On perfectly generated weight structures and adjacent t-structures

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On perfectly generated weight structures and adjacent t-structures. / Bondarko, Mikhail V.

In: Mathematische Zeitschrift, 16.08.2021.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{49d65ec1e2594b1f9f69cdfffc3ff064,

title = "On perfectly generated weight structures and adjacent t-structures",

abstract = "This paper is dedicated to the study of smashing weight structures (these are the weight structures {"}coherent with coproducts{"}), and the application of their properties to t-structures. In particular, we prove that the hearts of compactly generated t-structures are Grothendieck abelian categories; 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 opposites to) the ones right adjacent to compactly generated t-structures; they give injective cogenerators for the hearts of the latter. Moreover, we 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 that was also defined by Neeman) is perfectly generated; actually, we prove more than that. Furthermore, 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 and J. {\v S}{\v t}ov{\'i}{\v c}ek to arbitrary smashing triangulated categories. This gives a generalization of a t-structure statement due to B. Keller and P. Nicolas.",

keywords = "Brown representability, Compact object, Grothendieck abelian category, Heart, Perfect class, t-structure, Torsion theory, Triangulated category, Weight structure",

author = "Bondarko, {Mikhail V.}",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2021",

month = aug,

day = "16",

doi = "10.1007/s00209-021-02815-6",

language = "English",

journal = "Mathematische Zeitschrift",

issn = "0025-5874",

publisher = "Springer Nature",

}

RIS

TY - JOUR

T1 - On perfectly generated weight structures and adjacent t-structures

AU - Bondarko, Mikhail V.

PY - 2021/8/16

Y1 - 2021/8/16

N2 - This paper is dedicated to the study of smashing weight structures (these are the weight structures "coherent with coproducts"), and the application of their properties to t-structures. In particular, we prove that the hearts of compactly generated t-structures are Grothendieck abelian categories; 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 opposites to) the ones right adjacent to compactly generated t-structures; they give injective cogenerators for the hearts of the latter. Moreover, we 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 that was also defined by Neeman) is perfectly generated; actually, we prove more than that. Furthermore, 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 and J. Šťovíček to arbitrary smashing triangulated categories. This gives a generalization of a t-structure statement due to B. Keller and P. Nicolas.

AB - This paper is dedicated to the study of smashing weight structures (these are the weight structures "coherent with coproducts"), and the application of their properties to t-structures. In particular, we prove that the hearts of compactly generated t-structures are Grothendieck abelian categories; 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 opposites to) the ones right adjacent to compactly generated t-structures; they give injective cogenerators for the hearts of the latter. Moreover, we 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 that was also defined by Neeman) is perfectly generated; actually, we prove more than that. Furthermore, 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 and J. Šťovíček to arbitrary smashing triangulated categories. This gives a generalization of a t-structure statement due to B. Keller and P. Nicolas.

KW - Brown representability

KW - Compact object

KW - Grothendieck abelian category

KW - Heart

KW - Perfect class

KW - t-structure

KW - Torsion theory

KW - Triangulated category

KW - Weight structure

UR - http://www.scopus.com/inward/record.url?scp=85112491446&partnerID=8YFLogxK

U2 - 10.1007/s00209-021-02815-6

DO - 10.1007/s00209-021-02815-6

M3 - Article

AN - SCOPUS:85112491446

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

ER -

ID: 90570707