Relative derived equivalences and relative homological dimensions (Q265918)

Let \(\mathcal{A}\) be a small abelian category, and let \(F\) be a closed subbifunctor of \(\text{Ext}_{\mathcal{A}}^1 (-,-)\). In this setting, the relative derived category \({D}_F^{b}(\mathcal{A})\) has been constructed by A.~Buan by generalizing Verdier's construction of of the quotient category, the localization is with respect to \(F\)-acyclic complexes. This paper mainly deals with the module categories over the Artin \(R\)-algebras \(\Lambda\) and \(\Lambda\), where \(R\) is a commutative Artin ring. The first main result is a relative version of Rickard's Morita theory for derived categories, that is, a characterization of when the relative derived categories \({D}_F^{b}(\Lambda)\) and \({D}^{b}(\Lambda)\) are equivalent as triangulated categories is given in terms of one-sided \(F\)-tilting complexes. Next, the homological properties of relative derived categories are discussed, and the second main result reads as follows. Let \(L:{D}_F^{b}(\Lambda)\to {D}^{b}(\Lambda)\) be a relative derived equivalence with associated \(F\)-tilting complex \(T^\bullet\); then (1) gl.dim\(_F(\Lambda)-t(T^\bullet)\leq\) fin.dim\((\Gamma)\leq\) gl.dim\(_F(\Lambda)+t(T^\bullet)+2\); (2) fin.dim\(_F(\Lambda)-t(T^\bullet)\leq\) fin.dim\((\Gamma)\leq\) fin.dim\(_F(\Lambda)+t(T^\bullet)+2\). Finally, applications to Gorenstein algebras are given.