We establish when the partial orders ${\le }_{ext}$ and ${\le }_{deg}$ coincide for all modules of the same dimension from the additive category of a generalized standard almost cyclic coherent component of the Auslander-Reiten quiver of a finite-dimensional algebra.

Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given $\tau$-tilting module.

Let Λ be an artin algebra. We prove that for each sequence ${\left(h_i\right)}_{i\in \mathbb{Z}}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X{\in }^b\left(\Lambda \right)$, the bounded derived category of Λ, with $lengt{h}_{E\left(X\right)}{H}^i\left(X\right)=h_i$ for all i ∈ ℤ and E(X) the endomorphism ring of X in ${}^b\left(\Lambda \right)$ if and only if ${}^b\left(Mod\Lambda \right)$, the bounded derived category of the category $Mod\Lambda$ of all left Λ-modules, has no generic objects in the sense of [4].

We construct derived equivalences between generalized matrix algebras. We record several corollaries. In particular, we show that the $n$-replicated algebras of two derived equivalent, finite-dimensional algebras are also derived equivalent.