Derived endo-discrete artin algebras

Raymundo Bautista¹

¹ Instituto de Matemáticas UNAM, Unidad Morelia A.P. 61-3, Xangari, C.P. 58089 Morelia, Michoacán, México

Colloquium Mathematicum, Tome 105 (2006) no. 2, pp. 297-310

Résumé

Let ${\mit\Lambda} $ be an artin algebra. We prove that for each sequence $(h_{i})_{i\in \mathbb{Z}}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X\in \mathcal{D}^{\rm b}({\mit\Lambda} ),$ the bounded derived category of ${\mit\Lambda} $, with $\mathop{\rm length}\nolimits _{E(X)}H^{i}(X)=h_{i}$ for all $i\in \mathbb{Z}$ and $E(X)$ the endomorphism ring of $X$ in $\mathcal{D}^{\rm b}({\mit\Lambda} )$ if and only if $\mathcal{D}^{\rm b}(\mathop{\rm Mod}\nolimits {\mit\Lambda} )$, the bounded derived category of the category $\mathop{\rm Mod}\nolimits {\mit\Lambda} $ of all left ${\mit\Lambda} $-modules, has no generic objects in the sense of [4].

Détail

DOI : 10.4064/cm105-2-10

Keywords: mit lambda artin algebra prove each sequence mathbb non negative integers there only finite number isomorphism classes indecomposables mathcal mit lambda bounded derived category mit lambda mathop length nolimits mathbb endomorphism ring mathcal mit lambda only mathcal mathop mod nolimits mit lambda bounded derived category category mathop mod nolimits mit lambda mit lambda modules has generic objects sense