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 [midast ] 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 [midast ] G ⊆ K [midast ] 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 [midast ] G
to be semihereditary, maximal or Azumaya over V.