http://www.birs.ca/events/2011/5-day-workshops/11w5009
http://mathoverflow.net/questions/1440/freyd-mitchell-for-triangulated-categories
Thomason: The classification of triangulated subcategories. In addition to classification (relating subcats to subgroups of the Grothendieck group) and relation to the Nilpotence theorem, this contains some speculation on the “right” definition of algebraic n-cycles, which should be an object in some triangulated subcategory of the category of perfect complexes. These cycles should give a good notion replacing the Chow ring in cases such as singular algebraic varieties and schemes flat and of finite type over Spec Z. The article also reviews basic notions in triangulated categories, such as thick subcategories, the Grothendieck group, and more.
arXiv:0910.2539 General heart construction on a triangulated category (II): Associated cohomological functor from arXiv Front: math.CT by Noriyuki Abe, Hiroyuki Nakaoka In the preceding part (I) of this paper, we showed that for any torsion pair (i.e., -structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which we call the heart. Two extremal cases of torsion pairs are -structures and cluster tilting subcategories. If the torsion pair comes from a -structure, then its heart is nothing other than the heart of this -structure. In this case, as is well known, by composing certain adjoint functors, we obtain a cohomological functor from the triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes cohomological. In this paper, we unify these two constructions, to obtain a cohomological functor from the triangulated category, to the heart of any torsion pair.
arXiv:1101.3233 Report on locally finite triangulated categories from arXiv Front: math.CT by Henning Krause The basic properties of locally finite triangulated categories are discussed. The focus is on Auslander–Reiten theory and the lattice of thick subcategories.
arXiv:1101.5931 Does full imply faithful? from arXiv Front: math.CT by Alberto Canonaco, Dmitri Orlov, Paolo Stellari We study full exact functors between triangulated categories. With some hypothesis on the source category we prove that it admits an orthogonal decomposition into two pieces such that the functor restricted to one of them is zero while the restriction to the other is faithful. In particular, if the source category is either the category of perfect complexes or the bounded derived category of coherent sheaves on a noetherian scheme supported on a closed connected subscheme, then any non-trivial exact full functor is faithful as well. Finally we show that removing the noetherian hypothesis this result is not true.
arXiv:1102.2879 Stratifying derived categories of cochains on certain spaces from arXiv Front: math.CT by Shoham Shamir In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of localizing and thick subcategories in terms of subsets of the prime ideal spectrum of the given ring. In this paper two stratification results are presented: one for the derived category of a commutative ring-spectrum with polynomial homotopy and another for the derived category of cochains on certain spaces. We also give the stratification of cochains on a space a topological content.
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