The representation type of a category of complexes of fixed size and the strong global dimension

CHAIO, CLAUDIA; GONZALEZ CHAIO ALFREDO; PRATTI ISABEL

Marsella, Luminy

Conferencia; Advances in Representation theory of Algebras ARTA VI; 2017

Universidad Paris 7

Let A be a finite dimensional algebra over an algebraically closed field. We denote by mod A the finitely generated module category and by proj A the full subcategory of mod A of the finitely generated projective A-modules.In [BSZ], the authors defined and studied the categories C_n(proj A) of complexes of fixed size. We say that a category C_n(proj A) is representation-finite if there are a finite number of classes of isomorphic indecomposable complexes in C_n(proj A).The concept of strong global dimension has been introduced by Skowro\'nsky in [S]. Given a complex X \in {K}^{b}(proj A),$$ \cdots \rightarrow 0 \rightarrow 0\rightarrow X^r \rightarrow X^{r+1} \rightarrow \cdots \rightarrow X^{s-1} \rightarrow X^s \rightarrow 0 \rightarrow 0 \cdots$$with $X^r\neq 0$ and $X^s \neq 0$, 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} \}.In 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}(\rm{proj}\,A) is representation-finite if and only if $\mathbf{C_{\eta + 1}}(\rm{proj}\,A)$ is representation-finite.We also study implications which allow us to decide if for some positive integer $m$ the category $\mathbf{C_m}(\rm{proj} \,A)$ is representation-infinite taking into account different notions.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.