Derived equivalences of DG algebras are not governed by their cohomology rings

Abstract: Let $\mathscr{A}$ and $\mathscr{B}$ be two connected cochain DG algebra such that $\mathscr{A}^{#}=\mathscr{B}^{#}$ and the cohomology rings $H(\mathscr{A})$ and $H(\mathscr{B})$ are isomorphic. We give examples to show that $\mathscr{A}$ and $\mathscr{B}$ are not necessarily derived equivalent, thereby answering to a question of Dugas.