Fully Faithful Functors and Dimension

We define the countable Rouquier dimension of a triangulated category and use this notion together with Theorem 2 of [Ola21] to prove that if there is a fully faithful embedding $D^b_{coh}(X) \subset D^b_{coh}(Y)$ with $X, Y$ smooth proper varieties, then $\mathrm{dim}(X) \leq \mathrm{dim}(Y)$.