Relations Between the Strong Global Dimension, Complexes of Fized 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 $n\ge 2$. We find an algorithm to calculate the strong global dimension of $\Lambda$, when $\Lambda$ is a 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 $\Lambda$).

16 pages and 4 figures