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

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.

Degeneracy and decomposability in abelian crossed products

“…In [McK08,Section 3.3] examples of such algebras with index p n and exponent p for all p = 2 and all n 2 are given. An example is also given in the case p = 2 and n = 3.…”

“…In this paper we continue the study of the relationship between degeneracy and decomposability in abelian crossed products (K. McKinnie (2008) [McK08]). In particular we construct an indecomposable abelian crossed product division algebra of exponent p and index p 2 for p an odd prime.…”

“…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 N.A. Karpenko (1998) [Kar98] and K. McKinnie (2008) [McK08]. 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.…”

“…For example, in light of the present paper, we draw the readers attention to [McK08] and [Mou08] where the decomposability of generic abelian crossed product division algebras are studied. In [Mou08, Corollary 3.6] it is shown that a p-power degree generic abelian crossed product defined by a non-degenerate matrix (definition given below) is indecomposable.…”

“…In [Mou08, Corollary 3.6] it is shown that a p-power degree generic abelian crossed product defined by a non-degenerate matrix (definition given below) is indecomposable. [McK08] and [Mou08] leave open the question of whether any abelian crossed product defined by a non-degenerate matrix is indecomposable. In this paper we answer this question in the negative by constructing a decomposable abelian crossed product F A defined by a non-degenerate matrix (Theorem 2.12/5.2).…”