We prove that a finite-dimensional algebra
$ \Lambda $
is
$ \tau $
-tilting finite if and only if all the bricks over
$ \Lambda $
are finitely generated. This is obtained as a consequence of the existence of proper locally maximal torsion classes for
$ \tau $
-tilting infinite algebras.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.