On t-structures and torsion theories induced by compact objects

By M. Hoshino, Yoshiaki Kato, Jun-ichi Miyachi. 2000

109 citations recorded by the source corpus.

Abstract First, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Second, in an abelian category we show that a complex P · of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P · to be a tilting complex. Finally, in the case of artin algebras, we give a one-to-one correspondence between tilting complexes of term length two and torsion theories with certain conditions.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/03826d5418f92799b71268e86ff62d465e06170a

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Citations in the corpus

Compact corigid objects in triangulated categories and co-t-structures (2007, background): “Abstract: In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, C , of a triangulated category, T , which is rigid in the sense of Iyama and Yoshino, [12].”
Realisation functors in tilting theory (2015, background): “Compact silting objects were also considered in abstract triangulated categories ([1,17,40,43]) and, more recently, non-compact silting objects and their associated t-structures were studied in derived module categories ([8,78]).”
Discreteness of silting objects and t ‐structures in triangulated categories (2017, background): “In recent years the connection between silting theory and the theory of t-structures (and co-t-structures) has received much attention [4, 7, 14, 16, 38, 55, 62], and silting theory has been playing an increasingly important role in the study of triangulated categories of algebraic origin, due to su”
Silting Objects (2018, background): “An important role is played by the case of a two-term silting complex, which had already been investigated by Hoshino, Kato and Miyachi in [53].”
Homological and Homotopical Aspects of Torsion Theories (2007, result): “A similar approach was considered independently also in [63].”
Silting reduction and Calabi--Yau reduction of triangulated categories (2014, unknown): “We refer to [39, 23, 11, 2, 36, 41, 4, 50] for related results on this subject.”
Silting mutation in triangulated categories (2010, methodology): “As an application, we deduce a result of Hoshino–Kato–Miyachi [19], which associates a t-structure for each silting subcategory (Corollary 4.”
On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings (2012, methodology): “Pauksztello was motivated by the analogies between rational homotopy theory and homological methods in commutative algebra [5] and his goal was to find a version of results on t-structures from [32] which would work for the derived categories of cochain dg algebras.”
Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) (2007, background): “nd most of our main results are completely new. The only exception known to the author is that a part ofTheorem 4.5.2 (the one that concerns t-structures) is a slight generalization of Theorem 1.3 of [HKM02]. Recently weight structures were also (independently) introduced by D. Pauksztello (he calle”