ON SEMIHEREDITARY MAXIMAL ORDERS

Abstract

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[otimes ]_VV^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.