Wide subcategories and brick-finiteness for length categories

We extend to abelian length categories of finite rank a characterisation of brick-finiteness known for finite-dimensional algebras.

We prove that such a category is brick-finite if and only if every torsion class is generated by a wide subcategory and every torsionfree class is cogenerated by a wide subcategory.

The proof is based on an exchange relation for brick labels across wide intervals which is a shadow of the mutation of simple minded collections.

As a corollary,we extend the validity of the first Brick Brauer-Thrall conjecture to this setting.