On the infinite tame-wild dichotomy, the Brauer-Thrall 3 conjectures, and related problemns
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 for 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. $\dim({\rm Ext}^1(S,T))<\infty$ 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. We also show that these two properties are local in the sense of localization: a category of comodules is tame/not wild if and only if every "finite" localization is so, and give the relations to Simson's f.c.tame/f.c.wild dichotomy. We use the methods and various embeddings we obtain, to give a proof for the Brauer-Thrall 3 conjecture, also raised by Simson, for the class of all wild algebras, thus covering "almost" all algebras. We list several questions that seem to arise naturally from this study.