Is the colimit of regular rings still regular?

Question

Let $I$ be a directed set. $\{R_i\}_{i\in I}$ be a direct system of regular rings. Can we deduce the direct limit of $R_i$ is still regular? (Here by "regular" we means its localization at any prime ideal is a regular local ring.) Could you prove it or give a counterexample? Thanks!

Answer 1

This is true as long as the direct limit $R = \varinjlim_{i \in I} R_i$ is Noetherian; see [Swan 1998, Lemma 1.4]. You can also look at [Asgharzadeh 2018] for some related statements when $R$ is not necessarily Noetherian.