Non-standard derived equivalences over arbitrary fields

Abstract

In 1991 Rickard asked whether every derived equivalence between finite-dimensional algebras over a common field is standard. Hu, Xi and Zhang constructed non-standard derived equivalences over the field with two elements and conjectured that Rickard's question has a positive answer in characteristic different from two. We disprove this conjecture by constructing non-standard derived autoequivalences over every field. In fact, each field admits infinitely many finite-dimensional algebras with such autoequivalences. The proof is characteristic-free and is based on a supertrace identity for matrices over truncated polynomial algebras. We also give a second family over finite fields. Consequently, Rickard's question has a negative answer over every field.