Hisaaki Fujita

Frobenius Full Matrix Algebras and Gorenstein Tiled Orders

ABSTRACT For integers n ≥ 2, n × n full matrix algebras with structure systems are introduced in Fujita (2003) as a framework to study certain factor algebras of tiled orders. Such factor algebras of Gorenstein tiled orders are Frobenius full matrix algebras. In this article, we verify that for every integer n ≤ 7, the converse is true. For every integer n ≥ 8, we show that the converse is not true, that is, there are Frobenius n × n full matrix algebras having no corresponding Gorenstein tiled orders.

Neat idempotents and tiled orders having large global dimension

Abstract We study neat primitive idempotents in a semiperfect Noetherian ring, and as an application, we improve an example of a tiled order having large global dimension given by Jansen and Odenthal. Moreover, another two tiled orders having large global dimension are added and two questions on tiled orders of finite global dimension are posed.

A Tiled Order of Finite Global Dimension with No Neat Primitive Idempotent

Let R be a discrete valuation ring with residue class field F. For an arbitrary prime p, we construct a tiled R-order Λ such that global dimension gld Λ = 5 if characteristic char F ≠ p and gld Λ = ∞ if char F = p. If char F ≠ p, Λ is a tiled R-order of finite global dimension with no neat primitive idempotent, so that a question posed in Fujita (2002) is solved.