Abstract
AbstractAssume thatkis an algebraically closed field andAis a finite-dimensional wildk-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointedA-modules is undefined. Hence the result of M. Ziegler implies that there exists a super-decomposable pure-injectiveA-module, if the base fieldkis countable. Here we give a different and straightforward proof of this fact. Namely, we show that there exists a special family of pointedA-modules, called an independent pair of dense chains of pointedA-modules. This also yields the existence of a super-decomposable pure-injectiveA-module.
Publisher
Springer Science and Business Media LLC
Reference31 articles.
1. Crawley-Boevey, W.W.: Tame algebras and modules generic. Proc. London Math. Soc. 63, 241–264 (1991)
2. Drozd, Yu. A.: Tame and wild matrix problems. In: Representations and Quadratic Forms, Kiev, 39–74 (in Russian) (1979)
3. Erdmann, K., Skowroński, A.: Weighted surface algebras. J. Algebra 505, 490–558 (2018)
4. Gregory, L., Prest, M.: Representation embeddings, interpretation functors and controlled wild algebras. J. Lond. Math. Soc. (2) 94(3), 747–766 (2016)
5. Huisgen-Zimmermann, B.: Purity, Algebraic Compactness, Direct Sum Decompositions, and Representation Type. In: Infinite Length Modules (Bielefeld, 1998), pp 331–367. Birkhäuser, Basel, Trends Math. (2000)