t-Structures on stable derivators and Grothendieck hearts

Saorı́n, Manuel; Šťovíček, Jan; Virili, Simone

doi:10.1016/j.aim.2023.109139

Cited by 5 publications

(4 citation statements)

References 40 publications

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“…In this way, we generalize various results which appeared in the literature beforefor t-structures in presentable stable ∞-categories ([45, Remark 1.3.5.23], [46,Remark C.1.4.6]), for homotopically smashing t-structures in nice enough stable derivators ( [69], [41,Theorem 4.6]) or for compactly generated t-structures ([14, Theorem 0.2]).…”

Section: Theorem 12 Let D Be a Triangulated Category Which Has Arbitr...supporting

confidence: 69%

“…In analogy with Propotision 3.3, we prove that the last step in its factorization according to Corollary 7.8 is trivial. As a consequence, we obtain a counterpart of [69,Theorem C] for abstract triangulated categories without using their models. Corollary 7.9 Let D be a standard well generated triangulated category with the universal coproduct-preserving homological functor h pure : D −→ A pure (D) and let t = (U, V) be a t-structure such that the heart H is AB5 and H 0 t : D −→ H preserves coproducts.…”

Section: T-structures With Grothendieck Heartsmentioning

confidence: 99%

“…The last lemma says that PInj(V) is t-cogenerating if condition (2) holds. The subtle point is that the conclusion of Lemma 8.10 can be derived already from the a priori weaker condition (2') (which, however, can be stated only using an enhancement of D) and, thus, holds for any homotopically smashing t-structure in the language of [41,69].…”

Section: T-structures With Grothendieck Heartsmentioning

confidence: 99%

“…The problem of identifying the t-structures whose heart is a Grothendieck category has deserved a lot of attention since it first arose for the Happel-Reiten-Smalø tstructure associated to a torsion pair in a Grothendieck or module category [18,19]. For the general question, several strategies have been used to tackle the problem, including ad hoc arguments [8,60], functor categories [13,14] and suitable enhancements of the ambient triangulated category, such as stable ∞-categories [45,46] or derivators [41,69].…”

Section: Introductionmentioning

confidence: 99%

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t-Structures with Grothendieck hearts via functor categories

Saorín,

Št’ovíček

2023

Sel. Math. New Ser.

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We study when the heart of a t-structure in a triangulated category $$\mathcal {D}$$ D with coproducts is AB5 or a Grothendieck category. If $$\mathcal {D}$$ 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 $$\mathcal {D}$$ 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 $$\mathcal {D}$$ 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 $$\mathcal {D}$$ D . This allows us to show that any standard well generated triangulated category $$\mathcal {D}$$ 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.

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Section: Theorem 12 Let D Be a Triangulated Category Which Has Arbitr...supporting

confidence: 69%

Section: T-structures With Grothendieck Heartsmentioning

confidence: 99%

Section: T-structures With Grothendieck Heartsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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t-Structures with Grothendieck hearts via functor categories

Saorín,

Št’ovíček

2023

Sel. Math. New Ser.

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Gluing compactly generated t-structures over stalks of affine schemes

Hrbek,

Hu,

Zhu

2024

Isr. J. Math.

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Tilting Complexes and Codimension Functions Over Commutative Noetherian Rings

HRBEK,

NAKAMURA,

ŠŤOVÍČEK

2024

Nagoya Math. J.

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In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.

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t-Structures on stable derivators and Grothendieck hearts

Cited by 5 publications

References 40 publications

t-Structures with Grothendieck hearts via functor categories

t-Structures with Grothendieck hearts via functor categories

Gluing compactly generated t-structures over stalks of affine schemes

Tilting Complexes and Codimension Functions Over Commutative Noetherian Rings

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