On the derived categories and quasitilted algebras. (Q735232)

The authors prove the following results and derive from them some corollaries about quasi-tilted algebras: Theorem. Let \(A\) be an algebra. If every indecomposable complex in the bounded derived category of \(A\)-modules has length (the number of nonzero components) at most \(n\) for some positive integer \(n\), then the finitistic dimension of \(A\) is at most \(n\). Theorem. A gentle algebra over an algebraically closed field is quasi-tilted if and only if each indecomposable complex in the bounded derived category of \(A\)-modules has length at most two. Theorem. For an algebra \(A\) the following conditions are equivalent: (i) \(A\) is quasi-tilted. (ii) Every complex in the bounded derived category of \(A\)-modules is isomorphic to a complex, consisting of a module \(X\) such that from the existence of a finite path of nonzero morphisms in \(\text{ind\,}A\) from \(Y\) to \(X\) it follows that the projective dimension of \(Y\) is at most \(1\). (iii) Every complex in the bounded derived category of \(A\)-modules is isomorphic to a complex, consisting of a module \(Y\) such that from the existence of a finite path of nonzero morphisms in \(\text{ind\,}A\) from \(Y\) to \(X\) it follows that the injective dimension of \(X\) is at most \(1\).