Generically $τ$-regular irreducible components of module varieties

TL;DR

This paper develops a unified framework to study generically $ au$-regular irreducible components of module varieties. It proves that a module is $ au$-regular precisely when its minimal projective presentation attains maximal rank, refining Plamondon’s earlier results and linking rank conditions to generic component structure. It then shows that generic extensions by simple projectives preserve $ au$-regularity for triangular algebras, enabling a complete classification of generically $ au$-regular components in that setting and a clear criterion for when all components are generically $ au$-regular, namely heredity. The work also analyzes when $ ext{Irr}^ au(A)$ coincides with $ ext{Irr}^{ au^-}(A)$, giving concrete results for Jacobian, gentle, Nakayama, local, and Dynkin-type generalized species algebras, and establishing a robust correspondence between $ au$-regular and $ au^-$-regular data via tau-regular pairs and AIR-type dualities. Overall, the paper advances the understanding of generic representation theory by connecting homological, geometric, and combinatorial data in module varieties, with implications for cluster algebras and related categorifications.

Abstract

In the representation theory of finite-dimensional algebras, the study of projective presentations of maximal rank is closely related to the study of generically $τ$-regular irreducible components of varieties of modules over such algebras. We show that a module is $τ$-regular if and only if its minimal projective presentation is of maximal rank. This is a refinement of a theorem by Plamondon. We prove that generic extensions of generically $τ$-regular components by simple projective modules are again generically $τ$-regular. This leads to the classification of all generically $τ$-regular components for triangular algebras. We also show that an algebra is hereditary if and only if all irreducible components of its varieties of modules are generically $τ$-regular. Finally, we discuss when the set of generically $τ$-regular components coincides with the set of generically $τ^-$-regular components.

Theorems & Definitions (111)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Proposition 1.9
• Proposition 1.10