Is Tame Open? *

Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the BasicWild-Rank Conjecture. A covering criterion on the rank of a basic wild algebra is given, which can be effectively applied to verify the BasicWild-Rank Conjecture for concrete algebras. It makes all conjectures much reliable. 2000 Mathematics Subject Classification: 16G60, 16G10, 16G20 Throughout k denotes a fixed algebraically closed field. By an algebra we mean a finite-dimensional associative k-algebra with identity. By a module we mean a left module of finite k-dimension except in the context of covering theory. We denote by modA the category of finite-dimensional left A-modules. For terminology in the representation theory of algebras we refer to [2] and [31]. 1. Tame-Open Conjecture ∗Project 10201004 supported by NSFC.