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 $\varDelta^{ \centerdot}_{ T}$ which induces an equivalence between $\cat{ D}_{ A/AeA}(A)$ and $\cat{ D}_{ B/BfB}(B)$. Third, we apply the above to a symmetric algebra A. We show that a partial tilting complex $P^{ \centerdot}$ for A of length 2 extends to a tilting complex, and that $P^{ \centerdot}$ is a tilting complex if and only if the number of indecomposable types of $P^{ \centerdot}$ 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 $\cat{ Mod}A/AeA$ and $\cat{ Mod}B/BfB$.
Categories: math.RA
