Even Expansion And Tilt The Theory Of Induced Anti-change Limited Sub-areas

The contravariant finity of subcategories are closely associated with the represen-tation theory, Torsion theory and the tilting theory. In this article our main purpose is to study the contravariant finity of p∞(A) and Gpd∞(A). By discussing the properties of dual extension algebras and the tilting theory and derived equivalence, we study the relationship of p∞(A) and Gpd∞(A). The related results and their proofs are given.this paper is divided into four parts. Some background and basic properties are introduced in the first two parts, and the main results and their proofs are given in the third and forth part.