The complexity of module categories through purity and its connections to exact structures, schemes of modules, and matrix reductions

• Applicant: Kevin Schlegel
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 580964112

There are major problems in representation theory of finite-dimensional algebras concerning the complexity of module categories, such as the second Brauer-Thrall conjecture and achieving a deeper understanding of the tame-wild dichotomy. These problems revolve around describing the structure and distribution of indecomposable finite dimensional modules. While more is known when the underlying field is algebraically closed, many questions remain open in general. They have long been considered too difficult to tackle, and little progress has been made over the past forty years. Moreover, even over an algebraically closed field, a conceptual understanding is often lacking. In this project, we aim to advance these questions using modern techniques and new connections established during the applicant's PhD research. More precisely, we will build on the newly developed connections of purity with schemes of modules, exact structures, and matrix reductions. Of particular interest in the context of purity is the Ziegler spectrum of a ring, which originates in model theory. For a finite dimensional algebra, the finite dimensional indecomposable modules correspond to closed and open points in the Ziegler spectrum. Understanding their structure can be achieved by getting a better understanding of the whole Ziegler spectrum. A connection we developed bridges schemes of modules and purity, showing that the Ziegler spectrum behaves like the constructible topology of the scheme when the dimension of finite dimensional modules is bounded. We used this connection to prove a variant of the first Brauer-Thrall conjecture for a wide class of subcategories of the module category of a finitely generated algebra. As part of this project, we aim to advance variants of the second Brauer-Thrall conjecture and gain a better understanding of the tame-wild dichotomy using this geometric approach. Purity also connects to matrix reduction techniques and exact structures, as shown in the applicant's PhD research. Matrix reductions were famously used to establish Drozd's tame-wild dichotomy, and a connection to exact structures was conjectured, which we realized indirectly via purity. Guided by this insight, we aim to formalize matrix reductions in the language of exact structures. Once developed, methods from exact structures can serve as a fruitful toolset. The newly established connections of purity with schemes of modules, exact structures, and matrix reductions offer a promising approach to tackle central problems in representation theory of finite dimensional algebras. By combining these methods in a coordinated way, the project aims to advance our understanding of fundamental questions related to the complexity of module categories.

• DFG Programme: Position