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^{centerdot} to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements \{\cat{D}_{A/AeA}(A), \cat{D}(A), \cat{D}(eAe)} and ${\cat{D}_{B/BfB}(B), \cat{D}(B), \cat{D}(fBf)}$, where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex ${\Delta }_T^{\cdot }$ which induces an equivalence between $\text{cat}{D}_{A\mathrm{/}AeA}(A)$ and $\text{cat}{D}_{B\mathrm{/}BfB}(B)$. Third, we apply the above to a symmetric algebra A. We show that a partial tilting complex $P^{\cdot }$ for A of length 2 extends to a tilting complex, and that $P^{\cdot }$ is a tilting complex if and only if the number of indecomposable types of $P^{\cdot }$ 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 $\text{cat}ModA\mathrm{/}AeA$ and $\text{cat}ModB\mathrm{/}BfB$.