Bricks and $\tau$-tilting theory under base field extensions

Abstract

Let $K:k$ be a field extension and let $\Lambda$ be a finite-dimensional $k$-algebra. We investigate the relationship between $\Lambda$ and $\Lambda_K = \Lambda \otimes_k K$ with particular emphasis on various aspects of $\tau$-tilting theory and bricks. We show that many types of objects for $\Lambda$ lift injectively to the same type of object for $\Lambda_K$, and many common constructions in $\tau$-tilting theory commute with the process of extending the base field. One of our main applications is the construction of a faithful functor from the $\tau$-cluster morphism category $\mathfrak{W}(\Lambda)$ of $\Lambda$ to the $\tau$-cluster morphism category $\mathfrak{W}(\Lambda_K)$ of $\Lambda_K$. In particular, this establishes a faithful functor from $\mathfrak{W}(\Lambda)$ to a group whenever $k$ is of characteristic zero which has many important consequences. In the appendix, E. J. Hanson shows the analogous result whenever $k$ is a finite field. Moreover, we give some nontrivial examples to illustrate the behaviour of $\tau$-tilting finiteness under base field extension.