On Semihereditary Maximal Orders

Let A be an order integral over a valuation ring V in a centralsimple F-algebra, where F is the fraction field of V. We showthat (a) if (V^h, F^h) is the Henselization of (V, F), then Ais a semihereditary maximal order if and only if A_VV^h is a semihereditarymaximal order, generalizing the result by Haile, Morandi andWadsworth, and (b) if J(V) is a principal ideal of V, then asemihereditary V-order is an intersection of finitely many conjugatesemihereditary maximal orders; if not, then there is only onemaximal order containing the V-order. 1991 Mathematics SubjectClassification 16H05.     

A Valuation Theory for Nonassociative Quaternion Algebras

A nonassociative quaternion algebra over a field F is a 4-dimensional F-algebra A whose nucleus is a separable quadratic extension field of F. We define the notion of a valuation ring for A, and we also define a value function on A with values from a totally ordered group. We determine the structure of the set on which the function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function associated to it if and only if the valuation ring is integral and invariant under proper F-automorphisms of A.     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.     

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 Galoisextension of F with group G, and S the integral closure of Vin K. If, in the crossed product algebra K * G, the 2-cocycletakes 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 andZ. Yi on the homological dimension of crossed products, thispaper discusses necessary and/or sufficient valuation-theoreticconditions, on the extension K/F, for the V-order S * G to besemihereditary, maximal or Azumaya over V. 2000 MathematicsSubject Classification 16H05, 16S35.