标题： 砖块和$τ$-倾斜理论在基域扩张下 显示英文标题

标题： Bricks and $τ$-tilting theory under base field extensions

摘要： 设$K:k$是一个域扩张，设$\Lambda$是一个有限维$k$-代数。我们研究$\Lambda$与$\Lambda_K = \Lambda \otimes_k K$之间的关系，特别关注$\tau$-倾斜理论和砖块的各种方面。 我们证明了许多类型的对象对于$\Lambda$可以单射地提升为$\Lambda_K$同类型的对象，且许多在$\tau$-倾斜理论中的常见构造在扩展基域的过程中保持交换。 我们主要的应用之一是从$\tau$-cluster 丛态射范畴$\mathfrak{W}(\Lambda)$的$\Lambda$到$\tau$-cluster 丛态射范畴$\mathfrak{W}(\Lambda_K)$的$\Lambda_K$的忠实函子的构造。 特别是，这建立了一个从$\mathfrak{W}(\Lambda)$到一个群的忠实函子，当$k$的特征为零时，这有许多重要的后果。 在附录中，E. J. Hanson 展示了当$k$是有限域时的类似结果。 此外，我们给出了一些非平凡的例子，以说明$\tau$-倾斜有限性在基域扩张下的行为。 显示英文摘要

摘要： 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.