Constructible Subcategories and Unbounded Representation Type (with an appendix by Andres Fernandez Herrero)

Abstract: We show that bounded type implies finite type for a constructible subcategory of the module category of a finitely generated algebra over a field, which is a variant of the first Brauer-Thrall conjecture. A full subcategory is constructible if it consists of all modules that vanish on a finitely presented functor. Our approach makes use of the Ziegler spectrum of a ring and a connection, established in this work, with the scheme of finite dimensional modules. We also discuss a variant of the second Brauer-Thrall conjecture in this setting. It is shown to be true under additional assumptions on the algebra, but wrong in general. Lastly, it is proven that exact structures and matrix reductions give rise to constructible subcategories. Based on this, a translation from matrix reductions to reductions of exact structures is provided.