Prüfer modules over wild hereditary algebras

Frank Lukas

Published 2024-02-20, updated 2024-03-18Version 2

Let $A$ be a hereditary finite dimensional algebra over an algebraically closed field $k$. A brick is defined as a finitely generated module with a division ring as an endomorphism ring. Two non-isomorphic bricks $X,Y$ are said to be orthogonal if $Hom(X,Y)=Hom(Y,X)=0$ and a class of pairwise orthogonal bricks is called a semi-brick. We show that a semi-brick $\mathcal{X}$ allows the construction of Pr\"ufer modules. We consider the category $Filt(\mathcal{X})$ of modules with a filtration in $\mathcal{X}$ and show how to construct injective objects in this category. We call the infinite dimensional, indecomposable injective objects in $Filt(\mathcal{X})$, Pr\"ufer modules and show that they share many properties with the Pr\"ufer modules over tame hereditary algebras as defined by C. M. Ringel in \cite{ringel1979infinite}. The results of this paper can also be applied to tame hereditary algebras. The construction gives a strict filtration of the generic module $Q$. We give an alternative proof for the classification of torsion-free divisible modules in order to show how useful this filtration is. In two appendices we show that the module category of a wild hereditary algebra has large classes of semi-bricks. The elements $X$ of these semi-bricks have the property $dim_k Ext(X,X)\geq 2$. Bricks $X$ with $dim_k Ext(X,X)=1$ arise from full and exact embeddings of module categories of tame hereditary algebras.