Derived equivalences, matrix equivalences, and homological conjectures

Abstract: Centralizer matrix algebras were investigated initially by Georg Ferdinand Frobenius in the Crelle's Journal around 1877. By introducing three new equivalence relations on all square matrices over a field, we completely characterize Morita, derived and almost $\nu$-stable derived equivalences between centralizer matrix algebras in terms of these matrix equivalences, respectively. Thus the categorical equivalences are reduced to matrix equivalences in linear algebra. Further, we show that a derived equivalence between centralizer matrix algebras of permutation matrices induces both a Morita equivalence and additional derived equivalences for $p$-regular parts and for $p$-singular parts. As applications, we show that the finitistic dimension conjecture and Nakayama conjecture are valid for centralizer matrix algebras.