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

Let R be a discrete valuation ring with a unique maximal ideal πR and a quotient field K, and let F = R/πR be the residue class field. Let n ≥ 2 be an integer and {λij | 1 ≤ i, j ≤ n} a set of n integers satisfying λii = 0, λik + λkj ≥ λij , λij + λji > 0 (if i = j) for all 1 ≤ i, j, k ≤ n. Then Λ = (πijR) is a basic semiperfect Noetherian R-subalgebra of the full n× n matrix algebra Mn(K). We call such Λ a tiled R-order in Mn(K). Let S be a semiperfect Noetherian ring and e a primitive idempotent of S. Following Ágoston, Dlab and Wakamatsu [1], we call e a neat primitive idempotent if ExtiS(V, V ) = 0 for all i ≥ 1, where V is a simple right S-module with V e = 0 (see [5], too). It was proved by Jategaonkar [7] that for a fixed integer n ≥ 2, there are, up to isomorphism, only finitely many tiled R-orders of finite global dimension in Mn(K). The literature contains a number of papers concerned with determining tiled R-orders of finite global dimension. Tiled R-orders of global dimension two were studied by Roggenkamp and Wiedemann in connection with the interest of orders of finite lattice type (see [2], [11], [12], [20]). As for the problem to determine the maximum finite global dimension among tiled R-orders in Mn(K) for a fixed n, some authors studied tiled R-orders having large global dimension, but it is not known what is the maximum (see [4], [5], [6], [7], [8], [9], [14], [17], [18]). In such examples, neat primitive idempotents play an essential role when we compute global dimension inductively. Then in [5], we posed a question “Does any tiled R-order of finite global dimension have a neat primitive idempotent?”, which can be considered as an improved version of Jategaonkar’s conjecture disproved by Kirkman and Kuzmanovich [9] and [4] for all n ≥ 6. We notice that in those studies, almost all known results hold ifR is an arbitrary discrete valuation ring. However, among other things, Rump [14] proved that global dimension gl.dim Λ of a tiled R-order Λ = (πijR) is determined by the set {λij | 1 ≤ i, j ≤ n} and charF (characteristic of F ), and that if gl.dim Λ ≤ 2 then gl.dim Λ does not depend on charF , by using matroid theory (see Tutte [19]). Moreover, he provided an example of a tiled R-order Λ in Mn(K) such that gl.dim Λ = 3 if charF = 2, and gl.dim Λ = 4 if charF = 2, where n = 14. In accordance with matroid theory, Rump calls a tiled R-order regular if its global dimension does not depend on charF , and he added the following sentence: “For the present, at least, we have demonstrated that the problem to determine the tiled orders of finite global dimension can hardly be solved without a careful inspection of regularity.” In this report, we announce a new example of non-regular tiled R-orders. Namely, for an arbitrary prime p, we construct a tiled R-order Λ in Mn(K) such that gl.dim Λ = 5 if charF = p and gl.dim Λ = ∞ if charF = p, where n = 4p + 5. Moreover, in the