## Abstract

A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and _{R} M _{S} is an S-R-bimodule. In the main theorem of this paper we show that if T _{S} is a tilting S-module, then under certain homological conditions on the S-module M _{S}, one can extend T _{S} to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and T _{S}, and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M _{S} is finitely generated. In this case, (S′,R,M′)=(S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M _{S} has a finite projective resolution and Ext _{S} ^{n} (M _{S}, S)=0 for all n>0. In this case, (S′,R,M′)=(S, R, Hom _{S} (M, S)).

Original language | English |
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Pages (from-to) | 57-74 |

Number of pages | 18 |

Journal | Algebras and Representation Theory |

Volume | 14 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2011 |

Externally published | Yes |

## Keywords

- Derived equivalence
- Tilting complex
- Triangular matrix ring

## ASJC Scopus subject areas

- Mathematics (all)