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. Geiß, Labardini-Fragoso and Schröer 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.

Figures (21)

• Figure 1: Triangulation T of a sphere with 5 punctures.
• Figure 2: $Q(\mathbb{T})$
• Figure 3: $Q'$
• Figure 4: $Q$
• Figure 5: $\Gamma$ with $\rho= \{\alpha\gamma+\beta\gamma', \beta'\gamma+\alpha'\gamma'\}$

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5