I would like to understand the Tame-Wild dichotomy, and in particular why an algebra cannot be tame and (semi-)wild at the same time. I've looked in the papers by Drozd and Crawley-Boevey [D80, CB88].
In both the question of why an algebra cannot be both tame and wild is vaguely explained by the number of 'parameters' in the variety of indecomposable modules, and one is referred to [D77] for a more detailed proof.
I cannot locate this article, and even if I could, I don't speak Russian.
Question: Is there a modern reference that goes through the details of why an algebra cannot be both tame and wild?
In fact, I am mostly interested in why an algebra cannot be both tame and semi-wild. In [BD82] Bondarenko and Drozd state that this is proven in [D77], so presumably this is the same proof. But in case it is not, that is my main interest.
Edit: As requested here is a definition of semi-wild (over an algebraically closed field):
An algebra $A$ over $k$ is called semi-wild if there exist an exact functor $F\colon \operatorname{mod}k\langle x, y\rangle \to \operatorname{mod} A$, where $\operatorname{mod}$ means finite dimensional modules and $k\langle x, y\rangle$ is the free associative algebra. With the condition that for any $M \in \operatorname{mod}k\langle x, y\rangle$, there are up to isomorphism only finitely many $N \in \operatorname{mod}k\langle x, y\rangle$ such that $FM \cong FN$.
Note this is a generalisation of wild, because we don't require $F$ to preserve indecomposability or be essentially injective.
[D80] Drozd, J.A. (1980). Tame and wild matrix problems. In: Dlab, V., Gabriel, P. (eds) Representation Theory II. Lecture Notes in Mathematics, vol 832. Springer, Berlin, Heidelberg.
[CB88] Crawley-Boevey, W. W. (1988). On tame algebras and bocses. Proc. London Math. Soc. (3) 56 (1988), no. 3, 451–483. MR0931510
[D77] Drozd, Y.A. (1977). On tame and wild matrix problems. In: Matrix Problems [in Russian], Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, Kiev (1977).
[BD82] Bondarenko, V.M., Drozd, Y.A. (1982). Representation type of finite groups. Math Sci 20, 2515–2528 (1982).
Best Answer
A modern textbook which deals with the tame-wild theorem is "Differential Tensor Algebras and their Module Categories" by Bautista, Salmeron and Zuazua. It tries to be very detailed and even offers solutions to exercises but you sitll have to read a lot of techinical stuff to understand the full proof in chapter 27.