Global dimension two orders are quasi-hereditary
Let R be a complete discrete valuation ring with field of fractions K. Let \(\Lambda\) be a basic R-order in a separable K-algebra A, and let \(f_ 1,...,f_ s\) be a full set of primitive central idempotents of A. \(\Lambda\) is said to be quasi-hereditary if it contains primitive idempotents \(e_ 1,...,e_ s\) such that 1) \(f_ ie_ i=e_ i\) for each i; 2) gl.dim \(e_ i\Lambda e_ i<\infty\) for each i; and 3) For \(e=\sum e_ i\), the ring \(\Lambda\) /\(\Lambda\) \(e\Lambda\) is quasi- hereditary in the sense of \textit{E. Cline}, \textit{B. Parshall}, and \textit{L. Scott} [J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)]. The authors prove analogs for this situation of some results of \textit{V. Dlab} and \textit{C. M. Ringel} [Ill. J. Math. 33, 280-291 (1989; Zbl 0666.16014)]. To wit, if gl.dim \(\Lambda\leq 2\) then \(\Lambda\) is quasi- hereditary, and \(\Lambda\) is hereditary if and only if every chain of full, two-sided, idempotent ideals can be refined to a so-called heredity chain for \(\Lambda\). \{Reviewer's remark: The use of the noun ``heredity'' as an adjective is very unpleasant. If it is not already too late, a better term should be chosen.\}
scientific article; zbMATH DE number 4114816