Roland Lötscher: Essential dimension of separable algebras embedding in a fixed central simple algebra

Submission: 2014, Mar 10

In this paper we fix a central simple F-algebra A of prime power degree and study the essential dimension of separable algebras B over extensions K/F, which embed in A_K, such that the simple components of B_\alg{K} and of its centralizer in A_\alg{K} are all of the same degree d and r, respectively. This extends earlier work (preprint 403), where A was assumed to be a division algebra. Suppose d<=r. If ind(A)<=r/d the problem reduces to the computation of the essential dimension of the group (PGL_d)^m\rtimes S_m with m=deg(A)/(rd), which is extremely difficult in general. In case of ind(A)>r/d, however, we manage to compute the exact value of the essential dimension of the given class of algebras, except in case ind(A)=2 and r=d>1, where we provide lower and upper bounds on the essential dimension.

2010 Mathematics Subject Classification: 16W10, 16K20

Keywords and Phrases: essential dimension, central simple algebras, separable algebra, \'{e}tale algebras, non-split algebraic group

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