Let A = RG be a group ring which is given by a complete discrete
valuation ring R and a finite p-group G , such that A is an
isolated singularity in the sense of Auslander [Au 84]. Here, this is
equivalent to the condition that the prime number p is neither a unit
nor zero in R . Consider the problem of classifying all indecomposable
left A-lattices and, what is more, the problem of determining the
Auslander-Reiten quiver of A .
The present state of knowledge in this respect can be summarized
as follows. Denote by C n the cyclic group of order p and by v
the exponential valuation of R . Then A is of finite representation
type if and only if one of the following cases (i)-(iv)occurs:
(i) G = C2 ,
(ii) G = C and v(3) * 3 ,
(iii) G = C and v(p) 2 ,
(iv) G = C 2 and v(p) = 1 [Dr/Ro 67], [Ja 67].
Moreover, A is of tame representation type in each of the
following cases (v)-(vii):
(v) G = C x C and v(2) = 1 [Na 67],
(vi) G = Cg and v(2) = 1 [Ja 72],
(vii) G = C4 and v(2) = 2 [Ko 75].
In each of the cases (i)-(v) the Auslander-Reiten quiver of A is
known [Di 83a]. Furthermore, A is domestic tame in case (v), whereas
it is non-domestic tame in cases (vi) and (vii).
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