The strong global dimension of a peicewise hereditary algebra

CHAIO, CLAUDIA , ALFREDO GONZALEZ CHAIO, ISABEL PRATTI.

Quito

Congreso; Congreso Latinoamericano de algebras; 2017

Let $A$ be a finite dimensional algebra over an algebraically closed field. We denote by $m {mod},A$ the finitely generated module category and by $m{proj},A$ the full subcategory of $m{ mod},A$ of the finitely generated projective $A$-modules.In [BSZ], the authors defined and studied the categories $mathbf{C_n}(m{proj},A)$ of complexes of fixed size. We say that a category $mathbf{C_n}(m{proj},A)$ is representation-finite if there are a finite number of classes of isomorphic indecomposable complexes in $mathbf{C_n}(m{proj},A)$.The concept of strong global dimension has been introduced by Skowro´nsky in [S]. Given a complex $X in mathbf{K}^{b}(m{proy} ,A)$,$$ cdots ightarrow 0 ightarrow 0ightarrow X^r ightarrow X^{r+1} ightarrow cdots ightarrow X^{s-1} ightarrow X^s ightarrow 0 ightarrow 0 cdots$$oindent with $X^req 0$ and $X^s eq 0$, we define the length of $X$ as follows: $ell(X)= s-r$.The strong global dimension of $A$ is defined as: $$s.gl.mbox{dim}; A = ;Sup; { ell(X) mid X in K^{b}(proj ; A) ; mbox { is indecomposable} }.$$skip.2inIn this work, we prove that if $A$ is an algebra with $s.gl. emph{dim}, A=eta < infty$ then, for each $n geq 2$, $mathbf{C_n}(m{proj},A)$ is representation-finite if and only if $mathbf{C_{eta + 1}}(m{proj},A)$ is representation-finite.On the other hand, we consider some piecewise hereditary algebras $A$ and we prove how to read their strong global dimension taking into account their ordinary quiver with relations. Moreover, we also show the entries of the complexes of maximal length in such cases.