Recollement and tilting complexes.
In this very interesting paper the author studies compatibility between two constructions for derived categories of module categories. The first is the recollement of derived categories as it is defined by Beilinson-Bernstein and Deligne, and introduced into representation theory of algebras by Parshall and Scott. The second construction is the construction of equivalences between derived categories by tilting complexes due to Rickard and Keller. Recollement diagrams for module categories appear frequently using an idempotent \(e^2=e\) in an algebra \(A\), and then the category of \(A\)-modules is a recollement of the category of \(eAe\)-modules with the category of \(A/AeA\)-modules. Obviously, partial tilting complexes give rise to an idempotent and hence to a recollement diagram. The paper contains plenty of results. We can only describe some of the most striking. For a \(k\)-projective \(k\)-algebra \(A\) we denote by \(D(A)\) the derived category of complexes of \(A\)-modules. For an idempotent \(e\) of the \(k\)-projective \(k\)-algebra \(A\) the author constructs an unbounded complex \(\Delta_A(e)\) by some limit construction inducing a recollement of \(D(A)\) from \(D(eAe)\) and the derived category \(D_{A/AeA}(A)\) of complexes of \(A\)-modules with homology in \(A/AeA\)-modules. One of the main results of the paper is the characterization of the equivalence of the recollement of \(D(A)\) from \(D(eAe)\) and \(D_{A/AeA}(A)\) and the recollement of \(D(B)\) from \(D(fBf)\) and \(D_{B/BfB}(B)\) for another \(k\)-projective \(k\)-algebra \(B\) and an idempotent \(f\) of \(B\) in terms of very natural conditions on tilting complexes. Namely the tilting complex \(T\) over \(A\) is a direct sum of \(T_1\) and \(T_2\), \(f\) corresponds to the idempotent endomorphism of \(T\) given by projection on \(T_1\), \(T_1\) is formed by projective \(eA\)-modules, and the projection of \(P_1\) to \(D(eAe)\) is a tilting complex for \(eAe\). The behaviour of two-sided tilting complexes is studied as well. The paper contains also a detailed study of the special case of symmetric algebras over a field and yields very nice results there.
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