Geometric interactions between bricks and $\tau$-rigidity

Abstract

For finite-dimensional algebras over algebraically closed fields, we consider two fundamental classes of modules and their geometric counterparts: bricks and $\tau$-rigid modules, as well as brick components and $\tau$-regular components. We then apply our results in the study of some open conjectures. First, we investigate the situation where every brick is $\tau$-rigid. We prove that this occurs exactly when the algebra is locally representation-directed; a family of algebras introduced by Dr\"axler in the 1990s, which are always representation-finite. Then, we adopt a geometric perspective and analyze the brick and $\tau$-regular components of module varieties. In this greater generality, we establish new properties of such components. Inspired by some recent conjectures, we apply our results to the study of minimal brick-infinite algebras. Along the way, we construct some limits of rigid $g$-vectors, under a condition that we call the $\tau$-convergence property. This construction is novel and, in certain cases, yields an integral $g$-vector lying outside the $\tau$-tilting fan (a.k.a. $g$-vector fan). We show how our results provide new tools to the study of some open conjectures and particularly illustrate that for $E$-tame algebras.