Indecomposable p-algebras and Galois subfields in generic abelian crossed products

TL;DR

This paper investigates the structure of certain p-algebras over Henselian fields, showing that Galois subfields are inertial and establishing the indecomposability of specific generic abelian crossed product p-algebras, with implications for algebraic geometry.

Contribution

It proves that noncyclic generic abelian crossed product p-algebras defined by non-degenerate matrices are indecomposable, advancing understanding of their structure and construction.

Findings

Galois subfields of semi-ramified p-algebras are inertial.

Noncyclic generic abelian crossed product p-algebras are indecomposable.

Degeneracy in matrices relates to torsion in CH^2 of Severi-Brauer varieties.

Abstract

Let F be a Henselian valued field with char(F) = p and D a semi-ramified, "not strongly degenerate" p-algebra. We show that all Galois subfields of D are inertial. Using this as a tool we study generic abelian crossed product p-algebras, proving among other things that the noncyclic generic abelian crossed product p-algebras defined by non-degenerate matrices are indecomposable p-algebras. To construct examples of these indecomposable p-algebras with exponent p and large index we study the relationship between degeneracy in matrices defining abelian crossed products and torsion in CH^2 of Severi-Brauer varieties.