Controlled Wild Algebras

By Drozd's Tame and Wild Theorem, ®nite-dimensional algebras over an algebraically closed ®eld k may be divided into two disjoint classes: one class consists of tame algebras whose indecomposable modules can be parametrized using only one continuous parameter, the other class consists of wild algebras which have families of indecomposable modules depending on arbitrarily many continuous parameters (see [12, 13, 8]). However, a feature of `wildness' which is completely different from that described by Drozd was ®rst exhibited by Corner. He found that, for certain algebras A, all kinds of algebras, even those considered as very pathological, can be realized as endomorphism algebras of A-modules or factor algebras of endomorphism algebras of A-modules modulo nice ideals. This means that the given algebra has modules with nearly prescribed endomorphism algebras, in particular, modules with pathological decomposition behavior. In 1969, Corner showed that the 5-subspace problem is `wild' (see [7]). After the work of Corner, many other algebras were shown to be `wild' by Brenner (see [4, 5]). There is a common feeling that wild algebras (in the sense of Drozd) may be characterized by the property that any ®nite-dimensional algebra can be realized as a factor algebra of the endomorphism algebra of some module modulo some ideal (see [32]), that is, they are just the algebras of Corner's type (see [5]). Naturally this problem can be divided into two parts. One is whether this property implies wildness. Indeed, the motivation of [23] is to solve this part though it is not solved completely there. The other part of the problem is whether wildness implies this property or even another stronger property. It would be very good if such a stronger property could imply wildness. One of our motivations is to solve the second part of this problem. With this aim, we introduce the notion of socalled controlled wild algebras (De®nition (2.1)) which was ®rst given by Ringel. If A is a controlled wild algebra, then for any algebra B there exists an A-module M such that EndA…M† is a split extension of a nilpotent ideal I of EndA…M† by B (Proposition (2.3)), that is, the controlled wildness which implies wildness (Proposition (2.2)) is stronger than the above property. Thus the above problem is transformed into that given by Ringel in [33], that is, whether wildness coincides with (or implies) controlled wildness (in our terminology). It is very nice for an algebra A to be strictly wild, since the functor in the de®nition of strict wildness is a full representation embedding, that is, any morphism between images must be the image of some morphism. However, in general, the representation embedding functor in the de®nition of wildness need