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 “-algebra A“f=∑σ∈GSxσ in a natural way, which is a V “-order in the
crossed-product F-algebra (K∕F,G,f). If V is unramified and defectless in K, we
show that A“f is semihereditary if and only if for all σ,τ ∈ G and every maximal
ideal M of S, f(σ,τ)∉M2. If in addition J(V ) is not a principal ideal of
V , then A“f is semihereditary if and only if it is an Azumaya algebra over
V .
