Triangular Matrix Algebras Over Hensel Rings

Joseph A. Wehlen
1973 Proceedings of the American Mathematical Society  
Let (R, m) be a local Hensel ring and A an algebra over R which is finitely generated and projective as an /{-module. If A contains a complete set of mutually orthogonal primitive idempotents eu---,e" indexed so that eiNe^mA whenever liy", we show that A is isomorphic to a generalized triangular matrix algebra and that A is the epimorphic image of a finitely generated, projective /{-algebra B of Hochschild dimension less than or equal to one. Introduction. The class of residue algebras of
more » ... e algebras of semiprimary hereditary algebras has been thoroughly discussed in [5], [9], [2], [6] and [12] . They consist of those finitely generated algebras A over a field R which contain a complete set of mutually orthogonal primitive idempotents ex, • • • ,en which can be indexed so that ¿¿/Ve., = (0) whenever i\%j, where N denotes the Jacobson radical of A, and A/N is Ä-separable. All such algebras are isomorphic to generalized triangular matrix algebras. The purpose of this paper is to show that every finitely generated, projective algebra A over a local Hensel ring (R, m) satisfies the "triangular" idempotent condition e^Ne^ntA whenever i^j if and only if A is a residue algebra of a finitely generated, projective algebra B of Hochschild dimension less than or equal to one. We will call such algebras "almost one-dimensional." Such an algebra B, the maximal algebra for A, is usually neither semiprimary nor hereditary. In fact, if we assume that R is also noetherian, it has been shown in [11] that B is hereditary if and only if R is a field (Rdim B^l) or a DVR (Ä-dim 5=0); B is semiprimary if and only if R has a nilpotent radical-in which case every finitely generated, projective algebra over R has infinite global dimension-or R is a field. A generalization of a result of S. U. Chase [2] will provide a sort of converse to Theorem 2.7 of [10] in the case of a Hensel ring:
doi:10.2307/2038708 fatcat:33ulivam65exbpime2b5ywp6ky