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?
  • 3
    $\begingroup$ 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"! $\endgroup$
    – Tim Campion
    Apr 2 '19 at 19:34

Yes, they are regular (the maximal ideal is generated by a number of elements equal to its dimension) and therefore Cohen-Macaulay (Matsumura, Theorem 17.8). And a Cohen Macaulay ring is catenary (Matsumura, Theorem 17.4).

Matsumura, Commutative Ring Theory, CUP, 1986


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