On normality of ASL domains (Q1092127)

ASL stands for ``algebra with straightening laws'' in the sense of \textit{D. Eisenbud} [in Ring theory and algebra III, Proc. 3rd Okla. Conf., 1979, Lect. Notes Pure Appl. Math. 55, 243-268 (1980; Zbl 0448.13010)], alias ordinal Hodge algebra. The author gives some sufficient conditions for a graded ASL on a poset H over a field k to be a normal domain, assuming that the associated discrete ASL, k[H], is Cohen-Macaulay and that all maximal chains in H have the same length. His conditions are too technical to write here, but the arithmetic normality of Grassmann and Schubert varieties is easily derived from them. The key observation is that \textit{bad} prime ideals of height one of k[H] correspond with special subsets of H which he calls \textit{spindles} of H.