On automorphism groups induced by bimodules

Abstract

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 Out M (A) Out M (B) is algebraic, by showing in the way that every Outorbit 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.For any given mathematical object, its automorphism group is always an important object of study, even to understand the object itself. In the case of a finite-dimensional algebra A over an algebraically field K, as shown in [10], the appropriate context to study its group of automorphisms Aut(A), or the group Out(A) = Aut(A) Inn(A) of outer automorphisms of A, is that of algebraic groups. That scheme has been pursued by the authors in [6], [7] leading to a very accurate knowledge of Aut(A) and Out(A) when A is, for instance, monomial or, more generally, gradable by the radical. For the study of the category A Mod of all A-modules or the category A mod of all finitedimensional modules, the group Out(A) is very interesting since it can be embedded in the Picard group Pic(A), which can be thought of as the group of (natural isoclasses of) Morita self-equivalences A Mod 3 A Mod. Then, a natural question arises: what is the relation between Out(A) and Out(B) when A and B are well-related? A result attributed to Brauer ([10]) states that the identity components of Out(A) and Out(B) are isomorphic when A and B are Morita equivalent.The goal of this work is to study the relationship between Out(A) and Out(B) when there exists a finite dimensional faithfully balanced bimodule A M B and to use that in order to extend the above mentioned Brauers result. In order to get that extension, we use our two main results, Theorems 1.4 and 2.5, which state that the stabilizers of A M and M B under the

Hochschild Cohomology with Support

“…The localization sequence for Hochschild cohomology. We can now use (10) to obtain the following short exact sequence of Hochschild complexes:…”

“…Applications of type (1) can be seen as generalizations of results on the Hochschild cohomology of triangular matrix algebras, a topic which, since the work of Happel [11], has received a lot of attention [19,9,8,7,3,2,10]. An application of type (2) is the relation between the Hochschild complexes of Koszul dual algebras obtained in [12].…”

Automorphismes, graduations et catégories triangulées

“…Le résultat suivant aété obtenu indépendemment, et par des méthodes différentes, par Huisgen-Zimmermann et Saorìn [11]. Ce résultat avaitétéétabli pour deséquivalences dérivées particulières auparavant [9]. [24,28]).…”