Frobenius bimodules between noncommutative spaces

In this paper we study Frobenius bimodules between noncommutative spaces (quasi-schemes), developing some of their basic properties. If X and Y are spaces, we study those Frobenius X,Y-bimodules M satisfying properties that are natural in the context of noncommutative algebraic geometry, focusing in particular on cartain "local" conditions on M. As applications, we prove decomposition and gluing theorems for those Frobenius bimodules which have good local properties. Additionally, when X and Y are schemes we relate Frobenius X,Y-bimodules to the sheaf X,Y-bimodules introduced by Van den Bergh.