Superdecomposable pure-injective modules

Abstract

Abstract. Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations. We describe the relation in terms of pointed modules. We present methods for producing superdecomposable pure-injectives and give some details of recent work of Harland doing this in the context of tubular algebras.

On Wild Algebras and Super-Decomposable Pure-Injective Modules

“…On the level of infinite-dimensional modules, a fundamental characterization of the representation type is given in [1] in terms of generic modules. It is conjectured by M. Prest (see [18,Introduction]) that a finite-dimensional algebra A over an algebraically closed field is of domestic representation type if and only if there is no super-decomposable pureinjective A-module (these modules do not have indecomposable direct summands, so they are of infinite dimension). This paper is related to the conjecture of Prest.…”

Strongly simply connected algebras with super-decomposable pure-injective modules

“…It is conjectured by M. Prest (see [24], [25]) that a finite-dimensional algebra A over an algebraically closed field is of domestic representation type if and only if the Krull-Gabriel dimension KG(A) of A is finite (see [9,10] for definitions and [22] for a list of results supporting this conjecture). The second conjecture due to Prest states that an algebra A is of domestic representation type if and only if there is no super-decomposable pure-injective A-module (see for example [26]). Such a module is some special infinite-dimensional module.…”