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

TL;DR

This paper demonstrates that derived equivalences of DG algebras are not solely determined by their cohomology rings, providing counterexamples to a previously open question.

Contribution

It shows that isomorphic cohomology rings do not guarantee derived equivalence of DG algebras, addressing a question posed by Dugas.

Findings

Counterexamples where DG algebras with isomorphic cohomology rings are not derived equivalent

Clarification that cohomology rings alone do not govern derived equivalences

Answers an open question in the theory of DG algebras

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.