A family of noetherian rings with their finite length modules under control

Schmidmeier, Markus

Czechoslovak Mathematical Journal, Tome 52 (2002) no. 3, pp. 545-552.

Résumé

We investigate the category $\text{mod}\Lambda $ of finite length modules over the ring $\Lambda =A\otimes _k\Sigma $, where $\Sigma $ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$-algebra. Each simple module $E_j$ gives rise to a quasiprogenerator $P_j=A\otimes E_j$. By a result of K. Fuller, $P_j$ induces a category equivalence from which we deduce that $\text{mod}\Lambda \simeq \coprod _jbad hbox P_j$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda $ such that $\text{mod}A\simeq \text{mod}\Lambda $, (2) find twisted versions $\Lambda $ of algebras of wild representation type such that $\Lambda $ itself is of finite or tame representation type (in mod), (3) describe for certain rings $\Lambda $ the minimal almost split morphisms in $\text{mod} \Lambda $ and observe that almost all of these maps are not almost split in $\text{Mod}\Lambda $.

Détail

Classification : 16D50, 16D60, 16D90, 16G10, 16G20, 16G60, 16G70, 16P40
Keywords: V-ring; progenerator; almost split morphisms