Essential Dimension of Separable Algebras Embedding in a Fixed Central Simple Algebra

One of the key problems in non-commutative algebra is the classification of central simple algebras and more generally of separable algebras over fields, i.e., Azumaya-algebras whose center is étale over the given field. In this paper we fix a central simple F -algebra A of prime power degree and study seperable algebras over extensions K/F , which embed in AK . The type of such an embedding is a discrete invariant indicating the structure of the image of the embedding and of its centralizer over an algebraic closure. For fixed type we study the minimal number of independent parameters, called essential dimension, needed to define the separable K-algebras embedding in AK for extensions K/F . We find a remarkable dichotomy between the case where the index of A exceeds a certain bound and the opposite case. In the second case the task is equivalent to the problem of computing the essential dimension of the algebraic groups (PGLd) ⋊Sm, which is extremely difficult in general. In the first case, however, we manage to compute the exact value of the essential dimension, except in one special case, where we provide lower and upper bounds on the essential dimension.