May 29, 2025 — Padova, Torre Archimede 1BC45

\(\tau\)-tilting finiteness under base field extension

Maximilian Kaipel, University of Cologne

Abstract

Whenever a finite-dimensional associative algebra is representation finite, it is usually well-understood. On the other hand, for representation infinite algebras, a general understanding of all modules is beyond hope. Thus, it makes sense to restrict our attention to particular families of modules. In this talk, we focus on \(\tau\)-tilting modules, which are closely related to the theory of cluster algebras. As it turns out, many representation infinite algebras are \(\tau\)-tilting finite, in which case we call them \(\tau\)-tilting finite. Let \(L:K\) be a field extension. A theorem of Jensen-Lenzing from 1982 states that if a \(K\)-algebra \(A\) is representation finite, then the \(L\)-algebra \(A \otimes_K L\) is also representation finite, provided that \(L:K\) is ’nice enough’. In my talk I will discuss the question of whether \(\tau\)-tilting finite algebras are similarly preserved under base field extension, illustrating the theory on many examples. This is based on joint work with Erlend D. Børve.