Remarks on dimensions of triangulated categories

Abstract

In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.Applying the first assertion of this theorem, we deduce that for an isolated hypersurface singularity R, the stable category CM(R) is locally finite if and only if it has dimension zero, if and only if R has finite CM representation type (Corollary 2.14).The second subject of this paper, which is discussed in Section 3, is to find a generator G of a derived category D such that the dimension of D with respect to G in the sense of [AAITY] is as small as possible. This brings us a lower bound of the dimension of the derived category. The main result on this subject is the following.Theorem 1.2 (Theorem 3.2). Let A be an abelian category and X a full subcategory. Let M be an object of A admitting an exact sequence with X 0 , . . . , X n ∈ Xsuch that the corresponding element in Ext n+1 A (M, X n ) (resp. Ext n+1 A (X n , M)) is nonzero. Then M is outside ⊥ X n+1 (resp. X ⊥ n+1 ) in the derived category D(A) of A.