TY - JOUR

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

AU - Bondarko, Mikhail V.

N1 - Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022

Y1 - 2022

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

VL - 300

SP - 1421

EP - 1454

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 2

ER -