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

  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},
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
Compactly generated t-structures in the derived category of a commutative ring
We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of Alonso Tarrío et al. (J Algebra 324(3):313–346, 2010 )Expand
Torsion pairs in silting theory
In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purityExpand
On Torsion Theories, Weight and t-Structures in Triangulated Categories
We study triangulated categories and torsion theories in them, and compare two definitions of torsion theories in this work. The most important types of torsion theories—weight structures andExpand
Smashing localizations of rings of weak global dimension at most one
Abstract We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homologicalExpand
Gorenstein homological algebra and universal coefficient theorems
We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficientExpand
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previouslyExpand
Tilting theory via stable homotopy theory
We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these resultsExpand
$t$-Structures on stable derivators and Grothendieck hearts
We prove that given any strong, stable derivator and a $t$-structure on its base triangulated category $\cal D$, the $t$-structure canonically lifts to all the (coherent) diagram categories and eachExpand
$t$-Structures with Grothendieck hearts via functor categories
We study when the heart of a t-structure in a triangulated category $\mathcal{D}$ with coproducts is AB5 or a Grothendieck category. If $\mathcal{D}$ satisfies Brown representability, a t-structureExpand
Silting objects.
We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain theExpand


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
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