Tuesday, October 21, 14:00, 7.527.

Kevin Schlegel
Transcendental field extensions and 1-parameter families of modules.

Abstract:
We generalize the inductive step of the second Brauer-Thrall conjecture to a wide range of subcategories of the module category of a finitely generated algebra. That is, we prove that if there are infinitely many non-isomorphic indecomposable modules of a fixed dimension inside the subcategory, then there are infinitely many dimensions that each admit infinitely many non-isomorphic indecomposable modules inside the subcategory. More precisely, this is shown for the constructible subcategories of the module category, so those that can be expressed as all modules that vanish on a finitely presented functor. Key ingredients of the proof include the Ziegler spectrum of a ring, a new connection to schemes of modules, and a study of the behaviour of modules after passing to a transcendental field extension.