Study of three-dimensional algebras with straightening laws which are Gorenstein domains. I (Q1095976)

Over a field k (of arbitrary characteristic), the authors classify all 3- dimensional homogeneous Gorenstein ASL (algebras with straightening laws [cf. \textit{C. De Concini}, \textit{D. Eisenbud} and \textit{C.Procesi}, Astérisque 91 (1982; Zbl 0509.13026)] domains R by (1) describing all rank 3 posets H on which such an algebra can exist: it turns out there are 15 isolated cases plus a countable infinite family; (2) exhibiting- when k is infinite - a suitable R for each case. Preliminary steps to the main result include the determination of the posets on which 2-dimensional homogeneous Gorenstein ASL can exist, and the proof of the fact that every 3-dimensional homogeneous ASL domain is Cohen-Macaulay. Another interesting observation is that the realizations of the infinite family from (1) above include nonnormal ASL domains on wonderful posets: this disproves a conjecture 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)] to the effect that ASL domains defined by wonderful posets over fields of characteristic zero are normal with rational singularities. [See also the following review.]