Bricks and $\ddot{I}„$-tilting theory under base field extensions

Abstract

Let $K:k$ be a field extension and let $\hat{I}›$ be a finite-dimensional $k$-algebra. We investigate the relationship between $\hat{I}›$ and $\hat{I}{›}_K=\hat{I}›{\otimes }_{k}K$ with particular emphasis on various aspects of $\ddot{I}„$-tilting theory and bricks. We show that many types of objects for $\hat{I}›$ lift injectively to the same type of object for $\hat{I}{›}_K$, and many common constructions in $\ddot{I}„$-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 $\ddot{I}„$-cluster morphism category $\mathfrak{W}(\hat{I}›)$ of $\hat{I}›$ to the $\ddot{I}„$-cluster morphism category $\mathfrak{W}(\hat{I}{›}_K)$ of $\hat{I}{›}_K$. In particular, this establishes a faithful functor from $\mathfrak{W}(\hat{I}›)$ 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 $\ddot{I}„$-tilting finiteness under base field extension.

41 pages, comments welcome