Let $A$ be a finite-dimensional algebra over an algebraically closed field $K$. Then the classical Drozd's trichotomy asserts that the representation type of $A$ is either finite, tame or wild type.
I was convinced that this result still holds for any characteristic but I could not find any good references for it, so I have started to wonder if that is true or not. Is it true?
Anyway, what concerns more to me is if even if the trichotomy may not hold for arbitrary finite-dimensional algebras over an arbitrary field, is it true for hereditary algebras at least?
For an indecomposable finite dimensional hereditary algebra $H$ over an arbitrary field $K$, there is such a trichotomy.
The algebra $H$ has finite representation type if and only if the symmetric bilinear form on the Grothendieck group is positive definite, if and only if it is of Dynkin type, if and only if there are no regular modules. In this case all modules are both preprojective and preinjective, and taking the class of a module yields a bijection between the isoclasses of indecomposable modules and the set of positive roots of the Dynkin diagram.
The algebra $H$ has tame representation type if and only if the symmetric bilinear form is positive semi-definite (but not positive definite), if and only if it is of extended Dynkin type, if and only if the regular modules form a thick abelian subcategory. In this case taking the class of a module yields a surjection from isoclasses of indecomposable modules and the set of positive roots. For each positive real root, there is a unique such isoclass of indecomposables. For each multiple of the minimal positive imaginary root the set of such isoclasses has cardinality at least that of the field $K$. Moreover, the regular modules come in tubular families, and the tubes are indexed by the closed points of a regular normal curve over $K$.
The algebra $H$ has wild representation type if and only if the symmetric bilinear form is indefinite. In this general setting the Kac-type result on the classes of indecomposables is not known (it does hold over finite fields or algebraically closed fields though).