Essential Dimension of Central Simple Algebras when the Characteristic is Bad

This is a survey of the existing literature, the state of the art, and a few minor new results and open questions regarding the essential dimension of central simple algebras and finite sequences of such algebras over fields whose characteristic divides the degree of the algebras under discussion. Upper and lower bounds as well as a few precise evaluations of this dimension are included.

💡 Research Summary

The paper surveys and extends the theory of essential dimension (ed) and essential p‑dimension (edₚ) for central simple algebras (CSAs) in the “bad characteristic” situation, i.e. when the characteristic p of the base field k divides the degree d of the algebras under consideration. After recalling the general definitions of ed and edₚ for covariant functors F : Fields/k → Sets, the authors introduce the principal functor of interest, Alg_{d,e}, which assigns to each field F⊃k the set of isomorphism classes of CSAs of degree d and exponent dividing e. The distinction between the good characteristic case (p∤d) and the bad characteristic case (p|d) is emphasized throughout, because many phenomena change dramatically when p divides the degree.

The survey begins with a concise review of the classical theory of CSAs: Wedderburn–Artin decomposition, reduced trace and norm, Brauer groups, and the relationship between degree, index, and exponent. It then discusses p‑algebras (those whose Brauer class lies in the p‑torsion subgroup of the Brauer group) and recalls Albert’s theorem that every p‑algebra is Brauer‑equivalent to a cyclic algebra. The authors also present the Kato‑Milne cohomology framework, which replaces Galois cohomology in characteristic p and provides a convenient language for deriving lower bounds on essential dimensions.

A central component of the paper is Table 1, which collects all known bounds and exact values for ed(Alg_{n,n}) and related functors, both in bad and good characteristic. For example, the general upper bound ed(Alg_{n,n}) ≤ n²−3n+1 (from