The Equivalence of Two Notions of Discreteness of Triangulated Categories

Given an ST-triple (C, D, M) one can associate a co-t-structure on C and a t-structure on D. It is shown that the discreteness of C with respect to the co-t-structure is equivalent to the discreteness of D with respect to the t-structure. As a special case, the discreteness of D-b(mod A) in the sense of Vossieck is equivalent to the discreteness of K-b(proj A) in a dual sense, where A is a finite-dimensional algebra.