On Semihereditary Maximal Orders

Let A be an order integral over a valuation ring V in a central simple F ‐algebra, where F is the fraction field of V . We show that (a) if ( V h , F h ) is the Henselization of ( V , F ), then A is a semihereditary maximal order if and only if A ⊗ V V h is a semihereditary maximal order, generalizing the result by Haile, Morandi and Wadsworth, and (b) if J ( V ) is a principal ideal of V , then a semihereditary V ‐order is an intersection of finitely many conjugate semihereditary maximal orders; if not, then there is only one maximal order containing the V ‐order. 1991 Mathematics Subject Classification 16H05.