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

Schmidmeier, Markus. "A family of noetherian rings with their finite length modules under control." Czechoslovak Mathematical Journal 52.3 (2002): 545-552. <http://eudml.org/doc/30723>.

@article{Schmidmeier2002,
abstract = {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 $.},
author = {Schmidmeier, Markus},
journal = {Czechoslovak Mathematical Journal},
keywords = {V-ring; progenerator; almost split morphisms; V-rings; progenerators; almost split morphisms; categories of finite length modules; simple modules; category equivalences; Noetherian rings; wild representation type; finite representation type; tame representation type},
language = {eng},
number = {3},
pages = {545-552},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A family of noetherian rings with their finite length modules under control},
url = {http://eudml.org/doc/30723},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Schmidmeier, Markus
TI - A family of noetherian rings with their finite length modules under control
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 545
EP - 552
AB - 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 $.
LA - eng
KW - V-ring; progenerator; almost split morphisms; V-rings; progenerators; almost split morphisms; categories of finite length modules; simple modules; category equivalences; Noetherian rings; wild representation type; finite representation type; tame representation type
UR - http://eudml.org/doc/30723
ER -