Weak crossed - product orders over valuation rings

Kauta, John S. (2014) Weak crossed - product orders over valuation rings. Journal of Algebra, 402 . pp. 319-350. ISSN 0021-8693

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Abstract

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.

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