On Krull-Gabriel dimension of weighted surface algebras
Karin Erdmann, Alicja Jaworska-Pastuszak, Grzegorz Pastuszak
TL;DR
The paper determines the Krull-Gabriel dimension for weighted surface algebras and related classes by employing Galois coverings and orbit-category techniques, showing that weighted surface algebras have undefined (infinite) Krull-Gabriel dimension. It also analyzes hybrid algebras and algebras of generalized quaternion type, establishing infinite KG-dimension in many cases and providing finite KG-dimension examples in others. These results support Prest's conjecture for the studied families and illustrate how orbit-category methods transfer dimensional properties across coverings. Overall, the work clarifies the KG-dimension landscape for tame symmetric periodic and idempotent-structure algebras arising from triangulated surfaces and their generalizations, highlighting the boundaries between finite and infinite KG-dimensions.
Abstract
We determine the Krull-Gabriel dimension of weighted surface algebras, a class of algebras which recently appeared in the context of classification of tame symmetric periodic algebras of non-polynomial growth. Moreover, we consider Krull-Gabriel dimension of idempotent algebras of weighted surface algebras and generalize the result in some cases.