# Are the ring of power series and the ring of germs of holomorphic functions catenary?

I am wondering if the following rings are catenary:

1. If $$k$$ is a field, is the ring of formal power series $$k[[X_1,\dots,X_n]]$$ catenary?
2. Is the ring of complex power series with a non-zero radius of convergence $$\Bbb C\{X_1,\dots,X_n\}$$ (id est the ring of germs of holomorphic functions at zero) a catenary ring?
• Until a few minutes ago, the comments contained a link to the wikipedia article catenary ring. As that article makes clear, the catenary condition should be viewed as the weakest in a hierarchy of "smoothness" conditions on a ring, the strongest of which is the property of being a regular ring. In his answer, Leo Alonso observes that these rings are in fact regular. So basically, the question asks "are these rings nice in a weak sense", and the answer is "yes, and don't worry -- they're actually nice in the strongest possible sense"! Apr 2 '19 at 19:34