On t-structures and torsion theories induced by compact objects

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 one to one correspondence between tilting complexes of term length two and torsion theories with certain conditions.

Introduction to τ-tilting theory

“…We denote by 2-siltΛ the set of isomorphism classes of basic two-term silting complexes in K b ðproj ΛÞ. Two-term tilting complexes were studied by Hoshino et al (49). We have the following connection between support τ-tilting modules and two-term silting complexes.…”

Compact corigid objects in triangulated categories and co-t-structures

“…Hoshino, Kato and Miyachi,in [11], show that a natural t-structure is induced on T by a suitably nice compact object of T . In particular, they consider a compact object S of T which satisfies the following two conditions:…”

“…If S is a compact rigid object of T and {Σ i S | i ∈ Z} is a generating set for T , then the two halves of the t-structure obtained in [11] are given by X = {X ∈ T | Hom T (S, Σ i X) = 0 for i > 0}, Y = {X ∈ T | Hom T (S, Σ i X) = 0 for i < 0}.…”

“…Hence, the object S considered in [11] is analogous to a chain DGA and the two halves of the t-structure it induces are analogous to the full subcategories of DG R-modules whose cohomology vanishes in positive and negative degree, respectively. In the theory of DGAs, when one has a construction for chain DGAs it is natural to ask: what is the dual construction for cochain DGAs?…”

“…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]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on T whose heart is equivalent to Mod(End(C) op ).…”