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

Šťovíček, Jan; Pospíšil, David

doi:10.1090/tran/6561

Cited by 23 publications

(17 citation statements)

References 56 publications

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“…Following an idea from [, Theorem 4.10], we consider the following assignment. Theorem The map

normalΨ

which assigns to a torsion pair in

D (A)

generated by a set of compact objects

scriptS

the torsion pair in

D (A^{o p})

generated by

S^{*}

defines a bijection between (i)compactly generated torsion pairs in

D (A)

, (ii)compactly generated torsion pairs in

D (A^{o p})

, which restricts to a bijection between (i)compactly generated t‐structures in

D (A)

, (ii)compactly generated co‐t‐structures in

D (A^{o p})

, and maps intermediate t‐structures to cointermediate co‐t‐structures, and vice versa.…”

Section: Classification Resultsmentioning

confidence: 99%

See 1 more Smart Citation

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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“…Following an idea from [, Theorem 4.10], we consider the following assignment. Theorem The map

normalΨ

which assigns to a torsion pair in

D (A)

generated by a set of compact objects

scriptS

the torsion pair in

D (A^{o p})

generated by

S^{*}

defines a bijection between (i)compactly generated torsion pairs in

D (A)

, (ii)compactly generated torsion pairs in

D (A^{o p})

, which restricts to a bijection between (i)compactly generated t‐structures in

D (A)

, (ii)compactly generated co‐t‐structures in

D (A^{o p})

, and maps intermediate t‐structures to cointermediate co‐t‐structures, and vice versa.…”

Section: Classification Resultsmentioning

confidence: 99%

“…Remark Beware that the use of terminology and notation may vary in the literature. For example, torsion pairs as above are termed ‘complete Hom‐orthogonal pairs’ in , while torsion pairs according to [, Definition I.2.1] are in fact t ‐structures. Co‐ t ‐structures are called weight structures in .…”

Section: Torsion Pairsmentioning

confidence: 99%

Silting Objects

Hügel

2019

Bull. London Math. Soc.

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“…, for a non-positive ) are precisely the Milnor colimits of sequences of the form (4), where the cone of each , denoted by , is a coproduct of objects from (resp. of objects from ), for each (see the proof of [40, theorem 8.3.3] for , and [28, theorem 12.2] and [47, theorem 2] for ) (see also [57, theorem 3.7]).…”

Section: Preliminariesmentioning

confidence: 99%

Lifting of recollements and gluing of partial silting sets

Saorı́n¹,

Zvonareva²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

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“…(1) It is shown in [1] that, in a compactly generated triangulated category T , the orthogonal S ⊥ to a set S of compact objects (or, equivalently, the orthogonal S ⊥ to the pure-projective object S obtained as the coproduct of all objects in S) is a torsionfree class. In [52] it is then shown that the subcategory S ⊥ is also a torsion class if the category T is furthermore assumed to be algebraic. The statement (1) of the Corollary is a slight generalisation of this second fact.…”

Section: Torsion Pairs In Algebraic Triangulated Categoriesmentioning

confidence: 99%

Definability and approximations in triangulated categories

Laking,

Vitória

2018

Preprint

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We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.

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

Cited by 23 publications

References 56 publications

Silting Objects

Silting Objects

Lifting of recollements and gluing of partial silting sets

Definability and approximations in triangulated categories

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