Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein RR-algebras over a commutative Gorenstein ring RR. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein RR-algebra AA to have a Gorenstein RR-algebra BB as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein RR-algebra AA whose endomorphism algebra is not a Gorenstein RR-algebra.