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 satisfying properties that are natural in the context of noncommutative algebraic geometry, focusing in particular on cartain “local” conditions on . 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 in (J. Algebra 184 (1996) 435–490).