Systems of submodules and a remark by M.C.R. Butler
Abstract
Fix a poset
P
and a natural number
n
. For various commutative local rings
Λ
, each of Loewy length
n
, consider the category
sub
Λ
P
of
Λ
-linear submodule representations of
P
. We give a criterion for when the underlying translation quiver of a connected component of the Auslander-Reiten quiver of
sub
Λ
P
is independent of the choice of the base ring
Λ
. If
P
is the one-point poset and
Λ=Z/
p
n
for
p
a prime number, then
sub
Λ
P
consists of all pairs
(B;A)
where
B
is a finite abelian
p
n
-bounded group and
A⊂B
a subgroup. We can respond to a remark by M. C. R. Butler concerning the first occurence of parametrized families of such subgroup embeddings.