Super-decomposable pure-injective modules over some Jacobian algebras

Abstract

Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations over an algebra. Such an existence occurs when an algebra is non-domestic; a conjecture due to M. Prest. G. Puniski confirms the conjecture for non-domestic string algebras. Gei{\ss}, Labardini-Fragoso and Schr\"oer show that every Jacobian algebra associated with a triangulation of a closed surface with marked points is finite-dimensional and tame. We show that, excluding only the case of a sphere with four (or fewer) punctures, there exists a special family of pointed modules, called an independent pair of dense chains of pointed modules. In the process, we show the existence of such an independent pair in a non-domestic skew-gentle algebra and (skew) Brauer graph algebras by showing that the Galois semi-covering functor and trivial extension preserve such pairs. Then it follows from a result of M. Ziegler that there exists a superdecomposable pure-injective module if the algebraically closed field is countable.