An upper bound for the dimension of bounded derived categories

The derived dimensions and syzygy finite type

“…Note that ℓℓ tV (F n tV (X i )) = 0 for each i, we know that F n tV (X i ) ∈ F(V) by [17,Proposition 3.1]. Now by Lemma 3.7, we have Ω m+1…”

“…Remark 3.1. (see [17,Remark 3.16(2)]) If V is the set of all simple moduels, then the torsion pair (T V , F(V)) = (0, mod Λ), and ℓℓ tV (Λ) = 0 and pd V = gl.dim Λ. In this case, pd V + ℓℓ tV (Λ) = gl.dim Λ.…”

“…Proof. Two special cases are V = ∅ and ℓℓ tV (Λ Λ ) = 0, see the proof of [17,Theorem 3.8]. Now we will consider the case ℓℓ tV (Λ Λ ) = n 1 and pd V = m < ∞.…”

The derived dimensions of $(m,n)$-Igusa-Todorov algebras

“…Roughly speaking, it is an invariant that measures how quickly the category can be built from one object. Many authors have studied the upper bound of tri.dim T , see [3,5,7,9,12,14,17,18,22,25,23] and so on. There are a lot of triangulated categories having infinite dimension, for instance, Oppermann and Št'ovíček proved in [14] that all proper thick subcategories of the bounded derived category of finitely generated modules over a Noetherian algebra containing perfect complexes have infinite dimension.…”

“…The dimension of triangulated category plays an important role in representation theory( [3,5,7,9,14,17,18,23]). For example, it can be used to study the representation dimension of artin algebras ( [17,13]).…”