Remark on the review of: Brenner-Ringel.
Pathological modules over tame rings.
J. London Math. Soc. 14 (1976), 207-215.
The paper asserts: For all known fin.dim. tame algebras [A over a field k] there exists
a full subcategory of A-modules representation equivalent to k[t]-Mod.
In his review, Warfield says:
The author's results apply to many more rings than just $k[t]$. In
fact, if $A$ is a finitely generated algebra over a field $k$ and $A$
is not finite dimensional of finite module type (in which case the
module theory is very "non-pathological"), then it is believed (but
has not been proved) that the category of $k[t]$ modules has a full
exact imbedding into the category of $A$ modules. This imbedding is
possible (by folklore) in all known cases. (For commutative Artinian
rings, such a result was proved by the reviewer [Pacific J. Math. 60
(1975), no. 2, 289--302; MR0414534 (54 \#2635)].)
Note that for NO finite-dimensional local $k$-algebra $A$,
there is a full embedding of the category of $k[t]$ modules into the
category of $A$-modules.
We are endebted to Peter Vamos for drawing our attention to the review.
Claus Michael Ringel
Last modified: Sun Oct 31 18:30:58 CET 2004