Tameness of Λ/G under free action on indecomposables

Establish that for a locally bounded k-category Λ and a group G of k-linear automorphisms acting freely on the set [ind Λ] of isomorphism classes of indecomposable Λ-modules, if Λ is tame, then the orbit category Λ/G is tame.

Background

Section 3 investigates Galois coverings of locally bounded categories and their impact on representation type. A known result (Dowbor–Skowroński, 1985) asserts that if the quotient Λ/G is tame, then Λ is tame. This raises the question of whether the converse holds.

The note develops tools such as modules of the second kind and, under specific structural assumptions (e.g., weakly-G-periodic modules being linear or certain separating families), shows equivalences between the tameness of Λ and Λ/G. To address the general case, the authors record a conjecture positing that tameness descends from Λ to its orbit category under a free action on indecomposables.

The necessity of the free action assumption is underscored by examples where, if G does not act freely on [ind Λ], the quotient Λ/G can be wild (e.g., a skew-gentle algebra in characteristic 2 leading to a wild quotient). Confirming the conjecture would clarify tameness preservation under broad conditions for orbit categories of locally bounded k-categories.