Global dimensions of a class of tiled orders
The authors consider a tiled order \(\Lambda\) in \(M_ n(D)\), \(n>1\), where D is a commutative discrete valuation ring with maximal ideal (x). Tarsy's conjecture states that the \(gl\dim (\Lambda)<n\), provided it is finite. Recently H. Fujita has given a counterexample to this conjecture. In this paper the conjecture is verified in case \(x\cdot M_ n(D)\subset A\). The proofs depend on graphical techniques introduced by \textit{A. Wiedemann} and the reviewer [J. Algebra 80, 113-133 (1983; Zbl 0518.16001)], \textit{H. Fujita} [Tsukuba J. Math. 10, 121-130, 293-298 (1986; Zbl 0603.16006 and Zbl 0616.16003)] and the reviewer [Math. Z. 154, 225-238 (1977; Zbl 0335.16010); 160, 63-67 (1978; Zbl 0383.16002)]. In this connection the authors show the formula \[ rgl\dim (A)\leq pd(AeA_ A)+rgl\dim (A/AeA)+rgl\dim (eAe)+2. \] Finally they give quite an explicit formula for the global dimension of a `local idempotent ring corresponding to the vertex i' of a tiled order.
scientific article; zbMATH DE number 4120331
scientific article