Recollement and Tilting Complexes

Original · EN

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 Pcenterdot to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements {DA/AeA(A), D(A), D(eAe) and {DB/BfB(B), D(B), D(fBf), where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex ₜ which induces an equivalence between DA/AeA(A) and DB/BfB(B). 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 ModA/AeA and ModB/BfB.