This project is prompted by some open problems on the representation type of an algebra. The phenomena addressed by these problems concern small, i.e. finite length modules. But they are controlled by large modules which may have infinite length. We propose a novel approach that takes into account the interplay between small and large objects and is based on recent advances in silting theory and localization theory.
The representation type of an algebra appears to be related with properties of decompositions and localizations of the module category and its derived category, notably torsion pairs, t- structures, and ring epimorphisms. We aim to uncover and solidify these connections.
To this end, we plan to develop a theory of mutation at summands of cosilting objects and to investigate multiple interactions, e.g. with the geometric concept of stability, with a topological space originating in model theory called the Ziegler spectrum, and with Sylvester rank functions. This is complemented by concrete case studies aiming at classification results over specific classes of algebras.
The research combines homological, categorical and combinatorial techniques with tools from mathematical logic and algebraic geometry.
