Derived equivalences, new matrix equivalences, and homological conjectures

Based on the fact that every finite-dimensional algebra over a field is isomorphic to the centralizer of $\mathbf{\text{two}}$ matrices, we approach the representation theory of finite-dimensional algebras over fields by centralizers of matrices. The first fundamental question is to study the centralizer of a single matrix, called a centralizer matrix algebra. 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. 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 an application, we show that the finitistic dimension conjecture and the Nakayama conjecture are valid for centralizer matrix algebras.