$t$-Structures with Grothendieck hearts via functor categories

Abstract

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-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If $\mathcal{D}$ is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t-cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from $\mathcal{D}$ to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in $\mathcal{D}$. This allows us to show that any standard well generated triangulated category $\mathcal{D}$ possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t-structures in such triangulated categories.

arXiv comment60 pages; version 4: minor changes, final version; version 3: improvements in results (Thm. 1.5 and Prop. 6.9), several changes in the presentation, references added and updated; version 2: new sections 8.3 (a classification of t-structures with definable co-aisles in terms of suspended ideals of the compacts) and 8.4 (existence of right adjacent co-t-structures) added

References in corpus (17)

• Silting mutation in triangulated categories
• Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)
• On exact categories and applications to triangulated adjoints and model structures
• Algebraic versus topological triangulated categories
• Derived equivalences induced by big cotilting modules
• Torsion pairs in silting theory
• On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings
• Definability and approximations in triangulated categories
• The tilting-cotilting correspondence
• Injective DG-modules over non-positive DG-rings
• Cotilting Sheaves on Noetherian Schemes
• $t$-Structures on stable derivators and Grothendieck hearts
• Two results from Morita theory of stable model categories
• Covers and direct limits: a contramodule-based approach
• The t-structure induced by an n-tilting module
• Exactness of direct limits for abelian categories with an injective cogenerator
• On semi-infinite cohomology of finite dimensional graded algebras

Cited by in corpus (7)

• Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings
• Hearts for commutative noetherian rings: torsion pairs and derived equivalences
• Triangulated categories with a single compact generator and two Brown representability theorems
• Tilting complexes and codimension functions over commutative noetherian rings
• Singular equivalences to locally coherent hearts of commutative noetherian rings
• Gluing compactly generated t-structures over stalks of affine schemes
• Definable functors between triangulated categories