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 . 2000 Mathematics Subject Classification 16H05, 16S35.