Bimodule complexes via strong homotopy actions
Let \(A\) and \(B\) be two \(k\)-algebras over the commutative ring \(k\). A result of Rickard shows that the derived categories \(D(A)\) and \(D(B)\) are equivalent if and only if there is a complex \(T\) in the homotopy category of bounded complexes of finitely generated projective \(A\)-modules so that every homomorphism of \(T\) to some shifted copy is homotopic to 0, that \(T\) is big enough and so that the endomorphism ring of \(T\) is isomorphic to \(B\). Such a complex is called a tilting complex. It is shown in an earlier paper of the author that in case \(A\) is flat as \(k\)-module, then there is a complex \(X\) of \(A\)-\(B\)-bimodules so that the tensor product with \(X\) induces an equivalence which sends the rank one free \(B\) module to \(T\). The author gives an explicit construction of \(X\) with the only knowledge of \(T\) and the homotopies annihilating the homomorphisms of \(T\) to shifted copies. The basic tool for the intrinsic proof is the \(A_\infty\)-structure of \(T\). This is deduced by taking iterated homotopies of the operation of the endomorphism ring on \(T\). The author is making also intensive use of differential graded algebras and coalgebras which he already employed in the afore mentioned paper and in various other papers on related subjects. The paper under review also contains a complete and detailed description of the case of a two term and a three term tilting complex.
scientific article; zbMATH DE number 1572977