Relations Between the Strong Global dimension, Complexes of Fixed Size and Derived Category

Let $\mathbb{Z}$ be the integer numbers, $\mathbb{k}$ an algebraically closed field, $\mathrm{\Lambda }$ a finite dimensional $\mathbb{k}$-algebra, mod$\mathrm{\Lambda }$ the category of finitely generated right modules, proj$\mathrm{\Lambda }$ the full subcategory of mod$\mathrm{\Lambda }$ consisting of all projective $\mathrm{\Lambda }$-modules, and $C_n(proj\mathrm{\Lambda })$ the bounded complexes of projective $\mathrm{\Lambda }$-modules of fixed size for any integer $n\ge 2$. We found an algorithm to calculate the strong global dimension of $\mathrm{\Lambda }$, when $\mathrm{\Lambda }$ has finite strong global dimension and derived-discrete, using the Auslander-Reiten quivers of the categories $C_n(proj\mathrm{\Lambda })$. Also, we show the relation between the Auslander-Reiten quiver of the bounded derived category $D^b(\mathrm{\Lambda })$ and the Auslander-Reiten quiver of $C_{\eta +1}(proj\mathrm{\Lambda })$, where $\eta =s\mathrm{.}gl\mathrm{.}dim(\mathrm{\Lambda })$ (strong global dimension of $\mathrm{\Lambda }$).