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}f:G×G↦K∖{0} be a normalized two-cocycle such that f(G×G)⊆S∖{0}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∈GSxσAf=∑σ∈GSxσ with multiplication given by xσsxτ=σ(s)f(σ,τ)xστxσsxτ=σ(s)f(σ,τ)xστ for s∈Ss∈S, σ,τ∈Gσ,τ∈G. We characterize semihereditary and Dubrovin crossed-product orders, under mild valuation-theoretic assumptions placed on the nature of the extension K/FK/F.