Top-stable and layer-stable degenerations and hom-order

S. O. Smalø, and A. Valenta. "Top-stable and layer-stable degenerations and hom-order." Colloquium Mathematicae 108.1 (2007): 63-71. <http://eudml.org/doc/283943>.

@article{S2007,
abstract = {Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M <_\{deg\} N$ such that $^\{t\}M/^\{t+1\}M ≃ ^\{t\}N/^\{t+1\}N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $dim_\{k\}Hom(M,M) = dim_\{k\}Hom(P,M)$ if and only if M has no proper degeneration $M <_\{deg\} N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from our results. In particular, we find that her second result holds not just for modules with square-free top, but also for indecomposable modules in general.},
author = {S. O. Smalø, A. Valenta},
journal = {Colloquium Mathematicae},
keywords = {degeneration orders; classes of modules; hom orders; minimal degenerations; indecomposable modules; Auslander-Reiten quivers},
language = {eng},
number = {1},
pages = {63-71},
title = {Top-stable and layer-stable degenerations and hom-order},
url = {http://eudml.org/doc/283943},
volume = {108},
year = {2007},
}

TY - JOUR
AU - S. O. Smalø
AU - A. Valenta
TI - Top-stable and layer-stable degenerations and hom-order
JO - Colloquium Mathematicae
PY - 2007
VL - 108
IS - 1
SP - 63
EP - 71
AB - Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M <_{deg} N$ such that $^{t}M/^{t+1}M ≃ ^{t}N/^{t+1}N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $dim_{k}Hom(M,M) = dim_{k}Hom(P,M)$ if and only if M has no proper degeneration $M <_{deg} N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from our results. In particular, we find that her second result holds not just for modules with square-free top, but also for indecomposable modules in general.
LA - eng
KW - degeneration orders; classes of modules; hom orders; minimal degenerations; indecomposable modules; Auslander-Reiten quivers
UR - http://eudml.org/doc/283943
ER -