Crossed product orders over valuation rings II: Tamely ramified crossed product algebras

Let V be a commutative valuation domain of arbitrary K rull-dimension, with quotient field F, let K be a finite Galois extension of F with group G, and let S be the integral closure of V in K. Suppose that one has a 2-cocycle on G that takes values in the group of units of S. Then one can form the crossed product of G over S, S * G, which is a V-order in the central simple F-algebra K * G. If S * G is assumed to be a Dubrovin valuation ring of K * G, then the main result of this paper is that, given a suitable definition of tameness for central simple algebras, K * G is tamely ramified and defectless over F if and only if K is tamely ramified and defectless over F. The residue structure of S * G is also considered in the paper, as well as its behaviour upon passage to Henselization

Crossed product orders over valuation rings

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K * G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a 'crossed product order' S * G ⊆ K * G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S * G to be semihereditary, maximal or Azumaya over V

Weak crossed - product orders over valuation rings

Let F be a field, let V be a valuation ring of F of arbitrary Krull dimension (rank), let K be a finite Galois extension of F with group G, and let S be the integral closure of V in K. Let f:G×G↦K∖{0} be a normalized two-cocycle such that f(G×G)⊆S∖{0}, but we do not require that f should take values in the group of multiplicative units of S. One can construct a crossed-product V-order Af=∑σ∈GSxσ with multiplication given by xσsxτ=σ(s)f(σ,τ)xστ for s∈S, σ,τ∈G. We characterize semihereditary and Dubrovin crossed-product orders, under mild valuation-theoretic assumptions placed on the nature of the extension K/F

A valuation theory for nonassociative quaternion algebras

A nonassociative quaternion algebra over a field F is a 4-dimensional F-algebra A whose nucleus is a separable quadratic extension field of F. We define the notion of a valuation ring for A, and we also define a value function on A with values from a totally ordered group. We determine the structure of the set on which the function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function associated to it if and only if the valuation ring is integral and invariant under proper F-automorphisms of A