  We prove the tame-wild dichotomy conjecture, due to D. Simson, for infinite
dimensional algebras and coalgebras. The key part of the approach is proving
new representation theoretic characterizations local finiteness. Among other,
we show that the Ext quiver of the category ${\rm f.d.-}A$ of finite
dimensional representations of an arbitrary algebra $A$ is locally finite (i.e.
$\dim(\Ext^1(S,T))<\infty$ for all simple finite dimensional $A$-modules $S,T$)
if and only if for every dimension vector $\underline{d}$, the representations
of $A$ of dimension vector $\underline{d}$ are all contained in a finite
subcategory (a category of modules over a finite dimensional quotient algebra).
This allows one reduce the tame/wild problem to the finite dimensional case and
Drozd's classical result. Using this, we also prove a local-global principle
for tame/wild (in the sense of non-commutative localization): a category of
comodules is tame/not wild if and only if every ``finite" localization is so.
We give the relations to Simson's f.c.tame/f.c.wild dichotomy, and use the
methods and various embeddings we obtain to give connections to other problems
in the literature. We list several questions that naturally arise.
