What are your favorite properties of varieties over a field k (or more generally finite type k-schemes) that can be checked after base change to the algebraic closure of k? This is of course secretly about faithfully flat descent, but my question is partially pedagogical.

Sometimes the implication goes only one way (e.g. if something is true over the algebraic closure, then it is true for k); please be clear on that.

I would like later to add references, so people can know not just what is true, but also why it is true. If you know references off the top of your head (even somewhat vague ones, such as “I remember a nice proof in [reference here]”, please post them).

For simplicity, please put one per comment, so people can respond and comment further.