Relations Between the Strong Global Dimension, Complexes of Fized Size and Derived Category

TL;DR

This paper develops an algorithm to compute the strong global dimension of finite-dimensional algebras using Auslander-Reiten quivers and explores the relationship between derived categories and complexes of fixed size.

Contribution

It introduces a novel algorithm for calculating the strong global dimension of certain algebras and links the Auslander-Reiten quivers of derived categories with those of complexes of fixed size.

Findings

Algorithm for strong global dimension calculation

Relation between Auslander-Reiten quivers of derived categories and complexes

Application to finite strong global dimension and derived discrete algebras

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 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 $\Lambda$).