On a class of nontriangular representation-finite algebras forming an open Z-scheme

The article is related with the question if representation-finite algebras form an open Z-scheme in the sense of Jensen and Lenzing (1989) [12, Chapter 12]. We define a class TMCL of algebras and we give the positive answer to the question restricted to that class. This is carried out by applying van den Dries’ test. Let V be a valuation ring in an algebraically closed field K with the residue field R. Given a V-order A, we denote by the R-algebra obtained from A by reduction modulo the radical of V and A(K)=A⊗VK. One of the main results asserts that if the R-algebra is representation-finite and belongs to the class TMCL then the K-algebra A(K) is representation-finite and belongs to TMCL. It follows that the representation-finite algebras in TMCL form an open Z-scheme.