Triangulated equivalences and reconstruction of classifying spaces

In algebra such as algebraic geometry, modular representation theory and commutative ring theory, we study algebraic objects through associated triangulated categories and topological spaces. In this paper, we consider the relationship between such triangulated categories and topological spaces. To be precise, we explore necessary conditions for derived equivalence of Noetherian schemes, stable equivalence of finite groups, and singular equivalence of commutative Noetherian rings by using associated topological spaces.