Cotorsion modules over tame finite-dimensional hereditary algebras

Abstract

All modules are over tame finite-dimensional algebras. A module M is said to be cotorsion if and only if Ext (F,M)=0 for all torsion-free modules F. It is shown that every cotorsion module can be put in the form M = M₁ ∔ M₂ ∔ M₂ ∔ M₃ where M₁ is divisible, M₂ has no torsion-free direct summand and M₃ is reduced and torsion-free. The submodules M₁ and M₂ are uniquely determined by M. A module M is said to be pure injective if Pext (N,M)=0 for all modules N. The two classes of modules are connected by the following result: A torsion-free module with no preprojective direct summand is pure injective if and only if it is cotorsion. A pure injective module M can also be put in the form M = M₁ ∔ M₂ ∔ M₂ ∔ M₃ where M₁ is divisible, M₂ is reduced and nonsingular and M₃ is preprojective. As in abelian group theory M₁ and M₂ are completely described by sets of cardinal numbers. Under a restriction satisfied by Kronecker modules M₃ is also described by cardinal numbers.