On Weakly Negative Subcategories, Weight Structures, and (Weakly) Approximable Triangulated Categories

Bondarko, Mikhail V.; Vostokov, S. V.

doi:10.1134/s1995080220020031

Cited by 2 publications

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References 13 publications

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“…Remark 2.1.2 of ibid. ), in Theorem 3.1.3(2,3) of[BoS19], in §3.6 of[Bon14], and in (Remark 6.3(4) of)[BoV19b].…”

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confidence: 92%

On perfectly generated weight structures and adjacent $t$-structures

Bondarko¹

2019

Preprint

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This paper is dedicated to the study of smashing weight structures (one may say that these are weight structures "coherent with arbitrary coproducts"), and the application of their properties to t-structures. In particular, we prove that hearts of compactly generated t-structures are Grothendieck abelian; this statement strengthens earlier results of several other authors.The central theorem of the paper is as follows: any perfect (as defined by Neeman) set of objects of a triangulated category generates a weight structure; we say that weight structures obtained this way are perfectly generated. An important family of perfectly generated weight structures are the ones right adjacent to compactly generated t-structures; they give injective cogenerators for the hearts of the latter. We also establish the following not so explicit result: any smashing weight structure on a well generated triangulated category (this is a generalization of the notion of a compactly generated category defined by Neeman as well) is perfectly generated; actually, we prove more than that.Moreover, we give a classification of compactly generated torsion theories (these generalize both weight structures and t-structures) that extends the corresponding result of D. Pospisil D. and J. Šťovíček to arbitrary smashing triangulated categories.

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“…Remark 2.1.2 of ibid. ), in Theorem 3.1.3(2,3) of[BoS19], in §3.6 of[Bon14], and in (Remark 6.3(4) of)[BoV19b].…”

mentioning

confidence: 92%

On perfectly generated weight structures and adjacent $t$-structures

Bondarko¹

2019

Preprint

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On perfectly generated weight structures and adjacent t-structures

Bondarko

2021

Math. Z.

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We study t-structures (on triangulated categories) that are closely related to weight structures. A t-structure couple t = (C t≤0 , C t≥0 ) is said to be adjacent to a weight structure w = (C w≤0 , C w≥0 ) if C t≥0 = C w≥0 . For a triangulated category C that satisfies the Brown representability property we prove that t that is adjacent to w exists if and only if w is smashing (i.e., C w≥0 is C -closed). The heart Ht of this t is the category of those functors Hw op → Ab that respect products (here Hw is the heart of w); the result has important applications. We prove several more statements on constructing t-structures starting from weight structures; we look for a strictly orthogonal t-structure t on some C ′ (where C, C ′ are triangulated subcategories of a common D) such that C ′ t≤0 (resp. C ′ t≥0 ) is characterized by the vanishing of morphisms from C w≥1 (resp. C w≤−1 ). Some of these results generalize properties of semi-orthogonal decompositions proved in the previous paper, and can be applied to various derived categories of (quasi)coherent sheaves on a scheme X that is projective over an affine noetherian one. We also study hearts of orthogonal t-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal t-structures.The main tool of this paper is the notion of virtual t-truncations of (cohomological) functors; these are defined in terms of weight structures and "behave as if they come from t-truncations of representing objects" whether t exists or not.

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On Weakly Negative Subcategories, Weight Structures, and (Weakly) Approximable Triangulated Categories

Cited by 2 publications

References 13 publications

On perfectly generated weight structures and adjacent $t$-structures

On perfectly generated weight structures and adjacent $t$-structures

On perfectly generated weight structures and adjacent t-structures

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