On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings

@article{ovek2015OnCG,
  title={On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings},
  author={J. {\vS}ťov{\'i}{\vc}ek and David Posp{\'i}{\vs}il},
  journal={Transactions of the American Mathematical Society},
  year={2015},
  volume={368},
  pages={6325-6361}
}
We classify compactly generated co-t-structures on the derived category of a commutative noetherian ring. In order to accomplish that, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the literature) in triangulated categories that resembles Bousfield localization theory. Finally, we show that the category of perfect complexes over a connected commutative noetherian ring admits only the trivial co-t-structures and (de)suspensions of the canonical… Expand
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References

SHOWING 1-10 OF 73 REFERENCES
On t-structures and torsion theories induced by compact objects
Abstract First, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Second, in an abelian category we show that a complex P · ofExpand
Compactly generated t-structures on the derived category of a Noetherian ring
Abstract We study t -structures on D ( R ) the derived category of modules over a commutative Noetherian ring R generated by complexes in D fg − ( R ) . We prove that they are exactly the compactlyExpand
Tilting, cotilting, and spectra of commutative noetherian rings
We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolvingExpand
Generators and representability of functors in commutative and noncommutative geometry
We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces isExpand
Cotorsion pairs, model category structures, and representation theory
Abstract. We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have beenExpand
Homological and Homotopical Aspects of Torsion Theories
Introduction Torsion pairs in abelian and triangulated categories Torsion pairs in pretriangulated categories Compactly generated torsion pairs in triangulated categories Hereditary torsion pairs inExpand
The telescope conjecture for hereditary rings via Ext-orthogonal pairs
Abstract For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exactExpand
Weight structures and simple dg modules for positive dg algebras
Using techniques due to Dwyer-Greenlees-Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whoseExpand
Models for singularity categories
In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsionExpand
THE CO-STABILITY MANIFOLD OF A TRIANGULATED CATEGORY
Abstract Stability conditions on triangulated categories were introduced by Bridgeland as a ‘continuous’ generalisation of t-structures. The set of locally-finite stability conditions on aExpand
...
1
2
3
4
5
...