Degeneracy and decomposability in abelian crossed products

TL;DR

This paper constructs a specific indecomposable abelian crossed product division algebra with particular properties, providing new insights into degeneracy and decomposability in algebraic structures.

Contribution

It introduces a novel indecomposable generic abelian crossed product algebra that is decomposable as an underlying structure, advancing understanding of algebraic degeneracy.

Findings

Constructed an indecomposable abelian crossed product division algebra of exponent p and index p^2.

Showed the algebra is indecomposable without torsion in the Chow group.

Provided examples of Brauer classes with differing Dec group memberships.

Abstract

In this paper we study the relationship between degeneracy and decomposability in abelian crossed products. In particular we construct an indecomposable abelian crossed product division algebra of exponent $p$ and index $p^2$ for $p$ an odd prime. The algebra we construct is generic in the sense of Amitsur and Saltman and has the property that its underlying abelian crossed product is a decomposable division algebra defined by a non-degenerate matrix. This algebra gives an example of an indecomposable generic abelian crossed product which is shown to be indecomposable without using torsion in the Chow group of the corresponding Severi-Brauer variety as was needed in [Karpenko, Codimension 2 cycles on Severi-Brauer varieites (1998)] and [McKinnie, Indecomposable $p$-algebras and Galois subfields in generic abelian crossed products (2008)]. It also gives an example of a Brauer class which is in Tignol's Dec group with respect to one abelian maximal subfield, but not in the Dec group with respect to another.