Purity in compactly generated derivators and t-structures with Grothendieck hearts

Laking, Rosanna

doi:10.1007/s00209-019-02411-9

Cited by 29 publications

(64 citation statements)

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“…The equivalence (i) ⇔ (ii) is precisely [MV18, Theorem 3.13]. Also, [MV18, Theorem 3.14] implies that (i) is equivalent to V being definable (see [Lak18] for the definition), which is in our situation equivalent to (U, V) being homotopically smashing by [Lak18, Theorem 4.6].…”

Section: Intermediate T-structures and Cosilting Complexesmentioning

confidence: 96%

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Compactly generated t-structures in the derived category of a commutative ring

Hrbek

2019

Math. Z.

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We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of [ATLJS10] for noetherian rings. More specifically, we establish a bijective correspondence between the compactly generated t-structures and infinite filtrations of the Zariski spectrum by Thomason subsets. Moreover, we show that in the case of a commutative noetherian ring, any bounded below homotopically smashing t-structure is compactly generated. As a consequence, all cosilting complexes are classified up to equivalence.

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Section: Intermediate T-structures and Cosilting Complexesmentioning

confidence: 96%

“…The notion of a cosilting complex was introduced in [ZW17] as a dualization of the (big) silting complexes of [AHMV15]. Combining the recent works [MV18] and [Lak18] gives a useful characterization of t-structures induced by cosilting complexes. Before that, we need to recall a couple of definitions.…”

Section: Intermediate T-structures and Cosilting Complexesmentioning

confidence: 99%

Compactly generated t-structures in the derived category of a commutative ring

Hrbek

2019

Math. Z.

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“…We will say that

scriptT

is the underlying category , or the base, of the derivator

double-struckD

. We refer to for details. Here we only remark that all compactly generated triangulated categories which are algebraic, and in particular, all derived module categories, admit such an enhancement.…”

Section: Cosilting Objectsmentioning

confidence: 99%

“…In particular, it turns out that every nondegenerate compactly generated t ‐structure occurs in such a TTF‐triple and thus shares these properties, cf. .…”

Section: Introductionmentioning

confidence: 99%

Silting Objects

Hügel

2019

Bull. London Math. Soc.

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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 the interplay with t-structures and co-t-structures. We then focus on silting and cosilting objects in a triangulated category with coproducts and study the case of the unbounded derived category of a ring. We close the survey with some classification results over commutative Noetherian rings and silting-discrete algebras. Contents

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“…The following is an example of a partial silting object that is not silting (see also [26,Subsection 4.4]).…”

Section: 1mentioning

confidence: 99%