τ-tilting finiteness of non-distributive algebras and their module varieties

Abstract

We treat the τ-tilting finiteness of minimal representation-infinite algebras and particularly the non-distributive ones. Building upon the new results of Bongartz, we fully determine which algebras in this family are τ-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the minimal representation-infinite biserial algebras. Consequently, we obtain nontrivial explicit conditions for τ-tilting infiniteness of a large family of algebras. This also produces concrete families of “minimal τ-tilting infinite algebras”, recently studied in our work.We further use our results to establish a conjectural connection between the τ-tilting theory and two geometric notions (the dense orbit property and Schur-representation finiteness) introduced by Chindris, Kinser and Weyman while studying module varieties. We verify the conjectures for the algebras studied in this note: For the minimal representation-infinite algebras which are non-distributive or biserial, if Λ has the dense orbit property, then Λ is τ-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is τ-tilting finite. This gives a categorical interpretation of Schur-representation-finiteness over this family of algebras.