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

Abstract

Let $\mathbb{Z}$ be the integer numbers, $\mathbb{k}$ an algebraically closed field, $\Lambda$ a finite dimensional $\mathbb{k}$-algebra, mod$\Lambda$ the category of finitely generated right modules, proj$\Lambda$ the full subcategory of mod$\Lambda$ consisting of all projective $\Lambda$-modules, and $C_n(proj\Lambda)$ the bounded complexes of projective $\Lambda$-modules of fixed size for any integer $n\geq2$. We found an algorithm to calculate the strong global dimension of $\Lambda$, when $\Lambda$ has finite strong global dimension and derived-discrete, using the Auslander-Reiten quivers of the categories $C_n(proj\Lambda)$. Also, we show the relation between the Auslander-Reiten quiver of the bounded derived category $D^b(\Lambda)$ and the Auslander-Reiten quiver of $C_{\eta+1}(proj\Lambda)$, where $\eta=s.gl.dim(\Lambda)$ (strong global dimension of $\Lambda$).