproducts is a formal corollary. But in order to prove anything about these coproducts, for example the μ-compact generation, we need to have a handle on this adjoint. Also interesting is the counterexample at the end of the article: we produce a locally noetherian Grothendieck abelian category in which products of acyclic complexes need not be acyclic. It follows that D.A is not compactly generated. I believe this is the first known example of such a thing
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