Recollement and tilting complexes

First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P• to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements and , where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex ΔT• which induces an equivalence between and . Third, we apply the above to a symmetric algebra A. We show that a partial tilting complex P• for A of length 2 extends to a tilting complex, and that P• is a tilting complex if and only if the number of indecomposable types of P• is one of A. Finally, we show that for an idempotent e of A, a tilting complex for eAe extends to a recollement tilting complex for A, and that its standard equivalence induces an equivalence between and .