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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).

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    $\begingroup$ Since your question is about some technical notions and details, it could be helpful to precisely define the relevant terms (I know what tame and wild mean roughly speaking, but hadn't heard of "semi-wild"). $\endgroup$
    – Sam Hopkins
    Commented Feb 29 at 20:17
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    $\begingroup$ If by any chance you will want to see the paper (even though you do not speak Russian), this is the entire collection of papers "Matrix Problems": dropbox.com/scl/fi/drk0uvgn2jykmx4cpbni5/… $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 1 at 8:48
  • $\begingroup$ Thank you for the paper. It will be interesting to look at if I am able to translate it. $\endgroup$
    – Jacob FG
    Commented Mar 1 at 10:00

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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.

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    $\begingroup$ Thank you. The question of why an algebra cannot be both tame and wild, seems to be covered by Lemma 27.9 and section 33. $\endgroup$
    – Jacob FG
    Commented Mar 1 at 9:59

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