On \(t\)-structures and torsion theories induced by compact objects.

*(English)*Zbl 1006.18011The authors present in this paper a new approach and a generalisation of J. Rickard’s theory of tilting complexes. Rickard’s theory describes equivalences between derived module categories. The authors study in the present paper the relation with \(t\)-structures of triangulated categories, torsion theories of abelian categories, and the consequences for tilting theory given by two-term tilting complexes. The restriction to two-term complexes seems to be motivated by Okuyama’s method for proving Broué’s conjecture. Okuyama’s method uses two-term tilting complexes, and is at the moment the most successful way to prove the conjecture in examples.

The authors first show that for a compact object \(C\) in a triangulated category so that for positive \(n\), \(\text{Hom}(C,C[n])=0\), \(C\) generates a \(t\)-structure so that \(\text{Hom}(C,-)\) induces an equivalence between its core and the endomorphism ring of \(C\). In case of the derived category of an abelian category \(A\) and \(C\) being a two-term complex, the authors show when \(C\) defines a torsion theory of \(A\). The condition is related to the condition of being a tilting complex, but is weaker. As a consequence, if this happens the authors deduce an equivalence between the endomorphism ring of \(C\) and the core of the \(t\)-structure, as well as a torsion theory for the core. Finally, the authors make the case of a module category of an algebra, in particular of an Artin algebra, more precise. They provide an explicit interesting example for these phenomenons.

The authors first show that for a compact object \(C\) in a triangulated category so that for positive \(n\), \(\text{Hom}(C,C[n])=0\), \(C\) generates a \(t\)-structure so that \(\text{Hom}(C,-)\) induces an equivalence between its core and the endomorphism ring of \(C\). In case of the derived category of an abelian category \(A\) and \(C\) being a two-term complex, the authors show when \(C\) defines a torsion theory of \(A\). The condition is related to the condition of being a tilting complex, but is weaker. As a consequence, if this happens the authors deduce an equivalence between the endomorphism ring of \(C\) and the core of the \(t\)-structure, as well as a torsion theory for the core. Finally, the authors make the case of a module category of an algebra, in particular of an Artin algebra, more precise. They provide an explicit interesting example for these phenomenons.

Reviewer: Alexander Zimmermann (Amiens)

##### MSC:

18E30 | Derived categories, triangulated categories (MSC2010) |

16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |

18E40 | Torsion theories, radicals |

16G10 | Representations of associative Artinian rings |

##### Keywords:

\(t\)-structures of triangulated categories; module category of algebra; tilting complexes; equivalences between derived module categories; torsion theories of abelian categories; Broué’s conjecture; Artin algebra##### References:

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