Here's a theorem I find useful:

>> **Theorem.** Let $\phi \colon X \to Y$ be a smooth morphism of schemes of relative dimension $d$. Then there exists an open cover $X = \bigcup U_i$ of $X$ such that each $U_i \to Y$ factors as
$$U_i \stackrel \pi \to \mathbb A^d_Y \to Y,$$
with $\pi$ étale. **Mnemonic: smooth morphisms have étale coordinates.**

See [Tag 054L][1].


  [1]: http://stacks.math.columbia.edu/tag/054L