On automorphism groups induced by bimodules

摘要

Let R, S be rings and R M S a faithfully balanced bimodule. It is proved that the groups Out M (R) and Out M (S) of outer automorphisms of R and S which fix (up to isomorphism) the underlying modular structure of M are isomorphic. In case A, B are finite dimensional algebras and A M B is finite dimensional, it is proved that the isomorphism ${rm Out}^{M} {rm(A)}cong {rm Out}^{M} {rm (B)}$ is algebraic, by showing in the way that every Out-orbit of a finite dimensional module has a canonical structure of algebraic variety. The result is then applied to prove that the identity components of Out(A) and Out(B) are isomorphic when the Out-orbits of A M and M B are finite. That generalizes a result of Brauer and allows us to stablish the same conclusion for algebras which are iterated tilted from one another. The paper finishes with an identification of the iterated tilted algebras which have finite Picard group.