Two Remarks About Hereditary Orders

H. Jacobinski
1971 Proceedings of the American Mathematical Society  
In the first remark it is shown that, over a Dedekind ring, hereditary orders in a separable algebra are precisely the "maximal" orders under a relation stronger than inclusion (Theorem 1). At the same time simple proofs for known structure theorems of hereditary orders are obtained. In the second remark a complete classification is given of lattices over a hereditary order, provided the underlying Dedekind ring is contained in an algebraic number field and the lattices satisfy the Eichler
more » ... fy the Eichler condition (Theorem 2). • Let o be a Dedekind ring with quotient field k, A/k a separable finite-dimensional algebra over k and R an o-order in A (i.e. a finitely generated O-algebra in A, containing the identity and such that kR = A). An order R is hereditary, if every left ideal is a projective i?-module. It is a classical result-apart from terminology-that maximal orders are hereditary, but the converse of this is false: there are nonmaximal hereditary orders. Our first remark is, that if inclusion is replaced by a stronger relation, hereditary orders are characterized by the property of being locally maximal everywhere under this relation. To avoid confusion, we will use the term extremal orders instead. This characterization of hereditary orders can be used to give very simple proofs of some known properties of hereditary orders, which were obtained by Harada [4] and Brumer [2]. Since Brumer [2] is not available in print, we include proofs of the main results given there. In the complete local case, the structure of i?p-lattices is well known (Brumer [2] ). The basic fact is that indecomposable i?p-lattices are in fact lattices over a maximal order containing Rp. This does not hold globally and only partial results are known in that case. Using results of an earlier paper (Jacobinski [5]), we give a complete classification of lattices over a hereditary o-order, provided the quotient field of 0 is an algebraic number field. The local theory yields a classification of genera of ¿^-lattices. Our result is that the lattices in a restricted genus are isomorphic. This means that two i?-lattices M and N are
doi:10.2307/2037745 fatcat:quwob37vg5c25ieo34s2eisnay