Relations Between the Strong Global Dimension, Complexes of Fized Size and Derived Category
Let $\mathbb{Z}$ be the integer numbers, $\mathbb{K}$ an algebraically closed field, $Λ$ a finite dimensional $\mathbb{K}$-algebra, mod$Λ$ the category of finitely generated right modules, proj$Λ$ the full subcategory of mod$Λ$ consisting of all projective $Λ$-modules, and $C_n(projΛ)$ the bounded complexes of projective $Λ$-modules of fixed size for any integer $n\geq2$. We find an algorithm to calculate the strong global dimension of $Λ$, when $Λ$ is a finite strong global dimension and derived discrete, using the Auslander-Reiten quivers of the categories $C_n(projΛ)$. Also, we show the relation between the Auslander-Reiten quiver of the bounded derived category $D^b(Λ)$ and the Auslander-Reiten quiver of $C_{η+1}(projΛ)$, where $η=s.gl.dim(Λ)$ (strong global dimension of $Λ$).